# Difference between revisions of "2002 USA TST Problems"

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− | Consider the family of nonisoceles triangles <math>ABC</math> satisfying the property <math> | + | Consider the family of nonisoceles triangles <math>ABC</math> satisfying the property <math>AC^2 + BC^2 = 2 AB^2 </math>. Points <math>M</math> and <math>D </math> lie on side <math>AB </math> such that <math>AM = BM </math> and <math> \angle ACD = \angle BCD </math>. Point <math>E</math> is in the plane such that <math>D </math> is the incenter of triangle <math>CEM </math>. Prove that exactly one of the ratios |

<center> | <center> | ||

<math> | <math> |

## Latest revision as of 06:58, 3 August 2017

Problems from the 2002 USA TST.

## Contents

## Day 1

### Problem 1

Let be a triangle. Prove that

### Problem 2

Let be a prime number greater than 5. For any integer , define

.

Prove that for all positive integers and the numerator of , when written in lowest terms, is divisible by .

### Problem 3

Let be an integer greater than 2, and distinct points in the plane. Let denote the union of all segments . Determine if it is always possible to find points and in such that (segment can lie on line ) and , where (1) ; (2) .

## Day 2

### Problem 4

Let be a positive integer and let be a set of elements. Let be a function from the set of two-element subsets of to . Assume that for any elements of , one of is equal to the sum of the other two. Show that there exist in such that are all equal to 0.

### Problem 5

Consider the family of nonisoceles triangles satisfying the property . Points and lie on side such that and . Point is in the plane such that is the incenter of triangle . Prove that exactly one of the ratios

is constant (i.e., it is the same for all triangles in the family).

### Problem 6

Find in explicit form all ordered pairs of positive integers such that divides .