# Mock AIME I 2012 Problems

## Contents

## Problem 1

A circle of maximal area is inscribed in the region bounded by the graph of and the -axis. The radius of this circle is , where , , and are integers and and are relatively prime. What is ?

## Problem 2

A permutation of the numbers , , , , is called *golden* if every two consecutive numbers in the permutation are relatively prime. How many *golden* permutations are there?

## Problem 3

Triangle has , , . The trisection points of are and , with . Segments and are extended to points and such that and . The area of pentagon is , where and are relatively prime positive integers. Find .

## Problem 4

Consider the polynomial . Let and . The product of the roots of can be expressed in the form where and are relatively prime positive integers. Find the remainder when is divided by .

## Problem 5

Define a triple inequality to be an inequality of the form . Bradley has a list of 50 real numbers , but he will only give Batra triple inequalities consisting of 3 numbers in the list. For example, if Bradley gives Batra only the triple inequality , Batra knows that the 7th number is greater than the 2nd number which is greater than the 49th number, but nothing else. Let be the maximum number of (distinct) triple inequalities that Bradley could give Batra such that Batra cannot determine the order of the full list of 50 numbers. Find the remainder when is divided by 1000.

## Problem 6

Eli has 5 strings. He begins by taking one end of one of the strings, and tying it to another end of a string (possibly the same string) chosen at random. He continues on by tying two randomly selected untied ends together until there are no more untied ends, and only loops remain. The expected value of the number of loops that he will have in the end can be expressed as where and are relatively prime positive integers. Find .

## Problem 7

Let two circles with radii in the plane be centered at points , respectively. Consider a point in the plane such that . Denote the intersections of the line with as , and the intersections of the line with as . Let and intersect at points such that . If equals the area of , can be written in the form where are distinct squarefree integers. Find .

## Problem 8

Suppose that the complex number satisfies . If is the maximum possible value of , can be expressed in the form . Find .

## Problem 9

A class of seven students is doing a Secret Santa, for which all seven students have contributed a gift each. However, one of the seven students has not arrived yet. The teacher decides to randomly assign each of the six students present to a gift that is not their own. If is the probability that the seventh student is left with his own gift once he arrives, find .

## Problem 10

Consider the function . Find the sum of all for which , where is measured in degrees and .

## Problem 11

Triangle has sides of length 13, 14, and 15. Set is the set of all points for which the circle of radius centered at intersects exactly times. The points in enclose a finite region whose area can be expressed in the form , where , , and are positive integers such that and are relatively prime. Find .

## Problem 12

Let be a polynomial of degree 10 satisfying . Find the maximum possible sum of the coefficients of .

## Problem 13

Eduardo and Silvie play the Factor Game. Eduardo initially picks a number ; Silvie then subtracts a factor of from to get . Eduardo then subtracts a factor of from to get and so on. Neither player is allowed to subtract 1 at any time. The loser is the person who must subtract a number from itself and get zero. Let be the set of all positive integers less than 1000 which Eduardo can choose as the initial number to ensure that he wins (assuming optimal play). Find the remainder when the sum of the elements of is divided by 1000.

## Problem 14

Let be the set of complex numbers of the form such that for some integers and . Find the largest integer that must divide for all numbers in .

## Problem 15

Paula the Painter initially paints every complex number black. When Paula toggles a complex number, she paints it white if it was previously black, and black if it was previously white. For each , Paula progressively toggles the roots of . Let be the number of complex numbers are white at the end of this process. Find .