Basic Number Theory - Integers, Fractions, Decimals | AI tutor The No.1 Homework Finishing Free App

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

'[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.'

'Proof problems related to multiples of 85'

'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}'

'Leading digit'

'Show the properties of real numbers.'

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

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

'From the above, the smallest value of k is'

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

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

''

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

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

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

'a = 1 or (a ≠ 1 and b = 4a² − 4a)'

'Consider a sequence of natural numbers {an}.'

'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]'

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

'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.'

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

'What is the first odd number in the nth group?'

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

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

'Translate the given text into multiple languages.'

'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.'

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

'Mathematics II'

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

'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.'

'Which term is the first to become negative?'

'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}.'

'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.'

'Gaussian symbol and summation of series, recurrence relation'

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

'Consider the following sequence.'

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

'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.'

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

'(2) \ n=87 \'

'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.'

'A math exercise B 60'

'Math problem'

"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."

'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'

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

'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'

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

'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.'

'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?'

'M ≥ 1 / 4'

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

'One of (a>0, a=0, a<0) is true.'

'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 \'

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

'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.'

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

'Comprehensive'

'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.'

'(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 $d$.'

'Integer problems and mathematical induction'

'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.'

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

'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.'

'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'

'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.'

'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.'

'(1) 2/3 (2) 30 (3) 16/3'

"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\\)"

'In sequence (1) 1, 2 (2) 1, 0 (3) 2, 1 (4) 1, 1'

'Translate the given text into multiple languages.'

'(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'

'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\\]'

'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.'

'(3) In order 7, 15, 14, 12'

'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}.'

'\\( \\frac{1}{9} n(5 n+13) \\pi \\)'

"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."

'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.'

'Mathematics II'

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

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

'(2) 0 when n is even; 15 when n is odd'

"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."

'(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 \'

'(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.'

'\\(\\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)}\\)'

'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.'

'36 -1'

'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'

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

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

'(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'

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

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

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

"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)."

'(1) 23 / 3 (2) 19 / 24 (3) -32 + 20√5 / 3'

'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 $x=\\frac{\\pi}{10}, \\frac{\\pi}{2}, \\frac{9}{10}\\pi$.'

'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)'

'Therefore, when n=k+1, (1) holds.'

'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.'

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

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

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

'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.'

'The result of 5√10 / 18 is 5√10 / 18'

'944 \\sqrt{2}-4'

'(3) (a, b)=(-1,-1),(1,-1),(-1,1), (1,1)'

'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'

"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."

'Translate the given text into multiple languages.'

'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.'

'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.'

'(1) 5 (2) 4 (3) 7/3'

'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}.'

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

'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.'

'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)'

'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\.'

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

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

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

'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'

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

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

'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.'

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

'Prove the Remainder Theorem.'

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

'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.'

'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'

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

'Since 1 and 3 are solutions'

'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}.'

'Proof of 3 Inequalities (2)'

'Page 394'

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

'Example 59 | Comparison of powers and roots'

'Find the sum of the following sequence.'

'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.'

'Prove that x_{p+1}=x_{1}.'

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

'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.'

'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.'

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

'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.'

'170 multiplied by 9 divided by 2'

'174 divided by 4 is equal to 27'

'−11 ≤ P ≤ 401 / 9'

'The maximum value of 180 is $9^{2 \times 0}$'

'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.'

'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'

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

'Properties used in proving inequalities'

'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.'

'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 \,'

'Find the 5th term of sequence (1).'

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

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

'36 (1) x=2, y=-1 (2) x=4/5, y=-7/5'

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

'Proof of Equation (2)...with conditions'

'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\\]'

'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 $ab=6$.'

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

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

'Seat allocation in the Math Door election'

'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.'

'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?'

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

"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}\"

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

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

'186 k=-4,0'

'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}.'

'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 .'

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

'39 (ア) -3'

'Find the minimum value of x + 16/x when x > 0.'

'(4) \ x= \\pm 1, \\pm \\sqrt{2} \'

'106606 digits, 2'

'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, ......'

'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.'

'52 (1) 11/3 (2) 2/3 (3) -1/3'

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

'\ 68 a<-2,2<a \'

'Solve the following equations.'

'(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'

'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}.'

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

'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.'

'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.'

'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 \,'

'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.'

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

'Fibonacci sequence'

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

'Prove that the following inequalities hold.'

'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.'

'Find two numbers that satisfy the following conditions.'

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

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

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

'item 21, and -903'

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

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

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

'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?'

'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.'

'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|.'

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

'Using common logarithmic values.'

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

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

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

'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'

"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."

'48 (A) 3 (B) 2 (C) 11 (I) 25'

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

'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.'

'Using mathematical induction to prove the following equation'

'(2) $\\frac{1}{x+\\frac{1}{x-\\frac{1}{x}}}$'

'(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, ...'

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

"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 ●."

'Which term becomes negative for the first time?'

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

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

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

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

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

'61 (1) \x=-1, \\frac{1 \\pm \\sqrt{3} i}{2}\'

'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.'

'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'

'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.'

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

'Find the following values.'

'Find the 4th number from the left on line 10.'

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

'Find the first term of the sequence (1).'

'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'

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

'Complex fraction calculation'

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

'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.'

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

'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=.'

'1338'

'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'

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

'(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.'

'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.'

'Find the sum of the following geometric progressions.'

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

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

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

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

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

'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.'

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

'Find the 100th term.'

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

'(4) -5'

'Basic 5: Three numbers forming an arithmetic progression'

'63 (1) 3'

'Fractional equation'

'How to rewrite 183-5<a<27'

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

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

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

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

'Translate the given text into multiple languages.'

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

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

"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."

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

'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.'

'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.'

'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.'

"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."

'Regarding Definition 5'

"(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."

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

'(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.'

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

"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."

'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?'

'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.'

'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).'

'(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).'

"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."

'(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.'

'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'

'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.'

'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?'

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