Difference between revisions of "AoPS Wiki:Sandbox"
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− | + | = AIME 2000 II = | |
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== Problem 13 == | == Problem 13 == | ||
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Find the least positive integer <math>n</math> such that <center><math>\frac 1{\sin 45^\circ\sin 46^\circ}+\frac 1{\sin 47^\circ\sin 48^\circ}+\cdots+\frac 1{\sin 133^\circ\sin 134^\circ}=\frac 1{\sin n^\circ}.</math></center> | Find the least positive integer <math>n</math> such that <center><math>\frac 1{\sin 45^\circ\sin 46^\circ}+\frac 1{\sin 47^\circ\sin 48^\circ}+\cdots+\frac 1{\sin 133^\circ\sin 134^\circ}=\frac 1{\sin n^\circ}.</math></center> | ||
− | + | = AIME 2001 II = | |
== Problem 13 == | == Problem 13 == | ||
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Let <math>EFGH</math>, <math>EFDC</math>, and <math>EHBC</math> be three adjacent square faces of a cube, for which <math>EC = 8</math>, and let <math>A</math> be the eighth vertex of the cube. Let <math>I</math>, <math>J</math>, and <math>K</math>, be the points on <math>\overline{EF}</math>, <math>\overline{EH}</math>, and <math>\overline{EC}</math>, respectively, so that <math>EI = EJ = EK = 2</math>. A solid <math>S</math> is obtained by drilling a tunnel through the cube. The sides of the tunnel are planes parallel to <math>\overline{AE}</math>, and containing the edges, <math>\overline{IJ}</math>, <math>\overline{JK}</math>, and <math>\overline{KI}</math>. The surface area of <math>S</math>, including the walls of the tunnel, is <math>m + n\sqrt {p}</math>, where <math>m</math>, <math>n</math>, and <math>p</math> are positive integers and <math>p</math> is not divisible by the square of any prime. Find <math>m + n + p</math>. | Let <math>EFGH</math>, <math>EFDC</math>, and <math>EHBC</math> be three adjacent square faces of a cube, for which <math>EC = 8</math>, and let <math>A</math> be the eighth vertex of the cube. Let <math>I</math>, <math>J</math>, and <math>K</math>, be the points on <math>\overline{EF}</math>, <math>\overline{EH}</math>, and <math>\overline{EC}</math>, respectively, so that <math>EI = EJ = EK = 2</math>. A solid <math>S</math> is obtained by drilling a tunnel through the cube. The sides of the tunnel are planes parallel to <math>\overline{AE}</math>, and containing the edges, <math>\overline{IJ}</math>, <math>\overline{JK}</math>, and <math>\overline{KI}</math>. The surface area of <math>S</math>, including the walls of the tunnel, is <math>m + n\sqrt {p}</math>, where <math>m</math>, <math>n</math>, and <math>p</math> are positive integers and <math>p</math> is not divisible by the square of any prime. Find <math>m + n + p</math>. | ||
− | + | = AIME 2002 II = | |
== Problem 13 == | == Problem 13 == |
Revision as of 08:27, 12 April 2010
Contents
[hide]AIME 2000 II
Problem 13
The equation has exactly two real roots, one of which is , where , and are integers, and are relatively prime, and . Find .
Problem 14
Every positive integer has a unique factorial base expansion , meaning that , where each is an integer, , and . Given that is the factorial base expansion of , find the value of .
Problem 15
Find the least positive integer such that
AIME 2001 II
Problem 13
In quadrilateral , and , , , and . The length may be written in the form , where and are relatively prime positive integers. Find .
Problem 14
There are complex numbers that satisfy both and . These numbers have the form , where and angles are measured in degrees. Find the value of .
Problem 15
Let , , and be three adjacent square faces of a cube, for which , and let be the eighth vertex of the cube. Let , , and , be the points on , , and , respectively, so that . A solid is obtained by drilling a tunnel through the cube. The sides of the tunnel are planes parallel to , and containing the edges, , , and . The surface area of , including the walls of the tunnel, is , where , , and are positive integers and is not divisible by the square of any prime. Find .
AIME 2002 II
Problem 13
In triangle , point is on with and , point is on with and , , and and intersect at . Points and lie on so that is parallel to and is parallel to . It is given that the ratio of the area of triangle to the area of triangle is , where and are relatively prime positive integers. Find .
Problem 14
The perimeter of triangle is , and the angle is a right angle. A circle of radius with center on is drawn so that it is tangent to and . Given that where and are relatively prime positive integers, find .
Problem 15
Circles and intersect at two points, one of which is , and the product of the radii is . The x-axis and the line , where , are tangent to both circles. It is given that can be written in the form , where , , and are positive integers, is not divisible by the square of any prime, and and are relatively prime. Find .