# Difference between revisions of "2006 AIME I Problems"

(→Problem 9) |
(→Problem 10) |
||

Line 54: | Line 54: | ||

== Problem 10 == | == 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 accumilated 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 I Problems/Problem 10|Solution]] | [[2006 AIME I Problems/Problem 10|Solution]] |

## Revision as of 15:41, 25 September 2007

## Contents

## Problem 1

In convex hexagon , all six sides are congruent, and are right angles, and and are congruent. The area of the hexagonal region is Find .

## Problem 2

The lengths of the sides of a triangle with positive area are , , and , where is a positive integer. Find the number of possible values for .

## Problem 3

Let be the product of the first 100 positive odd integers. Find the largest integer such that is divisible by

## Problem 4

Let be a permutation of for which

An example of such a permutation is Find the number of such permutations.

## Problem 5

When rolling a certain unfair six-sided die with faces numbered 1, 2, 3, 4, 5, and 6, the probability of obtaining face is greater than 1/6, the probability of obtaining the face opposite is less than 1/6, the probability of obtaining any one of the other four faces is 1/6, and the sum of the numbers on opposite faces is 7. When two such dice are rolled, the probability of obtaining a sum of 7 is 47/288. Given that the probability of obtaining face is where and are relatively prime positive integers, find

## Problem 6

Square has sides of length 1. Points and are on and respectively, so that is equilateral. A square with vertex has sides that are parallel to those of and a vertex on The length of a side of this smaller square is where and are positive integers and is not divisible by the square of any prime. Find

## Problem 7

Find the number of ordered pairs of positive integers such that and neither nor has a zero digit.

## Problem 8

There is an unlimited supply of congruent equilateral triangles made of colored paper. Each triangle is a solid color with the same color on both sides of the paper. A large equilateral triangle is constructed from four of these paper triangles. Two large triangles are considered distinguishable if it is not possible to place one on the other, using translations, rotations, and/or reflections, so that their corresponding small triangles are of the same color.

Given that there are six different colors of triangles from which to choose, how many distinguishable large equilateral triangles may be formed?

## Problem 9

Circles and have their centers at (0,0), (12,0), and (24,0), and have radii 1, 2, and 4, respectively. Line is a common internal tangent to and and has a positive slope, and line is a common internal tangent to and and has a negative slope. Given that lines and intersect at and that where and are positive integers and is not divisible by the square of any prime, find

## 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 accumilated 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 11

A collection of 8 cubes consists of one cube with edge-length for each integer A tower is to be built using all 8 cubes according to the rules:

- Any cube may be the bottom cube in the tower.
- The cube immediately on top of a cube with edge-length must have edge-length at most

Let be the number of different towers than can be constructed. What is the remainder when is divided by 1000?

## Problem 12

Find the sum of the values of such that where is measured in degrees and

## Problem 13

For each even positive integer let denote the greatest power of 2 that divides For example, and For each positive integer let Find the greatest integer less than 1000 such that is a perfect square.

## Problem 14

A tripod has three legs each of length 5 feet. When the tripod is set up, the angle between any pair of legs is equal to the angle between any other pair, and the top of the tripod is 4 feet from the ground In setting up the tripod, the lower 1 foot of one leg breaks off. Let be the height in feet of the top of the tripod from the ground when the broken tripod is set up. Then can be written in the form where and are positive integers and is not divisible by the square of any prime. Find (The notation denotes the greatest integer that is less than or equal to )

## Problem 15

Given that a sequence satisfies and for all integers find the minimum possible value of