Difference between revisions of "2002 USA TST Problems"
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=== Problem 5 ===
=== Problem 5 ===
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> \ACD = \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
Latest revision as of 05:58, 3 August 2017
Problems from the 2002 USA TST.
Let be a triangle. Prove that
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 .
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) .
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.
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).
Find in explicit form all ordered pairs of positive integers such that divides .