# Difference between revisions of "User:Rowechen"

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==Problem 3== | ==Problem 3== | ||

− | + | <math>x</math>, <math>y</math>, and <math>z</math> are positive integers. Let <math>N</math> denote the number of solutions of <math>2x + y + z = 2004</math>. Determine the remainder obtained when <math>N</math> is divided by <math>1000</math>. | |

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− | + | ==Problem 8== | |

− | ==Problem | + | Find the number of sets <math>\{a,b,c\}</math> of three distinct positive integers with the property that the product of <math>a,b,</math> and <math>c</math> is equal to the product of <math>11,21,31,41,51,</math> and <math>61</math>. |

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− | < | + | [[2016 AIME II Problems/Problem 8 | Solution]] |

+ | ==Problem 9== | ||

+ | A special deck of cards contains <math>49</math> cards, each labeled with a number from <math>1</math> to <math>7</math> and colored with one of seven colors. Each number-color combination appears on exactly one card. Sharon will select a set of eight cards from the deck at random. Given that she gets at least one card of each color and at least one card with each number, the probability that Sharon can discard one of her cards and <math>\textit{still}</math> have at least one card of each color and at least one card with each number is <math>\frac{p}{q}</math>, where <math>p</math> and <math>q</math> are relatively prime positive integers. Find <math>p+q</math>. | ||

− | + | [[2017 AIME II Problems/Problem 9 | Solution]] | |

+ | ==Problem 7== | ||

− | == | + | Triangle <math>ABC</math> has side lengths <math>AB = 12</math>, <math>BC = 25</math>, and <math>CA = 17</math>. Rectangle <math>PQRS</math> has vertex <math>P</math> on <math>\overline{AB}</math>, vertex <math>Q</math> on <math>\overline{AC}</math>, and vertices <math>R</math> and <math>S</math> on <math>\overline{BC}</math>. In terms of the side length <math>PQ = w</math>, the area of <math>PQRS</math> can be expressed as the quadratic polynomial |

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+ | <cmath>\text{Area}(PQRS) = \alpha w - \beta \cdot w^2.</cmath> | ||

− | + | Then the coefficient <math>\beta = \frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>. | |

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− | [[ | + | [[2015 AIME II Problems/Problem 7 | Solution]] |

==Problem 7== | ==Problem 7== | ||

+ | For integers <math>a</math> and <math>b</math> consider the complex number <cmath>\frac{\sqrt{ab+2016}}{ab+100}-\left(\frac{\sqrt{|a+b|}}{ab+100}\right)i.</cmath> Find the number of ordered pairs of integers <math>(a,b)</math> such that this complex number is a real number. | ||

− | + | [[2016 AIME I Problems/Problem 7 | Solution]] | |

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+ | ==Problem 8== | ||

+ | A single atom of Uranium-238 rests at the origin. Each second, the particle has a <math>1/4</math> chance of moving one unit in the negative x-direction and a <math>1/2</math> chance of moving in the positive x-direction. If the particle reaches <math>(−3, 0)</math>, it ignites fission that will consume the earth. If it reaches <math>(7, 0)</math>, it is harmlessly diffused. The probability that, eventually, the particle is safely contained can be expressed as <math>\frac{m}{n}</math> for some relatively prime positive integers <math>m</math> and <math>n</math>. | ||

+ | Determine the remainder obtained when <math>m + n</math> is divided by <math>1000</math>. | ||

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==Problem 10== | ==Problem 10== | ||

+ | <math>ABCDE</math> is a cyclic pentagon with <math>BC = CD = DE</math>. The diagonals <math>AC</math> and <math>BE</math> intersect at <math>M</math>. <math>N</math> is the foot of the altitude from <math>M</math> to <math>AB</math>. We have <math>MA = 25</math>, <math>MD = 113</math>, and <math>MN = 15</math>. The area of triangle <math>ABE</math> can be expressed as <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Determine the remainder obtained when <math>m + n</math> is divided by <math>1000</math>. | ||

− | + | ==Problem 12== | |

− | + | <math>ABCD</math> is a cyclic quadrilateral with <math>AB = 8</math>, <math>BC = 4</math>, <math>CD = 1</math>, and <math>DA = 7</math>. Let <math>O</math> and <math>P</math> denote the circumcenter and intersection of <math>AC</math> and <math>BD</math> respectively. The value of <math>OP^2</math> can be expressed as <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime, positive integers. Determine the remainder obtained when <math>m + n</math> is divided by <math>1000</math>. | |

==Problem 11== | ==Problem 11== | ||

+ | For integers <math>a,b,c</math> and <math>d,</math> let <math>f(x)=x^2+ax+b</math> and <math>g(x)=x^2+cx+d.</math> Find the number of ordered triples <math>(a,b,c)</math> of integers with absolute values not exceeding <math>10</math> for which there is an integer <math>d</math> such that <math>g(f(2))=g(f(4))=0.</math> | ||

− | + | [[2020 AIME I Problems/Problem 11 | Solution]] | |

+ | == Problem 10 == | ||

+ | Seven teams play a soccer tournament in which each team plays every other team exactly once. No ties occur, each team has a <math> 50\% </math> chance of winning each game it plays, and the outcomes of the games are independent. In each game, the winner is awarded a point and the loser gets 0 points. The total points are accumulated to decide the ranks of the teams. In the first game of the tournament, team <math> A </math> beats team <math> B. </math> The probability that team <math> A </math> finishes with more points than team <math> B </math> is <math> m/n, </math> where <math> m </math> and <math> n </math> are relatively prime positive integers. Find <math> m+n. </math> | ||

− | [[ | + | |

+ | [[2006 AIME II Problems/Problem 10|Solution]] | ||

==Problem 14== | ==Problem 14== | ||

− | + | Centered at each lattice point in the coordinate plane are a circle radius <math>\frac{1}{10}</math> and a square with sides of length <math>\frac{1}{5}</math> whose sides are parallel to the coordinate axes. The line segment from <math>(0,0)</math> to <math>(1001, 429)</math> intersects <math>m</math> of the squares and <math>n</math> of the circles. Find <math>m + n</math>. | |

− | [[ | + | [[2016 AIME I Problems/Problem 14 | Solution]] |

− | == Problem | + | ==Problem 15== |

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− | + | Circles <math>\omega_1</math> and <math>\omega_2</math> intersect at points <math>X</math> and <math>Y</math>. Line <math>\ell</math> is tangent to <math>\omega_1</math> and <math>\omega_2</math> at <math>A</math> and <math>B</math>, respectively, with line <math>AB</math> closer to point <math>X</math> than to <math>Y</math>. Circle <math>\omega</math> passes through <math>A</math> and <math>B</math> intersecting <math>\omega_1</math> again at <math>D \neq A</math> and intersecting <math>\omega_2</math> again at <math>C \neq B</math>. The three points <math>C</math>, <math>Y</math>, <math>D</math> are collinear, <math>XC = 67</math>, <math>XY = 47</math>, and <math>XD = 37</math>. Find <math>AB^2</math>. | |

− | + | [[2016 AIME I Problems/Problem 15 | Solution]] | |

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==Problem 15== | ==Problem 15== | ||

+ | <math>ABCD</math> is a convex quadrilateral in which <math>AB \parallel CD</math>. Let <math>U</math> denote the intersection of the extensions of <math>AD</math> and <math>BC</math>. <math>\Omega_1</math> is the circle tangent to line segment <math>BC</math> which also passes through <math>A</math> and <math>D</math>, and <math>\Omega_2</math> is the circle tangent to <math>AD</math> which passes through <math>B</math> and <math>C</math>. Call the points of tangency <math>M</math> and <math>S</math>. Let <math>O</math> and <math>P</math> be the points of intersection between <math>\Omega_1</math> and <math>\Omega_2</math>. | ||

+ | Finally, <math>MS</math> intersects <math>OP</math> at <math>V</math>. If <math>AB = 2</math>, <math>BC = 2005</math>, <math>CD = 4</math>, and <math>DA = 2004</math>, then the value of <math>UV^2</math> is some integer <math>n</math>. Determine the remainder obtained when <math>n</math> is divided by <math>1000</math>. | ||

− | + | ==Problem 13== | |

+ | <math>P(x)</math> is the polynomial of minimal degree that satisfies | ||

+ | <cmath>P(k) = \frac{1}{k(k+1)}</cmath> | ||

− | < | + | for <math>k = 1, 2, 3, . . . , 10</math>. The value of <math>P(11)</math> can be written as <math>−\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively |

+ | prime positive integers. Determine <math>m + n</math>. | ||

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==Problem 14== | ==Problem 14== | ||

− | + | <math>3</math> Elm trees, <math>4</math> Dogwood trees, and <math>5</math> Oak trees are to be planted in a line in front of a library such that | |

− | + | i) No two Elm trees are next to each other. | |

− | + | ii) No Dogwood tree is adjacent to an Oak tree. | |

− | + | iii) All of the trees are planted. | |

+ | How many ways can the trees be situated in this manner? |

## Revision as of 10:03, 1 June 2020

Here's the AIME compilation I will be doing:

## Contents

## Problem 3

, , and are positive integers. Let denote the number of solutions of . Determine the remainder obtained when is divided by .

## Problem 8

Find the number of sets of three distinct positive integers with the property that the product of and is equal to the product of and .

## Problem 9

A special deck of cards contains cards, each labeled with a number from to and colored with one of seven colors. Each number-color combination appears on exactly one card. Sharon will select a set of eight cards from the deck at random. Given that she gets at least one card of each color and at least one card with each number, the probability that Sharon can discard one of her cards and have at least one card of each color and at least one card with each number is , where and are relatively prime positive integers. Find .

## Problem 7

Triangle has side lengths , , and . Rectangle has vertex on , vertex on , and vertices and on . In terms of the side length , the area of can be expressed as the quadratic polynomial

Then the coefficient , where and are relatively prime positive integers. Find .

## Problem 7

For integers and consider the complex number Find the number of ordered pairs of integers such that this complex number is a real number.

## Problem 8

A single atom of Uranium-238 rests at the origin. Each second, the particle has a chance of moving one unit in the negative x-direction and a chance of moving in the positive x-direction. If the particle reaches $(−3, 0)$ (Error compiling LaTeX. ! Package inputenc Error: Unicode char \u8:− not set up for use with LaTeX.), it ignites fission that will consume the earth. If it reaches , it is harmlessly diffused. The probability that, eventually, the particle is safely contained can be expressed as for some relatively prime positive integers and . Determine the remainder obtained when is divided by .

## Problem 10

is a cyclic pentagon with . The diagonals and intersect at . is the foot of the altitude from to . We have , , and . The area of triangle can be expressed as , where and are relatively prime positive integers. Determine the remainder obtained when is divided by .

## Problem 12

is a cyclic quadrilateral with , , , and . Let and denote the circumcenter and intersection of and respectively. The value of can be expressed as , where and are relatively prime, positive integers. Determine the remainder obtained when is divided by .

## Problem 11

For integers and let and Find the number of ordered triples of integers with absolute values not exceeding for which there is an integer such that

## Problem 10

Seven teams play a soccer tournament in which each team plays every other team exactly once. No ties occur, each team has a chance of winning each game it plays, and the outcomes of the games are independent. In each game, the winner is awarded a point and the loser gets 0 points. The total points are accumulated to decide the ranks of the teams. In the first game of the tournament, team beats team The probability that team finishes with more points than team is where and are relatively prime positive integers. Find

## Problem 14

Centered at each lattice point in the coordinate plane are a circle radius and a square with sides of length whose sides are parallel to the coordinate axes. The line segment from to intersects of the squares and of the circles. Find .

## Problem 15

Circles and intersect at points and . Line is tangent to and at and , respectively, with line closer to point than to . Circle passes through and intersecting again at and intersecting again at . The three points , , are collinear, , , and . Find .

## Problem 15

is a convex quadrilateral in which . Let denote the intersection of the extensions of and . is the circle tangent to line segment which also passes through and , and is the circle tangent to which passes through and . Call the points of tangency and . Let and be the points of intersection between and . Finally, intersects at . If , , , and , then the value of is some integer . Determine the remainder obtained when is divided by .

## Problem 13

is the polynomial of minimal degree that satisfies

for . The value of can be written as $−\frac{m}{n}$ (Error compiling LaTeX. ! Package inputenc Error: Unicode char \u8:− not set up for use with LaTeX.), where and are relatively prime positive integers. Determine .

## Problem 14

Elm trees, Dogwood trees, and Oak trees are to be planted in a line in front of a library such that i) No two Elm trees are next to each other. ii) No Dogwood tree is adjacent to an Oak tree. iii) All of the trees are planted. How many ways can the trees be situated in this manner?