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## Geometry and Measurement

### Solid Geometry - Properties of Three-Dimensional Shapes (Polyhedra, Cylinders, Cones, Spheres)

#### Q.01

'Find the locus of the center of the circle C, which is tangent to and circumscribes the circle given by (2) and touches the x-axis.'

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"Consider a cylinder inscribed in a sphere with a radius of 2, and let its height be 2x. (1) Express the radius a of the cylinder's base in terms of x. (2) Express the volume V of the cylinder in terms of x. (3) Find the maximum value of V. [Hokkaido Institute of Technology]"

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'2019 Shibuya Education Academy Makuhari Middle School 1st Attempt (26) The observer is facing the front of the cliff. The observer faces southeast at Cliff A, southwest at Cliff B, and north at Cliff C.'

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'Why not measure the elongation of a metal rod with a ruler? Explain briefly in about 20 words.'

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'According to (1), the layers in the region are dipping from south to north, so in the east-west direction, the layers are almost horizontally stacked. Therefore, on cliff C facing south, each layer appears to be almost horizontal.'

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'Why are raised floors used to prevent dampness and flooding?'

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'Prove that the shape given by the polar equation \ r=\\frac{2}{2+\\cos \\theta} \ is the same as the shape given by the complex number equation \ |z|+\\left|z+\\frac{4}{3}\\right|=\\frac{8}{3} \ and sketch the outline of this shape.'

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'Find the equation of a sphere that satisfies the following conditions: (1) Passes through the points A(1,2,4) and B(-5,8,-2) which are the ends of the diameter. (2) Passes through the point (5,1,4) and touches three coordinate planes.'

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'There are two spherical surfaces $S_{1}:(x-1)^{2}+(y-2)^{2}+(z+3)^{2}=5$ and $S_{2}:(x-2)^{2}+y^{2}+(z+1)^{2}=8$. Let the intersection of spherical surfaces $S_{1}, S_{2}$ be the circle $C$. Find:\n(1) The coordinates of the center $P$ and the radius $r$ of the circle $C$\n(2) The equation of the plane $α$ containing the circle $C$'

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'Let the point on the circle C with radius 1 centered at the origin as \\( (\\cos \\theta, \\sin \\theta) \\) be P. Let the circle that is tangent to circle C at point P and also tangent to the y-axis be S, and let the coordinates of its center Q be (u, v). (1) Express u and v in terms of \ \\cos \\theta \ and \ \\sin \\theta \, respectively. (2) Let the area of circle S be denoted by \\( D(\\theta) \\). Determine \\( \\lim_{\\theta \\to \\frac{\\pi}{2}-0} \\frac{D(\\theta)}{(\\frac{\\pi}{2}-\\theta)^{2}} \\).'

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'In the coordinate space with point O as the origin, let A(5,4,-2).'

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'Consider a cube with vertices O(0,0,0), A(1,0,0), B(0,1,0), C(0,0,1), D(0,1,1), E(1,0,1), F(1,1,0), G(1,1,1) in the coordinate space. Let point P be the trisection point of edge OA in the ratio 3:1, point Q be the division point of edge CE in the ratio 1:2, and point R be the division point of edge BF in the ratio 1:3. The plane passing through points P, Q, R is denoted as α.'

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'Find the equation of the sphere passing through the points (1,1,1), (-1,1,-1), (-1,-1,0), (2,1,0). Also, determine the coordinates of its center and the radius.'

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'In tetrahedron ABCD, AB²+CD²=BC²+AD²=AC²+BD², and ∠ADB=90°. Let G be the centroid of triangle ABC.'

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'Considering a rectangle ABCD in space where the coordinates of point A are (5,0,0) and the coordinates of point D are (-5,0,0), with the length of side AB being 5. Furthermore, both the y-coordinate and z-coordinate of point B are positive, and the length of the perpendicular projection from point B to the xy plane is 3. Please find the coordinates of points B and C.'

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'Find the area through which the line segment connecting a point on the curve represented by the polar equation $r=2(1+\\cos \\theta)(0 \\leq \\theta \\leq \\frac{\\pi}{2})$ and the pole $\\mathrm{O}$ passes. Basics 182, Mathematics $\\mathrm{C}$ p. 303 Reference'

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'Find the equation of a spherical surface passing through the points (0,0,0), (6,0,0), (0,4,0), and (0,0,-8). Also, determine the coordinates of its center and the radius.'

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'Calculate the number of faces (f), edges (e), and vertices (v) of the polyhedron formed by cutting all the vertices of a regular icosahedron with a plane passing through the midpoints of each edge.'

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'Chapter 3 Properties of Figures EX ⊕ 91 The figure on the right [1] is a polyhedron obtained by cutting out 8 vertices with a plane passing through the midpoints of each edge of a regular hexahedron. Let this polyhedron be denoted as X. The figure on the right [2] is a polyhedron obtained by cutting out vertices with a plane passing through the midpoints of each edge of polyhedron X. Let this polyhedron be denoted as Y. (1) Determine the number of faces, edges, and vertices of polyhedron X, respectively. (2) Determine the number of faces, edges, and vertices of polyhedron Y, respectively.'

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'Find the length of the side of a regular tetrahedron inscribed in a sphere of radius 1.'

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'Find the length of a side of a regular tetrahedron inscribed in a sphere of radius 1.'

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'Example problem 141 of finding the minimum value of a broken line on a tetrahedron\nGiven a tetrahedron ABCD with AB=BC=CA=8, AD=7. When cos∠CAD=11/14, find the following:\n(1) The length of side CD\n(2) The size of ∠ACD\n(3) For point E on side AC, find the minimum value of BE+ED'

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"The figure on the right [1] is a polyhedron obtained by cutting out eight vertices of a regular hexahedron with a plane passing through the midpoints of each edge. Let's call this polyhedron X. The figure on the right [2] depicts the polyhedron obtained by cutting out the polyhedron [1], which we'll call Y."

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'In the cuboid in the diagram to the right, AD=AE=1, EF=√3. (1) Find the edge perpendicular to edge BF.'

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'Consider a right circular cone with a base radius of 2 and a height of \ \\sqrt{5} \. Let \ \\mathrm{O} \ be the vertex of this cone, and let \ \\mathrm{A}, \\mathrm{B} \ be the two ends of the diameter of the base. Also, let \ \\mathrm{P} \ be the midpoint of the line segment \ \\mathrm{OB} \. Find the shortest distance on the lateral surface of the cone from A to P.'

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"In the figure on the right, the line AB is tangent to circles O and O' at points A and B, respectively. If the radii of circles O and O' are 5 and 4, and the distance between centers O and O' is 6, find the length of segment AB."

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'Paint each face of a regular tetrahedron and a regular hexahedron with paint. Only one color is painted on each face. Also, rotating it 323 degrees to match the coloring is considered the same. When there are 12 colors, there are 7 ways to paint the faces of the regular tetrahedron such that each face has a different color. When there are 8 colors, there are 億 ways to paint the faces of the regular hexahedron such that each face has a different color.'

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'The diagram on the right shows a solid figure formed by cutting a rectangular prism on a plane containing the edges DH, BF.'

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'Each face of a cube must be painted in such a way that adjacent faces have different colors. However, rotations of the cube that result in the same coloring are considered the same. (1) How many ways are there to paint using all 6 different colors? (2) How many ways are there to use all 5 different colors?'

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'Please find the number of faces f, edges e, and vertices v of the polyhedron formed by cutting all the vertices of a regular dodecahedron with a plane passing through the midpoints of each edge.'

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'Use the condition that the sum of opposite angles is 180° to prove that a quadrilateral is inscribed in a circle.'

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'Find the number of faces (f), edges (e), and vertices (v) of the polyhedron obtained by cutting off all the corners of a dodecahedron through a plane passing through the midpoint of each edge.'

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"In the figure on the right, the line AB touches the circles O and O' at points A and B respectively. If the radii of circles O and O' are denoted as r and r' (r < r'), and the distance between the centers of the two circles is d, then prove that AB = √(d^2 - (r' - r)^2)."

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'Determine the number of faces f, edges e, and vertices v for the following convex polyhedra:\n(1) A convex polyhedra consisting of 12 regular pentagons and 20 regular hexagons\n(2) A convex polyhedra formed by cutting all the corners with a plane passing through points that trisect each edge of a regular tetrahedron as shown in the figure to the right'

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'Light emitted from flashlights and the like spreads in a conical shape, but when illuminated at the right angle, the edge of the illuminated area becomes a parabola. This phenomenon occurs because cutting a cone parallel to its generatrix results in a parabola at the cutting edge.'

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"Position of Points in Space\nJust like points in a plane are represented by a pair of real numbers, points in space can also be described using coordinates, which consist of a triple of real numbers. Let's consider a point C in space and define three mutually orthogonal number lines at point O as shown in the diagram. These are called the x-axis, y-axis, and z-axis respectively, collectively known as the coordinate axes. Additionally, the point O is called the origin.\nThe plane determined by the x-axis and y-axis is known as the xy-plane, the plane determined by the y-axis and z-axis is the yz-plane, and the plane determined by the z-axis and x-axis is the zx-plane.\nIn a coordinate plane, there are two axes, the x-axis (horizontal) and the y-axis (vertical), but in coordinate space, the z-axis (height) is added. These three axes together are referred to as the coordinate plane."

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'Please answer (A to C) that fits in the following table.'

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'By passing a plane through the midpoints of each edge of the regular dodecahedron and cutting all the vertices, determine the number of faces f, edges e, and vertices v of the resulting polyhedron with 21 faces.'

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"For a convex polyhedron with 8 vertices and 6 faces, let's find the number of edges."

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'Calculate the number of faces f, edges e, and vertices v of the convex polyhedron formed by cutting all the vertices of a regular octahedron with a plane passing through the points that trisect each edge.'

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'Choose the term from the following mathematical terms that corresponds to regular polyhedron.'

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'Prove that in a right square pyramid A-BCDE where all edges have equal length, when the midpoint of edge AD is denoted as M, edge AD is perpendicular to the plane MEC.'

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'In the pentagonal prism ABCDE-FGHIJ shown in the diagram, answer the following questions.'

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"In problems related to polyhedra, Euler's polyhedron theorem is used to clarify the relationship between vertices, edges, and faces."

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'The Sine Rule is a theorem that expresses the relationship between the sine of the three internal angles of a triangle and the lengths of its three sides. To prove this theorem, the theorem of the inscribed angles learned in junior high school is used.'

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'For a right circular cone inscribed in a sphere of radius 1, find the height, base radius, and lateral area to maximize.'

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'Imagine cutting a rolled paper tube diagonally. How do you think the edge will look when the paper is unfolded? Here, assume the radius of the base is 1, and the angle between the cut edge and the base is π/4 (=45 degrees).'

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'In the right pyramid O-ABCD, with the length of one side of the base being 2a and the height being a. Find the following:\n(1) The length of the perpendicular line AE drawn from vertex A to edge OB\n(2) For point E in (1), find the measure of angle AEC and the area of triangle AEC'

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'To facilitate the calculation of the area by dividing'

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'When all vertices of a regular icosahedron W with side length 1 are on the surface of a sphere S, answer the following questions. A regular icosahedron has all faces congruent equilateral triangles, with each vertex shared by 5 triangles.'

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'Consider a right triangular prism T with base triangle ABC where AB=2, AC=3, and BC=t(1<t<5). A right triangular prism is defined as a prism where all edges are perpendicular to the base. Furthermore, suppose a sphere S is contained inside T and is tangent to all faces of T.'

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'Given that the height of the right triangular prism is 4, let the radius of the sphere be r, then 0 < r ≤ 2'

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'As shown in the figure, three points D, E, and F are taken outside △ABC in such a way that △ABD, △BCE, and △CAF form equilateral triangles. Let S be the area of △ABC, and the lengths of the three sides be BC=a, CA=b, AB=c. Answer the following questions: (1) Let ∠BAC=θ, express sinθ in terms of b, c, and S, and express cosθ in terms of a, b, and c. (2) Express DC² in terms of a, b, c, and S. It is permissible to use the general identity cos(60°+θ)=\\frac{cosθ-√3 sinθ}{2}. (3) Let the average area of the three equilateral triangles be T, express DC² in terms of S and T.'

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'Find the length of one side of a regular tetrahedron inscribed in a sphere of radius 1.'

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'There is a right circular cone with a base radius of 2 and a height of √5. Let O be the vertex of this cone, A and B be the endpoints of the base diameter. If the midpoint of segment OB is P, what is the shortest distance from A to P on the lateral surface?'

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'Finding the minimum value of a broken line on tetrahedron Example 141\nThere is a tetrahedron ABCD with AB=BC=CA=8, AD=7. When cos∠CAD=11/14, find the following:\n(1) The length of edge CD\n(2) The size of ∠ACD\n(3) For a point E on edge AC, find the minimum value of BE+ED\nBasics 121,137'

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'Example 127 Measurement Problem (Space)\nAs shown in the diagram on the right, a utility pole stands perpendicular to a plane containing points A, B, and C. When viewed from points A and B, the top of the pole D has angles of elevation of 60° and 45° respectively. Given that the distance between A and B is 6m, and ∠ACB = 30°, find the height of the pole CD. Assume no considerations for eye height.'

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"Tetrahedron and Sphere\nLet's consider a tetrahedron ABCD with edge length a.\n(1) Express the radius R of the sphere circumscribed about the tetrahedron in terms of a.\n(2) Express the radius r of the sphere inscribed in the tetrahedron in terms of a."

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'Find the length of the edge of a regular tetrahedron inscribed in a sphere of radius 1.'

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'There is a right circular cone with a base radius of 2 and a height of √5. Let O be the vertex of this cone, A and B be the two ends of the diameter of the base, and P be the midpoint of segment OB. Find the shortest distance from A to P on the lateral surface.'

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'In space, a point is typically defined by its coordinates and its distance to the origin O.'

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'Find the equation of a spherical surface with center at point (a, b, c) and radius r.'

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'Translate the coordinates of the center and the radius in order'

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'Find the real number a, and when the point P moves over the entire sphere S,'

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'As shown in the diagram, let S, T, U be the points of intersection of the plane z=t (0<t<2/3) with the circumference of disk D and the line segment CQ.'

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'Practice: Given the plane $\\alpha: x-2 y+2 z+3=0$ and two spheres $S_{1}:(x-1)^{2}+(y-2)^{2}+(z+3)^{2}=5$, $S_{2}:(x-2)^{2}+y^{2}+(z+1)^{2}=8$. Find the following. (1) The equation of a sphere passing through the origin that includes the intersection of the plane $\\alpha$ and the sphere $S_{1}$ (2) The equation of a plane that contains the circle $C$ of intersection between the spheres $S_{1}$ and $S_{2}$, along with the coordinates of the center $P$ of circle $C$ and the radius $r$'

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'What is the shape of the triangle ABC formed by the three points A(4,7,2), B(2,3,-2), C(6,5,-6)?'

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'The perpendicular dropped from the center C(a, b, c) of the sphere to the xy-plane passes through the center (5/6, 5/6, 0) of the circle. Therefore, the x and y coordinates of points C and the center of the circle are both equal to a=5/6, b=5/6. Additionally, the radius of the sphere S is OC = √(〖(5/6)〗^2+〖(5/6)〗^2+c^2) = √(c^2 + 25/18). Thus, the equation of sphere S is (x-5/6)^2 + (y-5/6)^2 + (z-c)^2 = c^2 + 25/18. When the point (t+2, t+2, t) lies on sphere S, (t+2-5/6)^2 + (t+2-5/6)^2 + (t-c)^2 = c^2 + 25/18, which simplifies to 9t^2 - 2(3c-7)t + 4 = 0. The necessary and sufficient condition for the line l to have points in common with sphere S is that the quadratic equation for t (1) has real solutions. Therefore, let D be the discriminant, and solving for D≥0 leads to (3c-1)(3c-13)≥0. From this, it follows that c≤1/3 or c≥13/3. Hence, the conditions that a, b, c must satisfy are a=b=5/6 and (c≤1/3 or c≥13/3).'

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'Assuming a>0. Find the following for the sphere passing through points O(0,0,0), A(0, a, a), B(a, 0, a), C(a, a, 0) with equation 54.'

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'Here we have obtained another solution.\n\n$$\\overrightarrow{\\mathrm{AP}}=(x-1, y, z)$$\n\n$$\\overrightarrow{\\mathrm{BP}}+2\\overrightarrow{\\mathrm{CP}}=(x, y-2, z)+2(x, y, z-3)=(3x, 3y-2,3z-6)$$\n\nTherefore, $$\\overrightarrow{\\mathrm{AP}} \\cdot(\\overrightarrow{\\mathrm{BP}}+2 \\overrightarrow{\\mathrm{\\mathrm{CP}}})=0$$ leads to $$ (x-1) \\times 3x + y \\times (3y-2) + z \\times (3z-6)=0$$\n\nHence $$ \\quad x^{2}-x+y^{2}-\\frac{2}{3} y+z^{2}-2z=0 $$ Therefore $$ \\left(x-\\frac{1}{2}\\right)^{2}+\\left(y-\\frac{1}{3}\\right)^{2}+(z-1)^{2}=\\frac{1}{4}+\\frac{1}{9}+1 $$ that is $$ \\quad\\left(x-\\frac{1}{2}\\right)^{2}+\\left(y-\\frac{1}{3}\\right)^{2}+(z-1)^{2}=\\frac{49}{36} $$\n\nThus, the set of points P is a spherical surface with center at $$ \\left(\\frac{1}{2}, \\frac{1}{3}, 1\\right) $$ and radius $$ \\frac{7}{6} $$.'

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'Explain the properties of a parabola and deduce the properties of a point P on the parabola.'

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'(1) Let the center of sphere $S_{1}$ be $\\mathrm{O}_{1}(1,2,-3)$ with radius $r_{1}=\\sqrt{5}$. The distance between plane $\\alpha$ and the center $\\mathrm{O}_{1}$ of sphere $S_{1}$ is $\\frac{|1-2 \\cdot 2+2 \\cdot(-3)+3|}{\\sqrt{1^{2}+(-2)^{2}+2^{2}}}=2 \\times \\sqrt{5}>2$, thus the plane $\\alpha$ intersects with the sphere $S_{1}$. Therefore, the common points between plane $\\alpha$ and sphere $S_{1}$ satisfy the following equation:\n\\[4pt]\n\\($k(x-2 y+2 z+3)+(x-1)^{2}+(y-2)^{2}+(z+3)^{2}-5=0$\\)\nEquation (1) represents the sphere. Since it passes through the origin, substituting $x=y=z=0$, we get\n\\[4pt]\n\$3 k+1+4+9-5=0$\\nTherefore, $k=-3$, substituting $k=-3$ into (1), we get\n\\[4pt]\n\\($-3(x-2 y+2 z+3)+(x-1)^{2}+(y-2)^{2}+(z+3)^{2}-5=0$\\)\nSimplifying, the resulting equation is $x^{2}+y^{2}+z^{2}-5 x+2 y=0$'

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'A regular hexagon ABCDEF with side length 1 is given. When point P moves on edge AB and point Q moves on edge CD independently, find the area through which point R, dividing the segment PQ in the ratio 2:1, can pass.'

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'(2) Since the sphere is tangent to each coordinate plane and passes through the point (5, -1, 4), the radius is denoted as r, and the coordinates of the center are (r, -r, r). Therefore, the equation of the sphere is'

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'Find the equation of the plane passing through the point (-1,2,3) and perpendicular to the line given by (4)(x-2)=(y+1)(-3)=z-3'

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'Prove that the points where line segments AG_A, BG_B, CG_C, and DG_D are internally divided in the ratios 3:1 coincide for the tetrahedron ABCD with centroids G_A, G_B, G_C, and G_D for the triangles BCD, ACD, ABD, and ABC respectively.'

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'Let the equation of the sphere be $x^{2}+y^{2}+z^{2}+2x-4y+4z=16$, and the plane $\\alpha: 6x-2y+3z=5$ intersecting it forms a circle $C$. Find the coordinates of the center and the radius of this circle.'

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'Let the surface obtained by rotating the line segment AB connecting the points A(0,0,2) and B(1,0,1) around the z-axis be denoted by S. As the points P on S and Q on the xy-plane move such that PQ = 2, let K be the range that the mid-point M of the line segment PQ can pass through. Find the volume of K.'

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'Find the coordinates of the vertices of the rectangular prism OABC-DEFG on the right, except for point O.'

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'Consider a circle with a radius of 1 centered at the origin on the xy-plane in coordinate space. Let S be the cone (including its interior) with this circle as its base and the point (0,0,2) as its vertex. Also, consider the point A(1,0,2).'

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Join our Discord#### Q.90

'(2) Let point B pass through and let α be the plane perpendicular to the x-axis. On the plane α, with point C(1,0,0) as the center and radius CB = √(3²+4²) = 5 being on the circle, let R be a moving point on the circle. Then, CB = CR, QB = √(QC² + CB²), QR = √(QC² + CR²). Therefore, QB = QR, so D(1,0,-5). Then, AQ + QB = AQ + QD ≥ AD. Since points A, Q, and D are on the zx plane, the minimum value of AQ + QD occurs when Q is on the line AD. Therefore, the minimum value of AQ + QB is AD = √((1-2)² + (0-0)² + (-5-3)²) = √65.'

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'In the cuboid ABCD-EFGH, prove that the midpoints of edges FB, BC, CD, DH, HE, and EF are all in the same plane.'

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'Consider a triangle ABC with three vertices A, B, C on curve K: y=1/x. Prove that the orthocenter H of triangle ABC lies on curve K.'

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'Find the volume of the part in coordinate space where the distance to the x-axis, y-axis, and z-axis are all less than or equal to 1.'

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'There is a circle C with radius 5 centered at the origin O on the coordinate plane. Let n=2 or n=3, and consider a circle C_{n} with radius n that is tangent to circle C and rotates without sliding. There is a point P_{n} on circle C_{n}. Initially, the center O_{n} of circle C_{n} is at (5-n, 0), and point P_{n} is at (5,0), with the center of circle C_{n} rotating counterclockwise n times inside circle C to return to its original position. Let the point of tangency between circles C and C_{n} be S_{n}, and let the angle formed by segment OS_{n} with the positive direction of the x-axis be t.\n(1) Express the coordinates of point P_{n} in terms of t and n.\n(2) Show that the curves described by point P_{2} and point P_{3} are the same.\n[Osaka University]'

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'In coordinate space, find the volume of the part where the distance to the x-axis, y-axis, and z-axis are all less than or equal to 1.'

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Join our Discord#### Q.96

'In coordinate space, find the volume of the region where the distance to both the x-axis and y-axis is less than or equal to 1.'

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'Find the area of the region on the plane z = t (-1 ≤ t ≤ 1) where the distances to the x-axis and y-axis are both less than or equal to 1.'

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'Let the polar coordinates of point A be (10,0), and let Q be any point on the circle C with diameter formed by the line segment connecting pole O and point A. Draw a perpendicular from pole O to the tangent of circle C at point Q, let the polar coordinates of point P be (r,θ), find the polar equation of its trajectory. Here, 0 ≤ θ < π.'

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'Exercise 1: Prove that the sum 1/OP^2 + 1/OQ^2 is constant when drawing two perpendicular half-lines from the center O of the ellipse to the intersection points P and Q.'

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"Points O, A', B' are on the xy-plane, so the figure formed by the intersection of the spherical surface S and the xy-plane is a circle passing through O, A', B'."

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'Equation of a sphere - a sphere of radius r centered at point (a, b, c) (x-a)^{2}+(y-b)^{2}+(z-c)^{2}=r^{2}'

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'Let \\( \\mathrm{A}(0,2,0) \\) be a point and \\( \\vec{d}=(1,1,-2) \\) be parallel to the line \ \\ell \.\n(1) Find the coordinates of the intersection point of the line \ \\ell \ and the plane \ 2x-3y+z=0 \.\n(2) Find the length of the segment cut by the line \ \\ell \ on the sphere \\( (x-4)^{2}+(y-2)^{2}+(z+4)^{2}=14 \\).'

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"(2) When t ranges over all real numbers, let the line determined by the point (t+2, t+2, t) in the xyz space be denoted by l. Given that the sphere S with center at C(a, b, c) passes through the points O(0,0,0), A'(2,1,0), B'(1,2,0) and shares a point with the line l, find the conditions for a, b, c. [Hokkaido University]"

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'In space, the set of points that are a constant distance r from a fixed point C is called a sphere with center C and radius r.'

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'Explain how the intersection of a cone cut by a plane that does not pass through its vertex forms a second-order curve, and describe the cases for an ellipse, hyperbola, and parabola.'

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'In tetrahedron $ABCD$, let $G_A$, $G_B$, $G_C$, $G_D$ be the centroids of triangles $BCD$, $ACD$, $ABD$, $ABC$ respectively. Prove that the points where the segments $AG_A$, $BG_B$, $CG_C$, $DG_D$ divide in the ratio $3:1$ coincide.'

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'There are four points A(4,0,0), B(0,8,0), C(0,0,4), D(0,0,2).'

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'Find the coordinates of the intersection points of the line passing through points A(3,1,-1) and B(-2,-3,2) with the xy-plane, yz-plane, and zx-plane.'

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'(2) The plane ax + (9-a)y - 18z + 45 = 0 touches a spherical surface centered at (3, 2, 1) with radius √5. Find the value of the constant a.'

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'Given that the center is (1, -3, 2) and the sphere passing through the origin intersects with the plane z=k to form a circle with a radius of √5. Find the value of k.'

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'Learned about vectors in the plane, now learning the basics of vectors in space, and understanding the equations of shapes (lines, spheres, etc.) in space coordinates.'

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'Find the center coordinates and radius of the circle formed by the intersection of a spherical surface with center (-1,5,3) and radius 4 and the plane x=1.'

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'Let \\( \\mathrm{A}(0,3,0), \\mathrm{B}(0,-3,0) \\) be the endpoints of a diameter of a spherical surface \ S \ in the coordinate space. When the point \\( \\mathrm{P}(x, y, z) \\) moves on the surface \ S \, find the maximum value of \ 3x+4y+5z \. Also, determine the coordinates of P at that point.'

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'For the points A(2,-1,3), B(5,2,3), C(2,2,0), prove that: (1) The triangle with vertices A, B, C is an equilateral triangle. (2) If the three vertices of a regular tetrahedron are A, B, C, find the coordinates of the fourth vertex D.'

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'Find the coordinates of the intersection points between the line passing through points A(2,4,0) and B(0,-5,6) and the sphere with center at (0,2,0) and radius 2.'

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'Given that the center of the sphere is (1, -2, 3a) and the radius is √13, when this sphere intersects the xy-plane, it forms a circle with a radius of 2. Find the value of a. Also, determine the coordinates of the center of this circle.'

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'(1) Find the equations of the shapes formed by the intersection of a sphere with center (-1,3,2) and radius 5 with the xy-plane, yz-plane, and zx-plane.'

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'In the regular pentagon ABCDE with side length of 1, let AB be vector b and AE be vector e.'

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'In the complex plane, let points A and B represent -1+2i and 3+i respectively. If AB is one side of a square, find the complex numbers representing the vertices C and D of the square ABCD.'

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'Consider the 3 points O(0,0,0), A(2,0,1), B(0,1,2). Let point P(x,y,z) move in such a way that |PO|=|PA|=|PB|.'

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'(2) The sphere with center (1,-2,3a) and radius sqrt(13) intersects the xy-plane to form a circle with a radius of 2. Find the value of a. Also, find the coordinates of the center of this circle.'

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'Find the equations of the shapes formed by the sphere with center (-1, 3, 2) and radius 5 intersecting the xy-plane, yz-plane, and zx-plane.'

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'For points A(2,-1,3), B(5,2,3), C(2,2,0): (1) Prove that triangle ABC is an equilateral triangle. (2) If A, B, and C are the three vertices of a regular tetrahedron, find the coordinates of the fourth vertex D.'

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'Let a>0. Find the following for the sphere passing through points O(0,0,0), A(0,a,a), B(a,0,a), C(a,a,0): (1) Coordinates of the center and radius (2) Equation of the intersection with the zx plane'

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'Given that a sphere with center (2, -3, 4) and radius r intersects the xy-plane to form a circle with a radius of 3. Determine the value of r.'

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'Next, consider a regular octahedron and a sphere tangent to all of its faces, and imagine a cross-section cut through a plane containing the points of contact, as shown in the diagram on the right [2]. If the radius of the sphere is r, then the area of the right-angled triangle in the mesh section is'

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'Explain how to find the height of a tetrahedron.'

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'Assuming the existence of a cube as shown in the right figure, the path from A to D is a permutation of 3 to the right, 2 up, and 1 up,'

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'Prove the following for the tetrahedron ABCD:\n1. Let M be the midpoint of edge AB.\n(A) Edge AB is perpendicular to plane CDM.\n(T) Edge AB is perpendicular to edge CD.\n2. Let the midpoints of edges BC, AC, AD, and BD be P, Q, R, S respectively, then the quadrilateral PQRS is a square.'

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'Determine the number of faces f, edges e, and vertices v of the polyhedron formed by cutting all the vertices of a regular icosahedron with a plane passing through the midpoints of each edge.'

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'Calculate the minimum number of colors required for a regular hexahedron.'

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'Please calculate the shortest distance in the unfolded diagram.'

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'Prove: A is the Foot of Perpendicular of Tetrahedron.'

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'Find the number of faces f, edges e, and vertices v of the polyhedron formed by cutting all corners through a plane passing through the midpoints of each edge of a regular dodecahedron.'

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'By the converse of the secant theorem, DA is a tangent to the circle passing through points A, E, and F.'

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'Consider the tetrahedron ABCD in space. Prove that there exists a spherical surface that passes through all 4 vertices A, B, C, D.'

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'One wants to color each face of a cube such that neighboring faces have different colors. However, rotations of the cube that result in the same coloring are considered the same.'

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Find the equation of the sphere with its center at the origin and radius r.

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Let a be a real number. In the xyz space, consider the four points A(0, a, 4), B(-2, 0, 3), C(1, 0, 2), and D(0, 2, 3), and place a light source at point P(1, 0, 6).
(1) The coordinates of the shadow of point A on the xy plane created by the light source are (アイ, ウ a, 0).
(2) The shadow of the triangle BCD on the xy plane created by the light source is also a triangle. The coordinates of the vertices of this triangle are 力 > ク.

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15 (2) \( (x-1)^{2}+(y-1)^{2}+(z-1)^{2}=1 \), \( (x-3)^{2}+(y-3)^{2}+(z-3)^{2}=9 \)

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Intersection of a sphere and planes
The intersection of the sphere \( (x+1)^{2}+(y-4)^{2}+(z-2)^{2}=3^{2} \) with the following planes is a circle. Find its center coordinates and radius.
(1) $x y$ plane
(2) $y z$ plane
(3) plane $y=4$

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A sphere centered at the origin with radius r
$x^{2}+y^{2}+z^{2}=r^{2}$

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(1) Find the coordinates of the center and the radius of the sphere $x^{2}+y^{2}+z^{2}-4 x-4 y-2 z+5=0$.
(2) Find the equation of the sphere passing through the points \( (2,0,0),(0,2,0),(0,0,2),(2,2,2) \).

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Find the equation of a sphere centered at the point (a, b, c) with radius r
\(\ (x-a)^{2}+(y-b)^{2}+(z-c)^{2}=r^{2}\)

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Equation of the sphere
Find the equations of the following spheres.
(1) A sphere centered at the point \( (3,-2,1) \) with radius 2
(2) A sphere centered at the origin that passes through the point \( (2,1,-3) \)
(3) A sphere with endpoints of the diameter at points \( \mathrm{A}(5,3,-2) \) and \( \mathrm{B}(-1,3,2) \)

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The intersection of the sphere \( (x-2)^{2}+(y+3)^{2}+(z-5)^{2}=10 \) with the following planes is a circle. Find the coordinates of the center and the radius of the circle.
(1) yz-plane
(2) zx-plane
(3) plane $z=3$

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Find the equations of the spheres as follows.
(1) A sphere centered at the origin with radius $2 \sqrt{2}$
(2) A sphere centered at point A(6,5,-3) passing through point B(2,4,-3)
(3) A sphere with endpoints A(-1,4,9) and B(7,0,1) of its diameter

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Shapes in coordinate spaces: Intersection of a sphere and a plane

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Drop a perpendicular $\mathrm{OH}$ from the origin $\mathrm{O}$ to the plane $\mathrm{ABC}$ determined by the three points \( \mathrm{A}(2,0,0), \mathrm{B}(0,1,0), \mathrm{C}(0,0,-2) \). Find the coordinates of point $\mathrm{H}$ and the length of segment $\mathrm{OH}$.