# Difference between revisions of "User:Rowechen"

(→Problem 8) |
(→Problem 13) |
||

Line 65: | Line 65: | ||

<cmath>P(k) = \frac{1}{k(k+1)}</cmath> | <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> | + | 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>. | prime positive integers. Determine <math>m + n</math>. | ||

## Revision as of 10:12, 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 , 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 , 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?