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## Numbers and Algebra

### Basic Number Theory - Integers, Fractions, Decimals

#### Q.01

'Find the number of integer pairs $(p, q)$ that satisfy $p^{2}-q^{2}=250$.'

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'[4]From (1/y) + (1/z) = (1/3) and (1/z) ≤ (1/y), we have (1/3) ≤ (2/y), hence y ≤ 6. Combining this with y ≥ 6, we get y = 6.\nSubstitute y = 6 into (1/z) = (1/3) - (1/6) = (1/6) to solve for z, yielding z = 6.'

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'The sum of natural numbers, squares, and cubes from 1 to n is represented as follows: (1) 1+2+3+...+n = \\frac{1}{2} n(n+1) (2) 1^{2}+2^{2}+3^{2}+...+n^{2} = \\frac{1}{6} n(n+1)(2n+1) (3) 1^{3}+2^{3}+3^{3}+...+n^{3} = \\left\\{\\frac{1}{2} n(n+1)\\right\\}^{2}'

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'Prove that the following inequalities hold for natural number n.'

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"What is the 'beginning' of mathematics? What should be considered as the beginning of mathematics?"

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'Find the discriminant D of the following quadratic equation and determine the type of its roots:'

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"In Greek mathematics, how was the proof of 'Even + Even = Even' established?"

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'Find the sum of terms 10 to 20 of an arithmetic sequence with the 8th term as 37 and the 24th term as 117.'

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'Find the general term of the given recurrence relation.'

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'(2) \ \\frac{1}{1-\\frac{1}{1-\\frac{1}{1+a}}} \'

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'Exercise 20 Grid Point Counting\n(1) Let k be a non-negative integer. The number of non-negative integer pairs \\( (x, y) \\) satisfying \ \\frac{x}{3} + \\frac{y}{2} \\leqq k \ is denoted as \ a_{k} \. Express \ a_{k} \ in terms of k.\n(2) Let n be a non-negative integer. The number of non-negative integer triples \\( (x, y, z) \\) satisfying \ \\frac{x}{3} + \\frac{y}{2} + z \\leqq n \ is denoted as \ b_{n} \. Express \ b_{n} \ in terms of n.\n[Yokohama National University]'

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'17 (1) multiplied by 10, product 29 (2) multiplied by 0, product 2 (3) multiplied by -4, product 4'

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'Find the sum of the given sequence of fractions by decomposing them into partial fractions to simplify the calculation. Use transformations like \\( \\frac{1}{k(k+1)} = \\frac{1}{k} - \\frac{1}{k+1} \\) for example.'

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'When real numbers x and y satisfy x^{2}+y^{2} ≤ 3, the maximum value of x-y-xy is □.'

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'When a>0, b>0, compare the sizes of (a+b)/2, √(ab), 2ab/(a+b), and √((a²+b²)/2).'

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'169 (1) \ \\frac{110}{3} \ (2) \ \\frac{37}{12} \ (3) \ \\frac{9}{8} \ (4) \ \\frac{14 \\sqrt{14}}{3} \'

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'Translate the given text into multiple languages.'

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'Since it is expressed as -2 = -2 + 0 \\cdot i, the complex conjugate of -2 is -2 -0 \\cdot i, which is -2. Therefore, the sum of -2 and -2 is -4, and the product of -2 and -2 is 4.'

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'Find the real and imaginary parts of the following complex numbers. (1) 2-√3 i (2) (-1+i)/2 (3) -1/3 (4) 4i'

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'45 \ \\frac{1}{\\tan \\frac{\\pi}{24}}-\\sqrt{2}-\\sqrt{3}-\\sqrt{6} \ is an integer. Find its value.'

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'Practice 42 \n When the equation of the given line is organized in terms of k, \n k(3x-2y-10) + x - 4y + 10 = 0 \n Find the conditions for which this equation holds true regardless of the value of k.'

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'Let {a_{n}} be a sequence with the initial term {a_{1}} up to the nth term {a_{n}} and the sum denoted as {S_{n}}. If {S_{n}+a_{n}=4 n+2}, then {a_{1}=} A {, a_{2}=} B. Expressing {a_{n+1}} in terms of {a_{n}} gives {a_{n+1}=C {a_{n}+} D}. Therefore, the general term of this sequence is {a_{n}=E}.'

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'Find the sum S of the arithmetic sequence from the first to the 100th term, with the first term being 1 and the common difference being -2.'

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'Gaussian symbol and summation of series, recurrence relation'

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'The sequence {an} satisfies a1=1, and for all natural numbers m, a2m=a2m-1+1, a2m+1=2a2m.'

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'Find the sum S of the arithmetic sequence 2, 17/6, 11/3, 9/2, ⋯⋯, 12.'

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'Drawing n chords on a circle, where any two chords intersect inside the circle and no three chords pass through the same point. The number of parts divided by these chords is denoted as D_{n}. In this case, D_{3}=口の□, D_{4}=1□, and D_{n}=ウ□. Furthermore, the number of parts that form polygons among D_{n} parts is denoted as d_{n}. When n is greater than or equal to 4, d_{n}=エ□.'

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'Prove by mathematical induction that for any natural number m, a_{3m} is a multiple of 5.'

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'Prove that for a sequence {an} (where {an} are greater than 0), if the relation (∑an)^2 = a1^3 + a2^3 + ... + an^3 holds, then an = n.'

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"Exercise 81 (1) |x| ≥ 1, so |t| ≥ 1. The slope of line OA is 1/t, the coordinates of the midpoint of line OA are (t/2, 1/2), so the equation of the perpendicular bisector of line OA is y-1/2=-t(x-t/2), i.e., y=-tx+(t^2+1)/2 (|t| ≥ 1). (2) y=-tx+(t^2+1)/2 gives t^2-2xt-2y+1=0. Let f(t)=t^2-2xt-2y+1, the required condition is {about real numbers t that make f(t)=0's discriminant D satisfy (1)}, so D/4=x^2+2y-1 ≥ 0, hence y ≥ -x^2/2+1/2. Real numbers t that satisfy (1) are all in -1<t<1, that is, satisfy {D ≥ 0 f(-1) > 0 f(1) > 0 -1 <x <1}, i.e., {y ≥ -x^2/2+1/2 y < x+1 y < -x+1 -1 <x <1}. Consider excluding the case of |t|<1 from all real number solutions that satisfy condition 1."

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'P ≥ 2√(a * 1/a) + 2√(b * 1/b) + 2√(c * 1/c) + 2√(abc * 1/abc) = 2 + 2 + 2 + 2 = 8 Therefore (a + 1/b)(b + 1/c)(c + 1/a) ≥ 8'

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'Find the general term of the sequence {a_n} determined by the following conditions.'

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'The first term is 96, the common ratio is -1/2, so the sum of the first 7 terms is 96{1-(-1/2)^7}/(1-(-1/2))=96/(3/2)(1+1/128)=64*129/128=129/2'

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'Which of the following sequences is a geometric progression?'

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'When throwing several dice at the same time, what is the minimum number of dice needed for the probability of the product of the numbers thrown to be even to be at least 0.994? Where log_{10} 2=0.3010, log_{10} 3=0.4771.'

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'23\\ n \\ 1,2,3 | 4,5,6,7,8 | 9,10,11,12,13,14,15 \\ mid 16, \\ cdots \\ cdots \\ n(1) \\ Find the first and last number in group \ n \. \\ n(2) \\ Find the sum of all numbers in group \ n \. \\ n(3) \\ In which group and at which position is 2014?'

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'Next, when 4^{10} is expressed in base 9, let the number of digits be denoted as n'

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'Math \ \\Pi \ 63 Therefore, \\( \\quad P(-1)=-a+b, P(1)=a+b \\) From (1), (2) we have \ -a+b=5, a+b=7 \ Solving simultaneously gives \ \\quad a=1, b=6 \ Therefore, the remainder we seek is \ \\quad x+6 \'

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'(2) \\( \\alpha=\eta=\\gamma=1 \\Leftrightarrow \\alpha-1=\eta-1=\\gamma-1=0 \\Leftrightarrow(\\alpha-1)^{2}+(\eta-1)^{2}+(\\gamma-1)^{2}=0 \\)'

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'Let integers a, b be not multiples of 3, and let f(x)=2 x^{3}+a^{2} x^{2}+2 b^{2} x+1. Find the remainders when f(1) and f(2) are divided by 3.'

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'(4) \\sqrt[4]{16}=\\sqrt[4]{2^{4}}= 2, \\quad \\sqrt[4]{625}=\\sqrt[4]{5^{4}}= 5 ,'

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'Find the sum of the numbers that satisfy the following conditions among two-digit natural numbers: (1) Numbers that leave a remainder of 3 when divided by 5. (2) Odd numbers or multiples of 3.'

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'(3) Just assuming that $c_{k+2}=c_{k+1}+c_{k}$ when $n=k$ is a multiple of $d$ is not sufficient for proof.\n[1] When $n=1,2$\n$c_{1}=a, c_{2}=b$, and since both $a$ and $b$ are multiples of $d$, $c_{1}, c_{2}$ are both multiples of $d$.\nTherefore, for $n=1,2$, $c_{n}$ is a multiple of $d$.\n[2] When $n=k, k+1$, assuming $c_{n}$ is a multiple of $d$, $c_{k}$ and $c_{k+1}$ are multiples of $d$, so using integers $l, m$, $c_{k}=d l, c_{k+1}=d m$ can be expressed. Consider $n=k+2$.\n$c_{k+2}=c_{k+1}+c_{k}=d l+d m=d(l+m)$\nSince $l+m$ is an integer, $c_{k+2}$ is a multiple of $d$. Therefore, when $n=k+2$, $c_{n}$ is also a multiple of $d$.\nFrom [1], [2], it can be concluded that for all natural numbers $n$, $c_{n}$ is a multiple of $d$.'

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'Find the general term of a geometric sequence where the 3rd term is 12 and the 6th term is -96. Assume the common ratio is a real number.'

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'(1) Since $\\frac{y}{x} > 0, \\frac{x}{y} > 0$, by the inequality of arithmetic mean and geometric mean, we have $\\frac{y}{x} + \\frac{x}{y} \\geq 2 \\sqrt{\\frac{y}{x} \\cdot \\frac{x}{y}} = 2$.'

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'For a natural number n, when √(2n)+1/2>1, an is an integer greater than 1. For a natural number m, when an = m, m ≤ √(2n) + 1/2 < m + 1, i.e., m - 1/2 ≤ √(2n) < m + 1/2. Since m - 1/2 > 0, then according to the above, (m - 1/2)^2 ≤ 2n < (m + 1/2)^2, we get m(m-1)/2 + 1/8 ≤ n < m(m+1)/2 + 1/8.'

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'Mathematics B\n287\nFrom (1) we have a=6\nSubstitute this into (2) we get 6(36-d^{2})=162\nTherefore d^{2}=9\nTherefore d=±3\nThus, the 3 numbers we seek are 3,6,9 or 9,6,3\nIn other words\n3,6,9\nSince the order of the 3 numbers is not specified, the answer can be one way.\nAnother solution is to let the sequence of 3 numbers forming an arithmetic sequence be denoted as a, b, c. Based on the conditions\n2b=a+c\na+b+c=18\nabc=162\nSubstitute (1) into (2) we get 3b=18, hence b=6\nAt this point, from (1) and (3) we obtain a+c=12, ac=27\nTherefore, a, c are two solutions of the equation x^{2}-12x+27=0. Solving (x-3)(x-9)=0 gives x=3,9\nIn other words\n(a, c)=(3,9),(9,3)\nThus, the 3 numbers we seek are 3,6,9'

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'The total number of terms from the 1st group to the 11th group is 66. Therefore, the 77th term of the sequence {an} is the number of 11th group which is (77-66=11). Therefore, based on (1), the 77th term of the sequence {an} is 12*11^2=1452.'

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'There are 2 red books and n blue books. Randomly line up these n+2 books on the shelf. Let X be the number of blue books between the two red books.'

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"Mathematics II\nThat is \\[ \\left(p, q\\right)=\\left(0,0\\),\\left(-2,-2\\) \\]\nWhen \\( \\left(p, q\\right)=\\left(0,0\\) \\), from (2) we get \ \\quad a=0 \\nSince this does not satisfy the condition \ a>0 \, it is not appropriate.\nWhen \\( \\left(p, q\\right)=\\left(-2,-2\\) \\), from (2) we get \ \\quad a=4 \\nThis satisfies the condition \ a>0 \.\nTherefore, the integer we seek \ a \ is \ \\quad a=4 \ \2\. Similar to \1\, considering two equations\n\ x^{2}+a x+b=0 \\n(4), \ y^{2}+b y+a=0 \\nall have integer solutions.\nLet's denote the two solutions of (4) as \ p, q \, according to the relationship between solutions and coefficients\n\ p+q=-a, p q=b \\nBecause \ a>0, b>0 \, therefore, \ \\quad p+q<0, p q>0 \\nTherefore, \ p, q \ are both negative integers, i.e., integers less than or equal to -1.\nTherefore, let \\( f(x)=x^{2}+a x+b \\), then the graph of \\( y=f(x) \\) has intersection points only with the part of the \ x \ axis which is -1 or below.\nThus \\( \\quad f(-1)=1-a+b \\geqq 0 \\) i.e., \ \\quad a \\leqq b+1 \\nCombining \ a>b \, it follows that \ \\quad b<a \\leqq b+1 \\nTherefore, in this case, (4) becomes \\( x^{2}+a x+(a-1)=0 \\), which leads to\n\\[ \\left(x+1\\right)\\left(x+a-1\\right)=0 \\]\nThen, the integer solutions are \ x=-1,-a+1 \.\nNext, let's consider (5) that is \\( y^{2}+b y+(b+1)=0 \\) and find the values of \ b \ for which it has integer solutions.\nAssuming the two solutions of (5) are \ r, s\\left(r \\leqq s\ \\), according to the relationship between solutions and coefficients we have\n\ r+s=-b \\]\n\\[ r s=b+1 \\nBy eliminating b from (7), (8) we get \ r s+r+s=1 \, therefore \\( \\quad \\left(r+1\\right)\\left(s+1\\right)=2 \\)\nSince \ r, s \ are integers, so are \ r+1, s+1 \.\nFurthermore, from (7) and (8), like \ p, q \, \ r, s \ are also integers less than or equal to -1, so\n\ r+1 \\leqq 0, s+1 \\leqq 0 \\nThus, from 9) we get\n\\[ \\left(r+1, s+1\\right)=\\left(-2,-1\\) \\]\nThat means\n\\[ \\left(r, s\\)\\right=\\left(-3,-2\\) \\]\nIn this case, from (7) we get \ \\quad b=5 \\nand therefore \ \\quad a=5+1=6 \\nThus, the integers we seek \ \\left(a, b\ \\ are \\) \\left(a, b\\)=\\left(6,5\\)"

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'Translate the given text into multiple languages.'

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'(1) When n ≥ 2, the number of numbers from the 1st group to the (n-1)th group is ∑_{k=1}^{n-1}(2 k+1)=2 ⋅ \\frac{1}{2}(n-1) n+(n-1)=n^{2}-1, therefore, the first number of the nth group is the {n^{2}-1+1}=n^{2} (term) of a sequence of natural numbers, and this is also true for n=1. Hence, the first number of the nth group is n^{2}, and the last number of the nth group matches the number of terms in the sequence of natural numbers included up to the nth group ∑_{k=1}^{n}(2 k+1)=2 ⋅ \\frac{1}{2} n(n+1)+n=n^{2}+2 n'

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'Arithmetic progression general term and sum\nGeneral term $a_{n}$ If the first term is $a$ and the common difference is $d$\n\\[\na_{n}=a+(n-1) d\n\\]\nArithmetic mean\nSequence $a, b, c$ is an arithmetic progression $\\Leftrightarrow 2 b=a+c$\nSum of an arithmetic progression Sum from the first to the $n$ th term $S_{n}$\n(1) First term $a$, $n$ th term (last term) $l$\n\\[\nS_{n}=\\frac{1}{2} n(a+l)\n\\]\n(2) First term $a$, common difference $d$\n\\[\nS_{n}=\\frac{1}{2} n\\{2 a+(n-1) d\\}\n\\]\nSum of natural numbers, sum of positive odd numbers\n\\[\n\egin{array}{l}\n1+2+3+\\cdots \\cdots+n=\\frac{1}{2} n(n+1) \n1+3+5+\\cdots \\cdots+(2 n-1)=n^{2}\n\\end{array}\n\\]'

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'The sequence a, b, c forms an arithmetic sequence, so 2b=a+c. The sequence b, c, a forms a geometric sequence, so c^2=ab. Since the product of a, b, c is 125, then abc=125. Substituting (2) into (3) gives c^3=125. As c is a real number, c=5. Substituting into (1), (2) gives 2b=a+5, ab=25. Eliminating b gives a(a+5)=50, so a^2+5a-50=0. Therefore, a=5, -10. From ab=25, we get b=25/a. Example: \\triangleleft 36-d^2=27. The arithmetic mean form of a series 2b=a+c is used. Two numbers with sum p and product q are the two solutions of the quadratic equation x^2-px+q=0 (Math II). 4 (common ratio)=(2nd term)/(1st term). 4 a_n=2*(-3)^n is incorrect. 4(-1)^{可效}=-1. It is allowed to consider r^3=-8 by dividing (2) by (1). 4 a_n=ar^{n-1}. Arithmetic mean form of an arithmetic sequence. Arithmetic mean form of a geometric sequence. Substituting ab=c^2 into (3). The first equation. Substituting (2nd equation) multiplied by 2.'

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'The sequence {a_n} is a geometric sequence with first term {a_1} = \\frac{1}{4}-\\frac{1}{3}=-\\frac{1}{12} and common ratio -\\frac{1}{8}, so find the general term of the sequence {a_n}.'

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"In mathematics A, I learned about the concept of 'permutations and combinations'. When arranging numbers from 1 to n in a row, if the k-th number from the left is not k, it is called a perfect permutation. Also, the number of perfect permutations of n items is denoted as W(n), known as the Monge-Montel number, where W(1)=0, W(2)=1, W(n)=(n-1){W(n-1)+W(n-2)} (n ≥ 3) holds (for more details, refer to Chart Math I+A p. 264). Here, let's consider expressing W(n) in terms of n based on the recursion formula. Note that, for simplicity, we will consider the recursion formula rewritten as follows."

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'Assuming the l-th term of the sequence {a_n} is equal to the m-th term of the sequence {b_n}, and given the equation 15l-2=7・2^{m-1}, solve for the variables l and m.'

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'Common ratio: A mathematical term referring to ratios and proportions.'

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'Using natural number n, compare the size of n! and 3^n.'

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"In round (1), the number of seats for parties B and C was a total of 5 seats, but as in round (2), by forming party E through merger and assuming that the total votes remain the same as before the merger, with no change in the votes of other parties, the number of seats became 6. Therefore, it is possible for the number of seats to change as parties merge, but the following properties are known regarding D'Hondt proportional representation."

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'(2) Let the two integer solutions of the quadratic equation \ x^{2}-x-m=0 \ be \\( \\alpha, \eta(\\alpha \\leqq \eta) \\) . From the relationship between the solutions and the coefficients\n\ \\alpha+\eta=1 \\nRearranging we get\n\\[ \\text { (1), } \\alpha \eta=-m \\]\nSince \ m \ is a natural number, it follows that \ \\alpha \eta<0 \, hence \ \\alpha \ and \ \eta \ have opposite signs, \ \\alpha<0, \eta>0 \.\nFrom (1), we have \ \\alpha=1-\eta \\nSince \ \\alpha<0 \, we have \ 1-\eta<0 \\nTherefore, \ \eta>1 \'

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'(3) 1 + 2 + 2^2 + ... + 2^n = 2^{n+1} - 1, so an = 2^{n+1} - 1. 2008 = 4 * 502, therefore, from (2), the remainder when dividing 2^{2008} - 1 by 17 is 0. Thus, 2^{2008} = 17k + 1 (where k is an integer). Hence an = 2^{2011} - 1 = 2^{2008} * 8 - 1 = (17k + 1) * 8 - 1 = 17 * 8k + 7. Therefore, the remainder when dividing an by 17 is 7, and since 2012 = 4 * 503, then an = 2^{4 * 503} - 1, so from (2), a_{2012} = 2^{2014} - 1.'

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'\\(\\frac{1}{b-a}\\left(\\frac{1}{x+a}-\\frac{1}{x+b}\\right) = \\frac{1}{b-a} \\cdot \\frac{(x+b)-(x+a)}{(x+a)(x+b)} = \\frac{1}{b-a} \\cdot \\frac{b-a}{(x+a)(x+b)} = \\frac{1}{(x+a)(x+b)}\\)'

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'When the sequence starts with a₁=1, it satisfies the conditions. When a₁=2, a₂=1, so it still satisfies the conditions. Next, consider the case when a₁>2. Assuming that for all natural numbers n, aₙ>2, we have a₁>a₃>a₅>a₇>⋯ from (2). Therefore, there exists a natural number m such that a₂ᵐ⁺¹≤2. This contradicts the assumption. Hence, there exists a natural number n such that 1≤aₙ≤2. If there exists a natural number n such that aₙ=1, it satisfies the conditions. If there exists a natural number n such that aₙ=2, then aₙ₊₁=1, which also satisfies the conditions. Therefore, regardless of the initial value of a₁, the sequence {aₙ} always contains an element with the value of 1.'

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'Among the two-digit natural numbers, the numbers that are divisible by 55 with a remainder of 3 are 5·2+3, 5·3+3, ..., 5·19+3. This forms an arithmetic progression with the first term as 13, the last term as 98, and 18 terms in total, therefore, the sum is 1/2·18(13+98)=999'

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'Calculate the total amount of principal and interest after depositing 200,000 yen with an annual interest rate of 5% for 7 years.'

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'Assuming the votes for Party 1, Party 2, and Party 3 are 300,000, 300,000, and 100,000 respectively.'

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'(2) 20=2^{2} \\cdot 5,10=2 \\cdot 5, therefore, 20^{x}=10^{y+1}, thus 2^{2 x-y-1}=5^{y+1-x} (1). Assuming y+1-x \\neq 0, then from (1) we get 2^{\\frac{2 x-y-1}{y+1-x}}=5 \\cdots\\cdots\\cdot(2). When x, y are rational numbers, 2 x-y-1, y+1-x are also rational numbers, and \\frac{2 x-y-1}{y+1-x} is also a rational number. Furthermore, from (2) we have 2^{\\frac{2 x-y-1}{y+1-x}}>1, so \\frac{2 x-y-1}{y+1-x}>0, therefore \\frac{2 x-y-1}{y+1-x}=\\frac{m}{n}(m, n are positive integers), which can be expressed as 2^{\\frac{m}{n}}=5. Multiplying both sides by n gives 2^{m}=5^{n}, the left side is a multiple of 2, whereas the right side is not a multiple of 2, leading to a contradiction. Hence y+1-x=0. In this case, from (1) we get 2^{2 x-y-1}=1, thus 2 x-y-1=0 (4), (5). Solving the system of equations yields x=0, y=-1'

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'When (1) holds, find the value of $\\frac{x y+y z+z x}{x^{2}+y^{2}+z^{2}}$.'

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'What is the formula to find the general term of an arithmetic sequence?'

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'Exercise 19 Gauss symbol and sum of sequences, recurrence formula'

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"Proportional allocation of seats method (1)..... D'Hondt method Introduction to the allocation of seats in Japan's national elections using the method of proportional representation. In proportional representation elections, the number of seats each party wins is determined using a calculation method known as the D'Hondt method based on the number of votes received by each party. Let's explain what this 'D'Hondt method' is and provide specific examples. *The 'D'Hondt method' is a method devised by Belgian mathematician Victor D'Hondt (1841-1902)."

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'Given $0 \\leqq x \\leqq \\pi$, we have $-\\frac{\\pi}{4} \\leqq \\frac{1}{2}\\left(5 x-\\frac{\\pi}{2}\\right) \\leqq \\frac{9}{4}\\pi$ and $\\frac{\\pi}{4} \\leqq \\frac{1}{2}\\left(x+\\frac{\\pi}{2}\\right) \\leqq \\frac{3}{4}\\pi$. This implies $\\frac{1}{2}\\left(5 x-\\frac{\\pi}{2}\\right)=0, \\pi, 2\\pi$ or $\\frac{1}{2}\\left(x+\\frac{\\pi}{2}\\right)=\\frac{\\pi}{2}$. Solving these equations we get $5 x-\\frac{\\pi}{2}=0, 2\\pi, 4\\pi$ or $x+\\frac{\\pi}{2}=\\pi$. Therefore, $x=\\frac{1}{5} \\cdot \\frac{\\pi}{2}, \\frac{1}{5} \\cdot \\frac{5}{2}\\pi, \\frac{1}{5} \\cdot \\frac{9}{2}\\pi$ or $x=\\frac{\\pi}{2}$. Hence, we conclude that $x=\\frac{\\pi}{10}, \\frac{\\pi}{2}, \\frac{9}{10}\\pi$.'

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'59 (1) 8^(1/8) < 2^(1/2) = 4^(1/4) (2) 2^30 < 3^20 < 10^10 (3) 6^(1/6) < √2 < 3^(1/3)'

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'Given an arithmetic sequence {a_{n}}, where the first term is a and the common difference is d, each term is represented as follows: \na, a+d, a+2d, a+3d, ..., a+(n-1)d. Find the nth term a_{n} of this sequence.'

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'Using a natural number n, compare the sizes of n² and 4^(n-2).'

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'Find the number of real solutions of f(x)=x^{3}+3 x^{2}-9 x-9.'

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'The coordinates of point R are from (-2+6)/2, (5-3)/2) to (2,1)'

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'Find the number of combinations b_n of integers x, y, z that satisfy the constraints x≥0, y≥0, z≥0, and (x/3)+(y/2)+z≤n.'

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'Find the sum of the numbers between 100 and 200 that satisfy the following conditions: (1) Numbers that leave a remainder of 2 when divided by 7. (2) Multiples of 4 or 6'

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"Explain the following properties of Pascal's Triangle:\n1. The numbers at the two ends of each row.\n2. Properties of each number except the ones at the ends.\n3. Array of numbers."

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'Translate the given text into multiple languages.'

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'Let real numbers p, q satisfy |p|≤1, |q|≤1, |p-q|≤1. Define the maximum of 0, p, q as M and the minimum as m. Prove the following inequalities hold.'

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'Two particles are located at vertex A of triangle ABC at time 0. These particles move independently, moving to a neighboring vertex with equal probability every 1 second. Let n be a natural number, and let the probability that these two particles are at the same point after n seconds be pn.'

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'From the given recurrence relation, for any natural number n, there exists a natural number a_{n} such that a_{n}<a_{n+1}. Therefore, when n \\geqq 2, a_{1}, ... a_{n-1} are not multiples of a_{n}, but a_{n} is a multiple of a_{n}. Next, for n \\geqq 2, using mathematical induction on m, it can be shown that for any natural number m, a_{n+m}-a_{m} is a multiple of a_{n}.'

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'Derive the real and imaginary parts from the following complex numbers.'

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'Show that the three inequalities a(1-b)>1/4, b(1-c)>1/4, c(1-a)>1/4 cannot hold simultaneously when a, b, c are all positive numbers less than 1.'

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'Comprehensive exercise 369 Since 2^{4n}-1 ≡ (-1)^n-1 (mod 17), when n is even, 2^{4n}-1 ≡ 0 (mod 17) and when n is odd, 2^{4n}-1 ≡ -2 ≡ 15 (mod 17) Therefore, the required remainder is 0 when n is even and 15 when n is odd (3) 2008=4 × 502, thus from (2) we have 2^{2008}-1 ≡ 0 (mod 17), meaning 2^{2008} ≡ 1 (mod 17) Hence, 2^{2011} ≡ 2^3 · 1 ≡ 8 (mod 17) 2^{2012} ≡ 2 · 8 ≡ 16 (mod 17) 2^{2013} ≡ 2 · 16 ≡ 32 ≡ 15 (mod 17) 2^{2014} ≡ 2 · 15 ≡ 30 ≡ 13 (mod 17) Therefore, a_{2010} ≡ 2^{2011}-1 ≡ 7 (mod 17) a_{2011} ≡ 2^{2012}-1 ≡ 15 (mod 17) a_{2012} ≡ 2^{2013}-1 ≡ 14 (mod 17) a_{2013} ≡ 2^{2014}-1 ≡ 12 (mod 17)'

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'Since \\(\\sum_{k=1}^{n}\\left(a_{k}-k\\right)^{2} \\geqq 0\\), the expression \1 \\cdot a_{1}+2 a_{2}+\\cdots \\cdots+n a_{n}\ is maximum when \\(\\sum_{k=1}^{n}\\left(a_{k}-k\\right)^{2}=0\\), which means \\(a_{k}=k (k=1,2, \\cdots \\cdots, n)\\). Therefore, the required sequence is \1,2,3, \\cdots \\cdots, n\.'

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'The quotient obtained by dividing the number of votes for each political party by 1, 2, 3, ... is as shown in the following table.'

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'Up to which term should the sum be taken from the initial term to maximize the sum? Also, find the sum at that point.'

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'Exercise (1) Prove the inequality |x+y+z| ≤ |x|+|y|+|z|.'

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'Calculate the following expressions using common logarithm tables and round the answers to two decimal places: (1) 2.37 × 3.79 (2) 7.67 ÷ 2.86'

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'Maximum value of 102 is 16, coordinates of point P are (5 / sqrt(26), 1 / sqrt(26)) or (-5 / sqrt(26), -1 / sqrt(26))'

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'62\n(1) (A) 3\n(B) \ -\\frac{5}{2} \\n(2) \ \\frac{13}{4} \'

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'Find the first term a and common difference d of an arithmetic sequence, where the sum of the first 5 terms is 125 and the sum of the first 10 terms is 500.'

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'Arrange the natural numbers as shown in the right diagram.'

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'Prove that when |x|<1 and |y|<1, |left|\\frac{x+y}{1+xy}|right|<1'

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'Let a and d be integers. Define the sequence {an} as an arithmetic sequence with first term a and common difference d. Let the sum of the first n terms of the sequence {an} be denoted by Sn.'

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'Since the first term is 2 and the common difference is 17/6-2=5/6, and if the 12th term is considered as the nth term, then 2+(n-1)・5/6=12, thus n=13. Therefore, the sum of the arithmetic series is calculated as S=1/2・13(2+12)=91'

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'Consider the sequence {Fn} determined by the following conditions.'

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'For a real number x, let [x] denote the largest integer that does not exceed x. Define the sequence {a_{k}} as a_{k}=2^[\\sqrt{k}] (k=1,2,3,......). For a positive integer n, find b_{n}=\\sum_{k=1}^{n^{2}} a_{k}.'

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'Prove that for all natural numbers n, 2^{n+1}+3^{2n-1} is a multiple of 7.'

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'Let x be a positive number. Prove that the inequality (x+1/x)(x+4/x) ≥ 9 holds true. Also, determine the conditions under which the equality holds.'

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'When k = 0, x = -1, 0, 4; when k = 12, x = -2, 2, 3'

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'Since $\\left(\\frac{1}{r}\\right)^{0}=1$ and $\\left(\\frac{1}{r}\\right)^{-1}=r$, from (1), it is conjectured that the general formula $a_{n}=n-1+\\sum_{k=1}^{n}\\left(\\frac{1}{r}\\right)^{k-2}$ holds. We will now prove this using mathematical induction.'

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'The condition for a point (x, y) to be inside ΔOAB is expressed as x>0, y>0, x+y<1. Let 2x+y=X (2), x+2y=Y (3), then 3x=2X-Y and 2×(3)-(2) leads to 3y=-X+2Y, thus x=(2X-Y)/3, y=(-X+2Y)/3. Substituting these into (1) we get x>0 implies 2X-Y>0, y>0 implies -X+2Y>0, and x+y<1 implies (X+Y)/3<1, meaning X+Y<3. Therefore, Y<2X, Y>1/2X, and X+Y<3. Thus, the range of the moving point (X, Y) or (2x+y, x+2y) is the region represented by the system inequalities y<2x, y>1/2x, x+y<3 when variables are changed to x, y. Hence, the desired range is the slanted line portion in the right diagram, excluding the boundary lines.'

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'Find the sum of the geometric progression (1) from the first term of the geometric sequence to the nth term Sn.'

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'If the sequence {a_{n}+b_{n}} has initial term {a_{1}+b_{1}=2} and common ratio 2 as a geometric progression, find its general term.'

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'Let {an} be a geometric sequence with a non-zero common ratio and initial term of 1. Also, let {bn} be an arithmetic sequence satisfying b1=a3, b2=a4, b3=a2.'

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'Find the following sums:\n(1) Sum of the arithmetic sequence 2, 8, 14, ..., 98\n(2) Sum of the arithmetic sequence with initial term 100 and common difference -8 from the first to the 30th term\n(3) Sum of the arithmetic sequence with the 8th term as 37 and the 24th term as 117 from the 10th to the 20th term'

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'Therefore, the desired maximum and minimum values are as follows:'

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'For the arithmetic sequence {an} with first term 77 and common difference -3, answer the following questions: 1. Find the general term an. 2. Which term becomes negative for the first time. 3. At which term from the first term does the sum become maximum and what is that sum.'

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'Chapter 1 Sequences - 219\nUsing the TR difference sequence, find the general term of the following sequence \ \\left\\{a_{n}\\right\\} \.\n19\n(1) \ 20,18,14,8,0 \, \ \\qquad \ (2) \ 10,10,9,7,4 \,'

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'Find the first five terms of the sequence represented by the following formulas.'

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'Find the sum of integers from 1 to 100 that are neither multiples of 3 nor multiples of 5.'

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'Find the sum of integers from 1 to 100 that are neither multiples of 3 nor multiples of 5.'

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'TRAINING 26\nLet \ n \ be a natural number. Using mathematical induction, prove the following equation:\n\\[\n1 \\cdot 4+2 \\cdot 5+3 \\cdot 6+\\cdots \\cdots+n(n+3)=\\frac{1}{3} n(n+1)(n+5)\n\\]'

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'By mathematical calculation, the arithmetic mean is greater than or equal to the geometric mean, we have $\\frac{ab}{4}+\\frac{9}{ab}+\\frac{13}{4} \\ge 2 \\sqrt{\\frac{ab}{4} \\cdot \\frac{9}{ab}}+\\frac{13}{4}=2 \\cdot \\frac{3}{2}+\\frac{13}{4}=\\frac{25}{4}$, so $\\left(\\frac{a}{4}+\\frac{1}{b}\\right)\\left(\\frac{9}{a}+b\\right) \\ge \\frac{25}{4}$. The equality holds when $ab>0$ and $\\frac{ab}{4}=\\frac{9}{ab}$, which means $ab=6$.'

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'Simplify the fractions in (1) and (2). Calculate the expressions in (3) to (5).'

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'Let n be an integer greater than or equal to 2. Using the binomial theorem, prove the following:'

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'Given an arithmetic sequence {an} with initial term 1 and common difference 4, and an arithmetic sequence {bn} with initial term -9 and common difference 6. Find the general term of the sequence {cn} formed by the terms common to both sequences, arranged in ascending order.'

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'In example 1, the number of votes for party B is 7000, and for party C is 6000. In example 2, what will happen in the case where the number of votes for party E is 13000?'

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"What is the minimum score for students ranking within the top 64000 in last year's test? Choose from the following 0-5 options."

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"Let's list some typical examples of fraction calculations.\n(1) Simplification\n......Simplification is dividing the numerator and denominator of a fraction by their common factor. A fraction that cannot be simplified further is called an irreducible fraction.\nExample:\n\\(\\frac{x^{2}+7x+12}{x^{2}+8x+15}=\\frac{(x+3)(x+4)}{(x+3)(x+5)}=\\frac{x+4}{x+5}\\)\n\\\frac{12}{15}=\\frac{3 \\cdot 4}{3 \\cdot 5}=\\frac{4}{5}\"

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'Dividing each side by 3 gives the following result.'

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'Find the first term and common ratio of the geometric sequence. The common ratio is a real number.'

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'Let {b_{k}} be a geometric sequence with first term 1 and common ratio 3. For each natural number n, let the largest b_{k} satisfying b_{k}≤n be denoted as c_{n}. Calculate Σ_{k=1}^{30} c_{k}.'

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'The 3rd degree equation x^3 + ax^2 + bx + 1 = 0, where coefficients a and b are integers, has 2 complex solutions and 1 negative integer solution. The number of integer pairs (a, b) that satisfy this condition is □.'

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'Find the general term and sum of a geometric series. Let the first term be a and the common ratio be r.'

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'The following sequences are geometric progressions. Find the values of x and y. \n(1) 3, x, 1/12, ......\n(2) 9, x, 4, y, ......'

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'In an arithmetic sequence with a first term of -83 and a common difference of 4, up to which term will the sum from the first term be the smallest? Also, determine the sum at that point.'

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'In the order of sum and product, the values are (1) 4, -3 (2) 3/2, 3 (3) -4/3, -5/3'

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'(2)(1/2)^{n} <0.001 Taking the common logarithm of both sides, we get n log_{10} 2>-3 Therefore, n>3/ \\log_{10} 2=9.96... The smallest natural number n that satisfies this inequality is n=10'

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'There is an arithmetic sequence {an} with the first term 7 and a common difference of 3, as well as an arithmetic sequence {bn} with the first term 8 and a common difference of 5. Let {cn} be the sequence formed by arranging the common terms of these two sequences in ascending order. Find the general term of the sequence {cn}.'

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'Prove that the following inequalities hold when a>0, b>0.'

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'When depositing 200,000 yen at the beginning of each year with an annual compound interest rate of 1%, calculate the total principal and interest at the end of the 10th year (i.e., the total amount of principal and interest at the beginning of each year). Use 1.01 raised to the power of 10 equals 1.105 for calculation.'

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'Given an annual interest rate r, annually saving a yen in compound interest for n years, find the total amount of savings at the end of n years.'

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'Chapter 1 Sequences - 219\nUsing the method of finite differences, find the general term of the following sequence \ \\left\\{a_{n}\\right\\} \.\n19\n(1) \ 20,18,14,8,0 \, \ \\qquad \ (2) \ 10,10,9,7,4 \,'

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'When the three points A(1,1), B(2,4), C(a,0) are the vertices of triangle ABC and form a right triangle, find the value of the constant a.'

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'Find the following values. (1) \ \\sqrt[4]{16} \ (2) \ -\\sqrt[3]{64} \'

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'Find the remainder when the polynomial $x^{1010} + x^{101} + x^{10} + x$ is divided by $x^3-x$.'

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'When the inequalities 4x+y≤9, x+2y≥4, and 2x-3y≥-6 are simultaneously satisfied, find the maximum and minimum values of x^2+y^2.'

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'Find two numbers that satisfy the following conditions.'

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'The sequence {a_n} is defined by the first term $a_1=1$ and the recursive formula $a_{n+1} = \\frac{3(n+1)}{n}a_{n}$.'

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'Prove the inequality 2^{n}>4 n+1 when n is an integer greater than or equal to 5.'

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'Determine whether the following sequences are arithmetic sequences or geometric sequences.\n1. Sequence 4, 7, 10, 13\n2. Sequence 3, 6, 12, 24'

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"Let's review the sum of natural numbers and the sum of arithmetic sequences!"

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'Find the general term of the sequence $\\ left \\ {a_ {n} \\ right \\}$ defined by $a_ {1} = 1, a_ {n + 1} = 2a_ {n} + 3 ^ {n}$.'

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'A student A, who commutes by bicycle, went to school at a speed of 12 km/h one day, and on the way back, walked with his friend at a speed of 6 km/h while pushing the bike. Now, on this day, at what average speed did student A travel?'

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'Find the number of terms n and the common difference d of an arithmetic sequence where the first term is 2, the last term is 38, and the sum is 200.'

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'For the sequence {an} where the sum from the initial term to the nth term is given by Sn=-n^2+24n (n=1,2,3,...), find the range of natural numbers n for which an<0, and calculate ∑_(k=1)^40|ak|.'

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'Find the general term of a geometric sequence {an} with initial term a and common ratio r.'

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'Find the numbers of an arithmetic sequence with a first term of 3 and a common difference of 4, up to the 5th term.'

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'\\sum_{k=1}^{n} k^{2}=\\frac{1}{6} n(n+1)(2 n+1)'

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'(4) \ \\\\sqrt[3]{54} \\\\times 2 \\\\sqrt[3]{2} \\\\times \\\\sqrt[3]{16} \'

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'Find the nth term of the following sequences:\n1. Arithmetic sequence with first term 3 and common difference 2\n2. Geometric sequence with first term 2 and common ratio 3'

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"Find the number of terms 'n' and the common difference 'd' of an arithmetic sequence with initial term -10, final term 200, and sum 2945."

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'Find the sum of an arithmetic series with first term 25, last term -10, and 16 terms.'

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'For an arithmetic sequence with initial term -0.2 and final term 0.6, if there are n terms between the initial and final terms, the sum is 405.'

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'Using mathematical induction to prove the following equation'

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'(1) Find the general term a_{n} of a geometric sequence with first term 7 and common ratio 1/2. (2) Find the common ratio and the general term a_{n} of the following geometric sequences. (a) 3, -3, 3, -3, ... (b) -16/27, 4/9, -1/3, 1/4, ...'

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'\ 65 a=-3, \\quad b=10 \, solution: \ x=-2,2 \\pm i \'

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"Create the pattern shown in Figure 1 using Pascal's triangle, where even numbers are represented by ○ and odd numbers are represented by ●. By following the four rules based on the properties of Pascal's triangle, mark the positions with ○ and ●."

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'Prove the inequality (A>B) by creating the difference (A-B). Use the following methods:'

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'Find the values of x and y in the arithmetic sequence.'

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'Standard 47: Determination of two numbers given their sum and product'

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'The star Vega (Weaving Maiden Star) is a zero-magnitude star'

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'Determine which column from the left the second digit of 2020 is located in.'

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'Find the sum of the following numbers for integers from 1 to 200: (1) Multiples of 4 (2) Numbers that are not multiples of 4.'

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'Find the first term and common ratio of a geometric sequence. The common ratio is a real number. (1) The third term is 18, and the fifth term is 162. (2) The second term is 4, and the fifth term is -32'

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'In the same election as example 1, party B and party C merged to form a new party E, while maintaining the same total number of votes before and after the merger. Assuming that the votes of the other parties remain the same, the votes for party A are 10000, party D are 4000, and party E are 15300.'

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'Understand the formula for the sum of a geometric series and conquer Example 13!'

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'Let TR be \ \\log _{10} 2=0.3010 \. Find the value of a natural number \ n \ that satisfies the following conditions.'

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'Find the first term and common ratio of a geometric sequence. The common ratio is a real number. (1) The 3rd term is -18, the 6th term is 486 (2) The 6th term is 4, the 10th term is 16'

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'Using the 2nd order difference sequence, find the general term of the following sequence {a_{n}}. (1) 20, 18, 14, 8, 0, ...'

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'Find the sum S of an arithmetic sequence with first term 25, last term -10, and 16 terms.'

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'Divide the sequence of natural numbers such that each group contains 2n numbers as follows: 1,2|3,4,5,6| 7,8,9,10,11,12 | 13,14, …… (1) Find the first number in the nth group. (2) Find the sum of all numbers in the nth group.'

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'Find the general term of the sequence {an}: 5, 11, 23, 41, 65, 95, ...'

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'For two distinct real numbers a, b, if a, 2, b form a geometric sequence in that order, and 1/2, 1/b, 1/a form an arithmetic sequence in that order, then a=□, b=□.'

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'Basic Example 3 Determination of the 4th series (1)...An arithmetic progression {an} in which the 5th term is 3 and the 10th term is 18'

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'Explain what an arithmetic progression is and find the 10th term of an arithmetic progression with initial term 5 and common difference 2.'

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'(4) A point (x, y) in the coordinate plane is called a lattice point when both coordinates are integers. In this problem, "inside the region" refers to including the interior and boundary of that region.'

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'Let a be a positive constant. Determine the range of values for a such that 2x^{2}+y^{2}-1=0, x^{2}+y^2-4x-4y+8-a=0 have common points.'

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'Find the sum of the following geometric progressions.'

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'Find the pattern in the following sequences and express the general term in terms of n that follows the pattern.'

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'Find the general term of the sequence {an} defined by a1=3, an+1 = an/(2an + 4).'

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'Find the 10th term of an arithmetic sequence with a first term of 5 and a common difference of 3.'

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'Basic 58: Use long division to find the quotient and remainder of a division operation.'

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'Find the sum of integers from 1 to 100 that are multiples of 6 and those that are not multiples of 6'

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'There are many glass plates of the same quality. When 10 glass plates are stacked and light passes through them, the intensity of the light becomes 2/5 of the original. How many more glass plates should be stacked to reduce the intensity of the transmitted light to below 1/8 of the original? Given that log10 2 = 0.3010 and log10 5 = 0.6990.'

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'Find the 5th term of a geometric sequence with first term 5 and common ratio 2.'

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'Using second order different sequence, find the general term of the sequence {an}. (2) 10,10,9,7,4, ...'

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'Basic 5: Three numbers forming an arithmetic progression'

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'Prove using mathematical induction that for all natural numbers n, 4n^3 - n is a multiple of 3.'

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'Proof of Equation (3)...condition is a proportionality'

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'Find the general term and sum of an arithmetic sequence.'

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'Prove that the inequality |1+ab| > |a+b| holds when |a| < 1, |b| < 1.'

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'Translate the given text into multiple languages.'

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'For the sentences X and Y regarding the underlined part f of question 6, choose the correct combination of true or false.'

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'Q. (2) Answer the following questions by providing appropriate values as integers.'

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"When the side length of the black square is 9 cm, what range of integers should be arranged in the white square's grid? Please list all possible options."

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'For sentences X and Y regarding the underlined part b in question 2, choose the correct combination of true or false from below.'

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'To completely react 11.2mL of hydrogen, at least 5.6mL of oxygen is required. The volume of air containing 5.6mL of oxygen can be determined from the percentage of air in Table 1, which is 5.6 ÷ 0.21 = 26.66, rounded to 26.7mL.'

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'Person A leaves school between 0 and 60 minutes later, arrives at station K between 12 and 72 minutes later, and arrives at station M between 14 and 74 minutes later. In addition, the train leaves station K at times divisible by 8, and the train leaves station M at times divisible by 5, as shown in the diagram 1. However, it is not possible to determine the difference in waiting time from diagram 1, which is why diagram 2 is created by shifting the M station diagram 2 minutes to the right to align the arrival times at the station. From diagram 2, it is apparent that the waiting times are equal when arriving at the station in the bold line section. In this case, if Person A determines the time to leave school at station M, it can be deduced from 45-14=31 minutes later to 50-14=36 minutes later (A is 45-2=43 minutes later). If determined at station K, the elapsed time can be narrowed down from 43-12=31 minutes later to the time seen in the figure.'

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'There is a stack of 144 cards with numbers 1, 2, 3, ..., 143, 144 placed on top of each other as a stack with a box next to it.'

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"Answer the questions regarding Data 2, 'Graph of atmospheric pressure changes'.(1) Select appropriate words or symbols to fill in the [ ] and enclose them in a circle.\nAround a typhoon, the closer to its center, the lower the atmospheric pressure. Therefore, it is understood that the graph created from our school's observation data is [(I) [ (low) ]. Additionally, from the graph in Data 2, we can determine the time when the typhoon's center approached each observation point. Comparing Tokyo and Choshi in the graph, it becomes clear that Tokyo's graph showed [(III) [ (low) ] as the first to approach the typhoon's center, while Choshi's graph showed [(V) [ (high) ] as the first."

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"(5) The sedimentation rate of the Chiba section is 2 meters per thousand years, so the time required to stack from the layer of volcanic ash formed 773,000 years ago to the layer 1.6 meters above is 1000×1.6/2=800 (years). Therefore, from 773,000-800=772,200 years ago, the Earth's magnetic field shifted to its current orientation."

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'When comparing the numbers at the same index of columns A and B, which number has the largest difference? List all possible answers.'

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'(6) The train heading from Makuhari Station to Makuharihongo Station travels 600m in the first 60 seconds, and then 20 × 17.5 = 350m in the remaining 17.5 seconds. Therefore, the position where the trains pass each other is at a distance of 950m from Makuhari Station.'

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'Events happened in 1428, 1392, and 1489, so in chronological order, it should be I-II.'

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"The average values of temperature and other data announced by the Japan Meteorological Agency are calculated by averaging the numbers from the years where the last digit of the year is '1' and continuing for 30 years. Starting from May 19, 2021, the data for the years 1991 to 2020 have replaced the previous data from 1981 to 2010."

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'There are points A and B upstream and downstream of a river, and boats P and Q sail back and forth between them. Boat P departs from upstream point A, arrives at B, and immediately returns to A. Boat Q departs from downstream point B, arrives at A, and immediately returns to B.\nBoats P and Q depart simultaneously from points A and B, meet at point C, then meet again at point D. The distance between A and C is in a 3:2 ratio with the distance between C and B, and C and D are 120 meters apart.\nIn still water, the speeds of boats P and Q are constant, with the speed of boat Q being 1.5 times the speed of boat P. Boat P takes 48 minutes for a round trip between A and B. Additionally, assume that the speed of the river current is constant.\nAnswer the following questions:\n(1) How many minutes does boat Q take for a round trip between A and B?'

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'When the side length of the black square is 14 cm, the number of white squares is (14+1) x 4 = 60. Thus, 60 can be represented as the product of two integers: 60 = 1 x 60, 2 x 30, 3 x 20, 4 x 15, 5 x 12, 6 x 10.'

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'Please fill in the blanks appropriately. The vertical axis value of point (2) represents the population number of generation (A), and the vertical axis value of point (3) represents the population number of generation (B).'

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'(1) Since the cross-sectional area inside the plastic tube is 0.25 cm^2, the volume of nitrogen at 20°C is 0.25 x 14.0 = 3.5 (cm^3), and the volume of oxygen is 0.25 x 30.0 = 7.5 (cm^3).'

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"Finding the number of cases\n(1) First, find the 20th number of Mr. A. As shown in Figure 1, when the digit in the thousands place is 1, there are 4 possibilities for the hundreds place, 3 possibilities for the tens place, and 2 possibilities for the ones place, so for a four-digit number, we find that the 24th number from the left is 1976. From here, drawing a tree diagram from largest to smallest as shown in Figure 2, we can determine that the 20th number from the smallest is 1947. Also, Mr. A's card number is 2938."

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'(7) (1)~(3) To accumulate layers of the same thickness of 1m, it takes 500 years in Chiba, while it takes 5000 years in Italy. Therefore, the speed of layer accumulation in Chiba is 10 times faster, calculated as 1/500 ÷ 1/5000 = 10.'

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'Mathematics: 100 points (estimated score)\n1. Each 7 points x 3\n2. (1) 8 points\n (2) to (4) each 5 points x 3 < Each fully answered > □\n3. □ Each 7 points x 8'

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'Arrange white squares with a side length of 1 cm around a black square with a side length of 1 cm. The diagram below shows white squares arranged around black squares with side lengths of 1 cm, 2 cm, 3 cm, and so on from left to right. Inside the white square grids, the integer A is used A times, and two or more consecutive different integers are arranged starting from a certain integer. For example, as shown on the left side of figure 1, when the side length of the black square is 2 cm, using 3 of 3, 4 of 4, and 5 of 5 can be arranged precisely. However, as shown on the right side of figure 1, it is not possible to arrange precisely 4 of 4, 5 of 5, and 6 of 6. Also, as shown in figure 2, when the side length of the black square is 8 cm, integers from 1 to 8 and from 11 to 13 can be arranged precisely.'

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'2020 Shibuya Educational Institute Makuhari Junior High School Mathematics 1st Test\n1 (3) What card is left on the mountain at the end of the operation?'

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'At a certain time, 12 lights were on. How many possible times are there?'

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'In relation to part c under question 3, the ancient battlefield of the Battle of Yashima is located in present-day Kagawa Prefecture. Kagawa Prefecture is the birthplace of former Prime Minister Masayoshi Ohira. Choose the correct combination of statements A to D regarding events in the 1970s when Masayoshi Ohira served as Foreign Minister and Prime Minister from the options below and answer by number.'

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'For sentences X and Y regarding the underlined part d in question 5, choose the correct combination of true or false as the answer.'

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'Volcanic ash layers serve as clues to compare distant layers. Choose the appropriate option in the square brackets to explain the reasons, and encircle it with ○.'

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'(3) Definition of Unit\nThe standard unit of "mass" began to be "kilogram prototype" at the end of the 19th century. The reason is that the mass of "1000 cm^3 of water" varies depending on the conditions of the water. The "kilogram prototype" is a solid metal, so its mass does not vary with conditions. Think of a condition that would change the mass of "1000 cm^3 of water" and write it down.'

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'When A is divided by 15, the quotient and remainder are denoted as □, so A ÷ 15 = □ remainder □. Now, when P ÷ Q = R remainder S, then P = Q × R + S. Therefore, A = 15 × □ + □ = (15 + 1) × □ = 16 × □, which means A is a multiple of 16. Similarly, when A is divided by 17, the quotient and remainder are denoted as △, so A ÷ 17 = △ remainder △. Hence, A = 17 × △ + △ = (17 + 1) × △ = 18 × △, showing that A is a multiple of 18. Therefore, A is a common multiple of 16 and 18. Additionally, from the calculation on the right, we find that the least common multiple of 16 and 18 is 2 × 8 × 9 = 144, and thus A is a multiple of 144. Finally, since dividing yields A ≤ 16 × 14 = 224, the only number that satisfies the conditions is 144.'

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'React 11.2 mL of gas 3 with air. Provide the minimum volume of air required to ensure that gas 3 does not remain, rounded to the first decimal place.'

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'If the liquid is continued to be poured at the same rate after figure 3, find the time it takes for containers A and B to fill up respectively, and answer which container will be filled first.'

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'2020 Shiba Education Academy Makuhari Middle School 2nd (2)\n(2) How many minutes after departure did ship P and ship Q meet at point D?'

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'(2) The illuminance at a distance of 100 cm from the light bulb is 120 lux, and at a distance of 50 cm, it is 500 lux. Therefore, 120 divided by 500 equals 0.24, so the illuminance at a distance of 100 cm is approximately one-fourth of the illuminance at a distance of 50 cm.'

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'Hara Takashi organized the Rikken Seiyukai as its president, and formed the first full-fledged party cabinet in 1918, which corresponds to the 7th year of Taisho.'

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"In Japan, television broadcasting began in 1953, followed by the start of a high economic growth period in the late 1950s. During this time, household electrical appliances began to spread to homes nationwide, with black and white televisions, electric washing machines, and electric refrigerators being popularly known as the 'Three Sacred Treasures'. Air conditioning and automobiles, along with color television, were referred to as '3C', and became common in the latter half of the high economic growth period. The Public Election Law and Maintenance of Public Order Law were enacted in the late Taisho era in 1925, the same year that radio broadcasting began."

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'Regarding the underlined part B in question 3, choose the correct combination of true or false for the following sentences X and Y'

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'There are 33 types of candles A, B, C. When you light 3 candles, they will burn at a certain rate. After lighting A, light B after 10 minutes, then C after another 5 minutes. Candle C burned out first, followed by candles A and B burning out simultaneously. The graph below shows the time it takes for all candles to burn out after lighting candle A, as well as the relationship between the longest and shortest candle lengths. The length of a burned-out candle is considered to be 0 cm. Answer the following questions.'

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'For sentences X and Y related to the underlined part b in question 3, choose one from the following combinations and answer with the number which is correct.'

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'For the sentence X and Y regarding the underlined part c in question 3, select the correct combination of true or false, and then choose one number from the options below to answer. X Yokohama City was one of the first designated government ordinance cities in Japan, along with Nagoya City, Osaka City, Kyoto City, and Kobe City. Due to historical reasons, Y in Yokohama City, dyeing industries such as silk handkerchiefs and scarves have become local industries.\n\egin{tabular}{|llllllllll|}\n\\hline 1 & \ \\mathrm{X} \ & True & \ \\mathrm{Y} \ & True & 2 & \ \\mathrm{X} \ & True & \ \\mathrm{Y} \ & False \\\\\n3 & \ \\mathrm{X} \ & False & \ \\mathrm{Y} \ & True & 4 & \ \\mathrm{X} \ & False & \ \\mathrm{Y} \ & False \\\\\n\\hline\n\\end{tabular}'

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'Due to its original green color, the color change of red cabbage solution is considered as red. This color change indicates that the pH of solution A is below 2.5. An acidic solution that exactly neutralizes the 5 mL of solution B has a pH of 3.5, calculated as 7-(10.5-7)=3.5. Assuming solution A has a pH of 2.5, compared to the acidic solution with a pH of 3.5, the acidity level is 10 times stronger, indicating that the volume of solution A needed to neutralize solution B is 1/10 of B. If the pH of solution A is lower than 2.5, the volume of solution A needed for neutralization will be even less.'

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'Reading the graduations on this scale, P’ is 4mm, Q is 26mm. Therefore, the length of P’Q is found to be 26 - 4 = 22mm.'

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'Math problem (1) at Shibuya Gakuen Makuhari Middle School in 2021'

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'For the following sentences X and Y regarding the underlined part c in question 3, choose one correct combination of true or false from the table below and answer with the corresponding number.'

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"When the side length of the black square is 14 cm, what range of integers should be lined up in the white square's grid of squares? Please answer with all possible options."

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"Shinichi walks from his home to his friend's house along a straight road. Initially, he was running towards his friend's house but got tired, so he started walking from the midpoint between his home and his friend's house. As a result, he arrived 20 minutes later than if he had run all the way. On his way back, his mother comes to pick him up by car. Shinichi walks to his friend's house, while his mother drives from home, both leaving at the same time. They meet on the way back, where Shinichi gets into the car, and they are supposed to return home together. However, Shinichi left his friend's house 10 minutes later than planned. His mother, who left as scheduled, continues driving until she meets Shinichi, picks him up, but it takes longer than planned. Shinichi's walking speed is x, running speed is 2x, and car speed is 5x."

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'Provide appropriate values for the following blanks.'

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"In this manner, set A's tens place to 9 and B's tens place to 9. Thus, the remaining card difference is 2-1=8-7=1, so the largest difference is found in the pairs (6491, 4392) and (6497, 4398) (both differences are 2099). Next, consider the cases where A becomes 6491 or 6497. From (1) and (2), we know that there are 24 integers with a thousandth place of 1 or 4. Additionally, there are 6 integers with a thousandth place of 6 and a hundredth place of 1. Arranging integers with a thousandth place of 6 and a hundredth place of 4 in ascending order gives {6417, 6419, 6471, 6479, 6491, 6497}, so we find that 6491 is the 59th number and 6497 is the 60th number."

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'Measure the weight of 12 seeds and find the weight of water lost from the 12 seeds.'

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'If train A runs 0.2 km/h slower than the actual speed, it arrived at station K 18 minutes late from the scheduled time.'

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'(1) "Meters per second" is a unit that represents the distance traveled in one second, so it is a unit of speed.'

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"In question 1, the blanks A to O will be filled with either 'low' or 'high'. Choose the correct symbol combination representing the blank filled with 'high' from the options below and answer with the corresponding number."

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'Question 1 is worth 3 points each × 4, Questions 2 to 5 are worth 4 points each × 4, Question 6 is 6 points, Question 7 is 4 points, Question 8 is 10 points, Question 9 is worth 3 points each × 2'

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'Question 7 pertains to the underlined part f, Owari Province was a country located in the western part of present-day Aichi Prefecture. The prefectural capital of Aichi is Nagoya, but the figure on the right, Figure 5, depicts the situation of the rice disturbance that erupted in Nagoya. Looking at this Figure 5, choose the correct combination of the following statements A～D concerning the rice disturbance.'

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'Which of the following is a solid at room temperature? (1) Sodium hydroxide (2) Aluminum (3) Salad oil (4) Disinfectant alcohol (5) Carbon dioxide (6) Oxygen'

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'2021 Shibuya Education Academy Makuhari Middle School 2nd time question (3) (2) It takes 12 minutes to cover the distance at a speed of 1 km per minute, so the distance between M station and K station is 1 * 12 = 12 km.'

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"The 3rd Abe Cabinet was a coalition cabinet of the Liberal Democratic Party (LDP) and Komeito, with ministers chosen from Komeito as well. Prime Minister Shinzo Abe resigned on September 16, 2020, with a total tenure of 3188 days, surpassing Taro Katsura's 2886 days to become the longest-serving prime minister in history. Furthermore, since the formation of the second cabinet on December 26, 2012, the continuous tenure has been 2822 days, surpassing Eisaku Sato's 2798 days, also making it the longest in history. Therefore, the statement is correct."

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'Question 1 In relation to underlined part a, during the Jomon period, there was a custom of burying the deceased as shown in the image on the right. What is this type of burial called? Answer in kanji.'

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'When various groups were divided to conduct experiments 1 and 2, some groups found that the mixed solution did not enter the round bottom flask vigorously. Select all applicable reasons from the following options and provide the corresponding symbols.'

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'2021 Shibuya Education Academy Makuhari Middle School 2nd (2)'

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'2 (2) ÷ C = 15 with a remainder of 15, therefore, B = C × 15 + 15 = 15 × (C + 1) so B is a multiple of 15. Similarly, B ÷ D = 17 with a remainder of 17, B = D × 17 + 17 = 17 × (D + 1) so B is a multiple of 17. Thus, B is a common multiple of 15 and 17, with the least common multiple of 15 and 17 being 15 × 17 = 255, so B is a multiple of 255. Furthermore, C is greater than or equal to 16, and D is greater than or equal to 18, therefore B is at least 17 × (18 + 1) = 323. Thus, when 999 is divided by 255 with a remainder of 234, the largest 3-digit integer is calculated to be 255 × 3 = 765.'

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'The fractions that cannot be simplified are in ascending order from small to large {1/2021, 2/2021, 3/2021, ...}, and in descending order from large to small {2020/2021, 2019/2021, 2018/2021, ...}. Adding them up in pairs, the sum of each pair is 1/2021+2020/2021=2/2021+2019/2021=3/2021+2018/2021=1. Moreover, as there are 2020-88=1932 fractions that cannot be simplified, the number of pairs is 1932÷2=966. Therefore, their sum is 1×966=966.'

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'The 73 lugworms eaten by sandpipers A ate double the organic matter of 2.19 x 2 = 4.38 grams in two days, 15th and 16th.'

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'Select the correct option regarding the underlined part a in the following sentences X and Y.'

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"(3) The density of water (weight per unit volume) is highest at 4°C, and decreases at temperatures higher or lower than 4°C. In other words, the mass of '1000 cm^3 of water' varies with temperature, making it unsuitable as a standard for weight."

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"Explain why, regardless of the length of a black square's side, it is not possible to arrange only two consecutive integers in the white square's grid."

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'In the photo of Figure 6, the main scale and vernier scale are aligned at the 3.5 mark of the vernier scale. What is the diameter of the button in millimeters? Please answer to two decimal places.'

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'As shown in the diagram, there is a stack of 144 cards numbered 1, 2, 3, ..., 143, 144, placed on top of each other, and next to it there is a box.'

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'For the sentence X・Y regarding the underlined part j in question 10, which combination of correct and incorrect is the correct one?'

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'With 1 km = 1000 m and 1 hour = 60 minutes = 3600 seconds, 72 km/h is equal to 72 × 1000 ÷ 3600 = 20 (m/s).'

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'What is the maximum time interval from when you hear the sound of A to when you hear the sound of B (rounded to one decimal place)?'

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'White and black stones are arranged in a single row from left to right without having the same color stones appear consecutively more than 3 in a row. The diagram on the right was drawn to consider the ways in which 4 stones can be arranged using white and black stones combined. (1) How many ways are there to arrange the stones using a total of 6 white and black stones?'

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'Question 5 a Byodo-in Phoenix Hall is a hall built by Fujiwara no Yorimichi in 1053, in the latter half of the 11th century. b In 784, towards the end of the 8th century, Emperor Kanmu relocated the capital from the strong Buddhist influence of Heian-kyo to Nagaoka-kyo in Kyoto.'

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'Answer the questions about the heat released during the combustion of methane, propane, and butane.\n(1) Perform the following calculations:\n a) The amount of heat released when 0.7g of methane combusts\n b) The weight of 1L of propane and the heat it releases\n c) The weight of 1L of butane and the heat it releases'

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'Choose the appropriate content in the following [], and mark it with ○.'

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'Calculate the amount of heat required to raise the temperature of ice from -20°C to 0°C.'

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"The United Nations, established in October 1945 with 51 founding member countries after the end of World War II, has its headquarters in New York City in the eastern United States. As of the end of 2021, 193 countries are members of the organization. The expenses required for the UN's activities are mainly covered by contributions from member countries. These contributions are allocated every three years based on factors such as each country's economic power, and are decided by the General Assembly. Japan's share of contributions has ranked second after the United States for many years, but in recent years, it has dropped to third place, after the United States and China."

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'These are equal to 6 times K or N-number or 3N. Identify one of them.'

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'Which of the following is the same as the gas or precipitate obtained through operations a and b? Please answer with symbols.'

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'By counting the sub-scale between PQ, you can determine the length between PQ. How long is PQ in millimeters? Please answer up to two decimal places.'

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'In 607 AD, Ono no Imoko was dispatched to Sui (China) as a envoy during the time of Empress Suiko. In 804 AD, Kukai crossed over to Tang as a scholar monk on a mission ship, learned Esoteric Buddhism, returned to Japan, and became the founder of Shingon sect in Japan by building Kongobuji on Mount Koya (Wakayama Prefecture).'

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'I happened in 1936, II in 1925, III in 1914, IV in 1918. The Taisho era lasted until December 1926, after which the Showa era began, so it should be II-III-I-III.'

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'What is the distance between the smallest graduation lines on the scale in millimeters? Please answer up to two decimal places.'

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'(5) As the temperature rises by 1 degree Celsius, kerosene increases by 0.14% of the standard, and nitrogen gas increases by 0.36%. Therefore, 0.36 ÷ 0.14 = 2.57..., which is approximately 2.6 times.'

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'In the second exam of 2020, you scored 203 points. Does this mean you have reached the passing score?'

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'Using the 2021 Shibuya Education Academy Makuhari middle school (2nd round) (26) question, calculate the answer up to the third decimal place.'

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'Question 6. Choose the correct combination of true or false for the following sentences X and Y regarding the underlined part e.'

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'It can be inferred that many school facilities were built around 1978 based on the fact that most of these facilities are over 40 years old. After the end of World War II in the late 1940s, there was a baby boom, and by the early 1970s when that generation became parents, a second baby boom occurred. It is predicted that when the children born during this time go to school, there will be a shortage of school facilities such as classrooms and school buildings, leading to many local governments carrying out new constructions or renovations.'

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'When assigning teams to a tournament bracket, the arrangement where the match-ups in the first round are all the same and the potential match-ups in the second round are also all the same is considered to be the same. For example, the assignments in Figures 2, 3, 4, and 5 are considered the same, while the assignments in Figures 2 and 6 are considered different.'

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'In the food chain of the intertidal zone, consider migratory birds utilizing organic matter. Observing a sample of Eastern Curlew (hereafter, Curlew) that visits the intertidal zone, the amount of organic matter consumed by them was determined. Calculate the appropriate numerical values (to two decimal places) for the following sentences.'

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'2020 Shibuya Educational Academy Makuhari Middle School 第1次 (24) (3) For figure 2, answer the suitable numbers for the following parentheses.'

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'(2) There are 6 possible combinations of 4 seats. In each case, there are 24 ways for 4 people to sit (4 × 3 × 2 × 1), so the total is 24 × 6 = 144 ways.'

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'Starting from 1/2022, where the denominator decreases by 1 and the numerator increases by 1, a total of 2022 fractions are lined up. Look for fractions that can be simplified, such as 4/6=2/3. Answer the following questions: (1) Which position from the left is the first one that can be simplified? (2) Which position from the left is the third one that can be simplified? (3) Which position from the left is the 25th one that can be simplified?'

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'For the sentences X and Y regarding the underlined part b in question 2, choose the correct combination of true or false from the options below and answer with the corresponding number.'

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'Problem regarding the speed and transmission of sound'

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'When A leaves S Middle School between 2:00 PM and 3:00 PM, the waiting time for the train at any station remains the same. At what time does A leave S Middle School from 2:00 PM to what time?'

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'From (5) and (4), the length of D can be calculated as 4 + 0.55 = 4.55(mm).'

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"Question 1: For the sentence X and Y regarding the underlined part 'a', choose one correct combination of true or false from the options below."

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'For question 2, choose the correct combination of true or false statements regarding the underlined part b in relation to the following explanations X・Y.\nX A revision of the Public Offices Election Law, including an increase of 6 seats, was passed to eliminate the disparity in votes in the House of Councillors elections.\nY A revision to the Imperial Household Law was made to allow the abdication of the Emperor for one time.\n1. X - True, Y - True\n2. X - True, Y - False\n3. X - False, Y - True\n4. X - False, Y - False'

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'[Math] 100 points (estimated score) 8 points each for 1 and 2 × 6, 7 points each for 3 and 4 × 4 boxed 5 × 8 points each for 3'

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'Question 56 points question 6 to question 8 each 4 points (3 questions)'

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'In the Pleiades star cluster, we can see two types of red-colored stars. Bright red stars and dark red stars. How can bright red stars be perceived differently from dark red stars? Fill in the blank to complete the sentence.'

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'Question 9. Choose one correct combination from the following options regarding the explanation about the underlined part h (X) and Y.'

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'When dividing the seating arrangements for 3 seats into 5 cases, each case has 6 ways for 3 people to sit (due to permutations). Therefore, the total number of combinations is calculated as 22x6=132.'

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'Ship Q arrived at A after 36 ÷ 2 = 18 minutes from departure. By that time, Ship P had advanced 6 minutes from B, so the distance between both ships when Q arrived at A was 36 - 1 × 6 = 30. Therefore, the two ships met at D after Ship Q turned back from A, which is 30 ÷ (4+1) = 6 minutes later. This is 24 minutes after departure.'

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'Find the ratio of water depth when the same amount of water is poured into A and B.'

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'From <Experiment 2>, calculate the weight of 12 grains of seeds for each group and the weight of water lost when the 12 grains of seeds turn into popcorn, rounded to the first decimal place.'

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'You have four lights that can illuminate red, blue, yellow, and green, arranged in a row. Each time you press the switch, the colors of these four lights change according to a certain rule. Starting from the initial state, how many different rules can there be that result in the four lights illuminating different colors?'

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'Choose one correct combination of X and Y statements regarding the time when the work in question 3, part c, was completed.'

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'In condition 4, 11.2mL of hydrogen and 22.4mL of oxygen (33.6 - 11.2 = 22.4) are mixed, so 5.6mL of oxygen is used in the reaction, leaving 16.8mL.'

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'Question 9 1) Both prefectural governors and local council members have a term of 4 years. 2) The right to stand for election for senators and prefectural governors is granted to those aged 30 and above, while the right for representatives, mayors, and local council members is granted to those aged 25 and above. 3) In 2015, an amendment to the Public Offices Election Act lowered the age for suffrage from 20 to 18 for national parliamentarians, mayors, and local council members. 4) Prefectural governors have the authority to dissolve the prefectural assembly, but do not have the authority to dissolve city (ward) and town/village councils.'

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'For a non-zero integer c, calculate 8 △ c. Find the largest possible value of 8 △ c.'

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'Question 12 (Example) Part stipulating that the number required to convene an extraordinary session is at least one-fourth of the total members of any of the houses of the legislature.'

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'Choose one option from the list below and answer with a number.'

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'For the statement X and Y regarding the policy in the underlined part of question 5, choose one correct option as the correct and incorrect combination from the table below.'

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'If A leaves S Middle School between 2 p.m. and 3 p.m., there is less waiting time for the train at the station to go to K Station than to go to M Station. How many minutes in total does A spend from leaving S Middle School at 2 p.m. until 3 p.m.?'

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"(4) 1 day is 24 hours, 1 hour is 60 minutes, 1 minute is 60 seconds, so 1 day is 24×60×60=86400 seconds. The number '86400' is derived from this."

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'Read the sentence in (), use explanatory text and numerical values in the table to find the integer that fits in ().'

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"Since it took 12 minutes for boat P to travel from point A to point B, the downstream speed of boat P is 1800 divided by 12 equal to 150 meters per minute. Furthermore, the ratio of boat P's downstream speed to the river flow speed is 3:1, thus the river flow speed is 150 multiplied by 1/3 equal to 50 meters per minute. Converting this to speed per hour, it becomes 50 multiplied by 60 divided by 1000 equal to 3 kilometers."

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'In 2019, at the Shibuya Education Academy Makuhari Middle School, the following experiment was conducted: 5 mL of aqueous solutions of substances A to F commonly used in daily life were taken in test tubes and their colors were observed. A was toilet cleaner, B was water with shirataki (konjac) noodles, C was carbonated water, D was laundry bleach, E was rice vinegar, F was mirin. A was green in color, while B to F were almost colorless and transparent, with D to F slightly yellow. It was also discovered that the green color of A does not change with acidity or alkalinity. Then, equal amounts of purple cabbage solution were added to the A to F aqueous solutions, resulting in the following color changes. Question 1. (2): Which of A to F are alkaline aqueous solutions? Select all and provide the symbols.'

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'When the total number of black stones is 1000, how many rounds at most can the first white stone be surrounded by black stones?'

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'(5) The definition of a meter is based on the speed of light, which is 299,792,458 meters per second. The speed of light was first measured in 1676 through observations of the moons of Jupiter.'

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'You want to set the readable length with an interval of 0.02mm. You need to change the length of the sub-scale to 49mm. How many parts should you divide 49mm into?'

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'Generations 1 to 3 increase to 10→99→690. Then, continuing the same process from point 3 onwards, as generations progress, the number of individuals fluctuates and eventually approaches a certain fixed number (the intersection of the individual number graph and line L is 570). Furthermore, the amplitude of these fluctuations gradually decreases. Therefore, () and (セ) are options to choose.'

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'When divided by 9, 60 gives a quotient and a remainder of 6 each, which are equal. Also, when divided by 11, 60 gives a quotient and a remainder of 5 each, which are equal.'

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'[Math] 100 points (estimated score)\n1 (1) 2 points each × 3\n(2),\n(3) 7 points each × 2<(3) is a complete answer > '

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"Question 9 has the word 'national tax' just after the underlined part. In addition, it is the travelers who bear the tax of 1000 yen, but the tax is collected by the airline or shipping company by adding it to the ticket price 'as a general rule', so it is the airline or shipping company that pays taxes to the country, which is an indirect tax where the burden and payer of the tax are different. Income tax in 1 is a direct tax as national tax, resident tax in 2 is a direct tax as local tax, liquor tax in 3 is an indirect tax as national tax, and local consumption tax in 4 is an indirect tax as local tax, so 3 is chosen."

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'Problem of finding teams that have the potential to become runners-up'

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'Problem to find the number of ways of allocating when A enters I\nConsider a 1st round of 4 matches as I to IV, and consider the case where A enters I. In this case, there are 7 ways to choose the other team that enters I. Furthermore, the two teams entering I will be chosen from the remaining 6 teams, so there are 6×5÷2×1=15 possible combinations for I. Also, the two teams entering {II, III, IV} will be chosen from the remaining 4 teams, so there are 4×3÷2×1=6 possible combinations for II, but swapping II and IV results in the same combination, making the combinations for II and IV to be 6÷2=3. Therefore, the total number of different allocations is 7×15×3=315.'

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'For candles A, B, C, find the length before lighting them.'

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'Choose the correct combination of X and Y sentences regarding events from 1960 to 1965 related to the underlined part k, and answer with the correct number from below.'

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'When student A returns home from S middle school, they can choose to use either K station on the Seaside Railway or M station on the Wakaba Railway. K station is located south of S middle school and takes 12 minutes to walk from S middle school. In addition, M station is located north of S middle school and takes 14 minutes to walk from S middle school. At K station, trains depart every 8 minutes after 2 p.m., and at M station, trains depart every 5 minutes after 2 p.m. Answer the following questions.'

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'Compare the number of lights turned on at a certain time with the number of lights turned on one minute later. At what time does the number of lights turned on increase the most after one minute. List all possible times.'

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'Question 1 is 3 points each for a total of 4 questions; Question 2 is 9 points; Question 3 and 4 are 5 points each for 2 questions; Question 5 is 11 points; Question 6 and 7 are 3 points each for a total of 4 questions.'

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'For the sentence X and Y regarding the underlined part f in question 8, choose the correct combination of true or false from the options below.'

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'Arrange white and black stones in a row without having more than three stones of the same color in a row. The diagram on the right shows the possible arrangements using a total of 4 white and black stones.'

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'(5) The train from Makuhari station to Makuharihongo station travels 600m in 60 seconds, and the train from Makuharihongo station to Makuhari station travels from Figure 7 for 60 seconds, at a speed of 20 × 40 ÷ 2 + 20 × (60-40) = 400 + 400 = 800m. Therefore, at this time, the distance between the two trains is 2100 - (600+800) = 700m. As both trains are moving towards each other at 20 m/s, they will meet after 700 ÷ (20+20) = 17.5 seconds. Hence, the time taken for the two trains to meet after departing is 60 + 17.5 = 77.5 seconds.'

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'To a solution of B with pH 10.5, 5mL of A solution was added. How many milliliters are needed for exactly neutralization? Encircle the appropriate answer in the following square brackets.'

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'At 20°C, what is the volume of gas in 20 cm^3? Provide your answer to one decimal place for nitrogen and oxygen respectively.'

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'Solve the following problem regarding a sequence and a pattern.'

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"In the Chinese historical book 'Records of the Three Kingdoms,' it is written that in the early 3rd century, in 239 AD, Queen Himiko of Yamatai-koku sent envoys to Wei (China) and was bestowed with the title of 'Queen Who Is Loyal to Wei' and a bronze mirror by the emperor."

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'By doubling the amount of water in A midway, find the depth of the water at 32.5 minutes and the time at that moment.'

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'At room temperature, a small amount of the following substances were added to water and stirred with a glass rod. Select all substances that do not dissolve in water and write down their symbols.'

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'Which container, A or B, doubled in amount per minute? Please provide the time as well.'

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'2020 Shibuya Gakuen Makuhari Junior High School Mathematics 1st Exam\n1 (1) Use operations P and Q to manipulate cards from 1 to 144. What is the card to be placed in the box at the 42nd position?'

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'Among the gases produced in the underlined parts (1) to (6), there is only one type of gas that is different. Choose from (1) to (6) and also provide the name of the gas produced.'

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'From (3) more than (2), the distance between AD is known to be 4 × 6 = 24. Also, the distance between AC is 36 × 3/(3+2) = 21.6, so the distance between CD is 24-21.6 = 2.4. This corresponds to 120 m, so the distance of 1 is 120 ÷ 2.4 = 50 m, and the distance between AB is 50 × 36 = 1800 m, which is calculated to be 1800 ÷ 1000 = 1.8 km.'

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"Another example of a 'unit of assembly' is [meters per second], which is obtained by dividing distance by time. What physical quantity is measured in meters per second?"

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'In Basic Example 40, we are dealing with a sequence starting from the 0th term.'

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'Define a square S_n and a circle C_n (n=1,2,⋯⋯) as follows. C_n is inscribed in S_n, and S_{n+1} is inscribed in C_n. If the side length of S_1 is a, find the total circumference.'

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'Prove that the inequality √(ab) < (b-a)/(log b-log a) < (a+b)/2 holds when 0 < a < b.'

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'Let α, β be the two solutions of the equation x^2-2px-1=0, with |α|>1.'

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'If the inequality -4 ≤ x ≤ a holds, and the maximum value of y=√(9-4x)+b is 6, while the minimum value is 4. In this case, what are the values of a and b?'

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"Explain what is meant by the term 'initial term', provide its definition."

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'Solve the inequality $\\frac{ax+b}{2x+1}>x-2$ when the function $y=\\frac{ax+b}{2x+1}$ takes the value found in (1).'

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'Let the line with the smaller slope be denoted as \\ell, drawn from point (2,1) to the parabola y=\\frac{2}{3}x^{2}-1.'

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'Let a, b be natural numbers. Prove that if ab is a multiple of 3, then either a or b is a multiple of 3.'

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'(1) \ \\frac{4}{5}<x<4 \ (2) \ x \\leqq-2, \\quad 1 \\leqq x \ (3) \ 1<x<4 \'

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'When a=4, b=6, the largest integer x that does not satisfy inequality (1) is x= ◻.'

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'There are 2 ways to choose 3 numbers whose sum is a multiple of 3 from 0, 1, 2, 3, 4: [1] {0,1,2}, {0,2,4} [2] {1,2,3}, {2,3,4}. [1] Since the hundreds digit is not 0, there are 4 different 3-digit numbers for each group.'

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'Explain how to calculate the expected value of scores and find the expected value.'

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'Math I\nD = a^{2}-4 \\cdot 1 \\cdot\\left(-a^{2}+a-1\\right)=5 a^{2}-4 a+4 \\\n=5\\left(a-\\frac{2}{5}\\right)^{2}+\\frac{16}{5}>0\n\nTherefore, D>0 always holds.\n-3<-\x0crac{a}{2}<3 implies -6<a<6\nf(-3)=-a^{2}-2 a+8 f(-3)>0 implies\n a^{2}+2 a-8<0\nSolving which gives -4<a<2\nf(3)=-a^{2}+4 a+8 f(3)>0 implies\n a^{2}-4 a-8<0\nThe roots of a^{2}-4 a-8=0 are a=2 \\pm 2 \\sqrt{3}\nHence 2-2 \\sqrt{3}<x<2+2 \\sqrt{3}\n\n(a+4)(a-2)<0\nwhen a= -(-2)\\pm \\sqrt{(-2)^{2}-1 \\cdot(-8)}\n\nFind the common range of (1), (2), (3)\n2-\\sqrt{3}<a<2'

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'Prove that the product of three consecutive integers m-1, m, m+1, (m-1)m(m+1) is a multiple of 6. Similarly, (n-1)n(n+1) is a multiple of 6. Therefore, prove that m^3n - mn^3 is a multiple of 6.'

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'When using the numbers 1, 1, 1, 2, 2, 2, 3, 3 to form an 8-digit integer, how many integers can be formed?'

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'Find the number of two-digit natural numbers that satisfy the inequality 6x + 8(6 - x) > 7.'

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'Find all the integer solutions to the following system of equations.'

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'Basic Problem 39 Determining the Elements of a Set'

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'A and B work part-time together for 4 days a week. Show that there is at least one day each week when A and B work together.'

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'Find the smallest fraction that, when multiplied by 34/5, 51/10, and 85/8, results in a natural number product.'

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'Let a be a natural number. If a+5 is a multiple of 4 and a+3 is a multiple of 6, prove that a+9 is a multiple of 12.'

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'Let p, q, r be three consecutive odd numbers (p<q<r). Prove that pqr + pq + qr + rp + p + q + r + 1 is divisible by 48.'

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'Find the non-negative integer value of k for which the equation in x, k x^{2}-2(k+3) x+k+10=0, has real solutions.'