Difference between revisions of "2014 USAMO Problems"
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===Problem 3=== | ===Problem 3=== | ||
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+ | Prove that there exists an infinite set of points <cmath>\ldots,\,\,\,\,P_{-3},\,\,\,\,P_{-2},\,\,\,\,P_{-1},\,\,\,\,P_0,\,\,\,\,P_1,\,\,\,\,P_2,\,\,\,\,P_3,\,\,\,\,\ldots</cmath> in the plane with the following property: For any three distinct integers <math>a,b,</math> and <math>c</math>, points <math>P_a</math>, <math>P_b</math>, and <math>P_c</math> are collinear if and only if <math>a+b+c=2014</math>. | ||
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[[2014 USAMO Problems/Problem 3|Solution]] | [[2014 USAMO Problems/Problem 3|Solution]] | ||
Revision as of 18:43, 29 April 2014
Contents
Day 1
Problem 1
Let be real numbers such that
and all zeros
and
of the polynomial
are real. Find the smallest value the product
can take.
Problem 2
Let be the set of integers. Find all functions
such that
for all
with
.
Problem 3
Prove that there exists an infinite set of points in the plane with the following property: For any three distinct integers
and
, points
,
, and
are collinear if and only if
.