# Mock Geometry AIME 2011 Problems

## Contents

## Problem 1

Let be a unit square, and let be its image after a degree rotation about point The area of the region consisting of all points inside at least one of and can be expressed in the form where are positive integers, and shares no perfect square common factor with . Find

## Problem 2

Eleven nonparallel lines lie on a plane, and their pairwise intersections meet at angles of integer degree. How many possible values are there for the smallest of these angles?

## Problem 3

In triangle Points and are located on such that The circumcircle of cuts at respectively. If then the ratio can be expressed in the form where are relatively prime positive integers. Find

## Problem 4

In triangle Let and be points on such that is equilateral. The perimeter of can be expressed in the form where are relatively prime positive integers. Find

## Problem 5

In triangle The bisector of angle meet at and the circumcircle at different from . Calculate the value of

## Problem 6

Three points are chosen at random on a circle. The probability that there exists a point inside an equilateral triangle such that can be expressed in the form where are relatively prime positive integers. Find

## Problem 7

In trapezoid and There is a point on side such that the circumcircle of triangle is tangent to If can be expressed in the form where are positive integers and are relatively prime. Find

## Problem 8

Two circles have center and radius respectively. The smallest distance between a point on with a point on is . Tangents from to meet at and tangents from to meet at such that are on the same side of line meets at and meets at Q. The length of can be expressed in the form where are relatively prime positive integers. Find

## Problem 9

is a right pyramid with square base edge length 6, and The probability that a randomly selected point inside the pyramid is at least units away from each face can be expressed in the form where are relatively prime positive integers. Find

## Problem 10

Circle is defined by the equation where is a positive real number. Circle passes through the center of and its center lies on the line Suppose that one of the tangent lines from the origin to circles and meets and at respectively, that where is the origin, and that the radius of is . What is ?

## Problem 11

is on a semicircle with diameter and center Circle radius is tangent to and arc and circle radius is tangent to and arc . It is known that . The ratio can be expressed where are relatively prime positive integers. Find

## Problem 12

A triangle has the property that its sides form an arithmetic progression, and that the angle opposite the longest side is three times the angle opposite the shortest side. The ratio of the longest side to the shortest side can be expressed as , where are positive integers, is not divisible by the square of any prime, and and are relatively prime. Find .

## Problem 13

In acute triangle is the bisector of . is the midpoint of . a line through parallel to meets at respectively. Given that the sum of all possible values of can be expressed as where are positive integers. What is ?

## Problem 14

The point is a focus of a non-degenerate ellipse tangent to the positive and axes. The locus of the center of the ellipse lies along graph of, where are positive integers with no common factor other than . Find

## Problem 15

Two circles radius respectively intersect at . is on and on such that are collinear. Tangents to at respectively meet at . Suppose The length of can be expressed in the form where are positive integers and is not divisible by the square of any prime. Find