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 |
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<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
.