# Difference between revisions of "2002 USA TST Problems"

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=== Problem 4 === | === Problem 4 === | ||

+ | |||

+ | Let <math> \displaystyle n</math> be a positive integer and let <math> \displaystyle S</math> be a set of <math> \displaystyle 2^n+1</math> elements. Let <math> \displaystyle f </math> be a function from the set of two-element subsets of <math> \displaystyle S </math> to <math>\{0, \dots, 2^{n-1}-1\}</math>. Assume that for any elements <math> \displaystyle x, y, z</math> of <math> \displaystyle S </math>, one of <math> \displaystyle f(\{x,y\}), f(\{y,z\}), f(\{z, x\}) </math> is equal to the sum of the other two. Show that there exist <math> \displaystyle a, b, c </math> in <math> \displaystyle S </math> such that <math> \displaystyle f(\{a,b\}), f(\{b,c\}), f(\{c,a\}) </math> are all equal to 0. | ||

[[2002 USA TST Problems/Problem 4 | Solution]] | [[2002 USA TST Problems/Problem 4 | Solution]] | ||

=== Problem 5 === | === Problem 5 === | ||

+ | |||

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

+ | <center> | ||

+ | <math> | ||

+ | \frac{CE}{EM}, \quad \frac{EM}{MC}, \quad \frac{MC}{CE} | ||

+ | </math> | ||

+ | </center> | ||

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

[[2002 USA TST Problems/Problem 5 | Solution]] | [[2002 USA TST Problems/Problem 5 | Solution]] | ||

=== Problem 6 === | === Problem 6 === | ||

+ | |||

+ | Find in explicit form all ordered pairs of positive integers <math> \displaystyle (m, n)</math> such that <math> \displaystyle mn-1 </math> divides <math> \displaystyle m^2 + n^2 </math>. | ||

[[2002 USA TST Problems/Problem 6 | Solution]] | [[2002 USA TST Problems/Problem 6 | Solution]] |

## Revision as of 13:49, 4 April 2007

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 $\ang ACD = \ang BCD$ (Error compiling LaTeX. ! Undefined control sequence.). 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 .