# Mock AIME I 2015 Problems

## Contents

## Problem 1

David, Justin, Richard, and Palmer are demonstrating a "math magic" concept in front of an audience. There are four boxes, labeled A, B, C, and D, and each one contains a different number. First, David pulls out the numbers in boxes A and B and reports that their product is . Justin then claims that the product of the numbers in boxes B and C is , and Richard states the product of the numbers in boxes C and D to be . Finally, Palmer announces the product of the numbers in boxes D and A. If is the number that Palmer says, what is ?

## Problem 2

Suppose that and are real numbers such that and . The value of can be expressed in the form where and are positive relatively prime integers. Find .

## Problem 3

Let be points in the plane such that , , and . Semicircles with diameters and intersect at a point with . Find the length of line segment .

## Problem 4

At the AoPS Carnival, there is a "Weighted Dice" game show. This game features two identical looking weighted 6 sided dice. For each integer , Die A has probability of rolling the number , while Die B has a probability of rolling . During one session, the host randomly chooses a die, rolls it twice, and announces that the sum of the numbers on the two rolls is . Let be the probability that the die chosen was Die A. When is written as a fraction in lowest terms, find the sum of the numerator and denominator.

## Problem 5

In an urn there are a certain number (at least two) of black marbles and a certain number of white marbles. Steven blindfolds himself and chooses two marbles from the urn at random. Suppose the probability that the two marbles are of opposite color is . Let be the smallest possible values for the total number of marbles in the urn. Compute the remainder when is divided by .

## Problem 6

Find the number of digit numbers using only the digits such that every pair of adjacent digits is no more than apart. For instance, and are acceptable numbers, while and are not.

## Problem 7

For all points in the coordinate plane, let denote the reflection of across the line . For example, if , then . Define a function such that for all points , denotes the area of the triangle with vertices , , and . Determine the number of lattice points in the first quadrant such that .

## Problem 8

Let be consecutive terms (in that order) in an arithmetic sequence with common difference . Suppose and are roots of a monic quadratic with . Then for positive relatively prime integers and . Find the remainder when is divided by .

## Problem 9

Compute the number of positive integer triplets with that satisfy the following properties:

(a) is a perfect square,

(b) is a power of ,

(c) is a multiple of .

## Problem 10

Let be a function defined along the rational numbers such that for all relatively prime positive integers and . The product of all rational numbers such that can be written in the form for positive relatively prime integers and . Find .

## Problem 11

Suppose , , and are complex numbers that satisfy the system of equations If for positive relatively prime integers and , find .

## Problem 12

Alpha and Beta play a game on the number line below.

Both players start at . Each turn, Alpha has an equal chance of moving unit in either the positive or negative directions while Beta has a chance of moving unit in the positive direction and a chance of moving unit in the negative direction. The two alternate turns with Alpha going first. If a player reaches at any point in the game, he wins; however, if a player reaches , he loses and the other player wins. If is the probability that Alpha beats Beta, where and are relatively prime positive integers, find .

## Problem 13

Let be a hexagon inscribed inside a circle of radius . Furthermore, for each positive integer let be the midpoint of the segment , where . Suppose that hexagon can also be inscribed inside a circle. If and , then can be written in the form where and are positive relatively prime integers. Find .

## Problem 14

Consider a set of pennies laid out in the formation of an equilateral triangle with "side length" . You wish to move some of the pennies so that the triangle is flipped upside down. For example, with , you could take the top penny and move it to the bottom to accomplish this task, as shown:

Let be the minimum number of pennies for which this can be done in terms of . Find .

## Problem 15

Let be a triangle with , , and . Let denote its circumcenter and its orthocenter. The circumcircle of intersects and at and respectively. Suppose where and are positive relatively prime integers. Find .