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Usamo 2014: Day 1 - 29 April 2014

The document contains 6 problems from the 2014 USA Mathematical Olympiad. The problems cover a range of topics including real polynomials, functions on integers, geometry problems involving points and triangles, and number theory involving greatest common divisors.

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212 views2 pages

Usamo 2014: Day 1 - 29 April 2014

The document contains 6 problems from the 2014 USA Mathematical Olympiad. The problems cover a range of topics including real polynomials, functions on integers, geometry problems involving points and triangles, and number theory involving greatest common divisors.

Uploaded by

SeanGee
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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USA

USAMO
2014

Day 1 - 29 April 2014

1 Let a, b, c, d be real numbers such that b d 5 and all zeros x1 , x2 , x3 , and x4 of the
polynomial P (x) = x4 + ax3 + bx2 + cx + d are real. Find the smallest value the product
(x21 + 1)(x22 + 1)(x23 + 1)(x24 + 1) can take.

2 Let Z be the set of integers. Find all functions f : Z Z such that

f (x)2
xf (2f (y) x) + y 2 f (2x f (y)) = + f (yf (y))
x
for all x, y Z with x 6= 0.

3 Prove that there exists an infinite set of points

. . . , P3 , P2 , P1 , P0 , P1 , P2 , P3 , . . .

in the plane with the following property: For any three distinct integers a, b, and c, points
Pa , Pb , and Pc are collinear if and only if a + b + c = 2014.

This file was downloaded from the AoPS Math Olympiad Resources Page Page 1
http://www.artofproblemsolving.com/
USA
USAMO
2014

Day 2 - 30 April 2014

4 Let k be a positive integer. Two players A and B play a game on an infinite grid of regular
hexagons. Initially all the grid cells are empty. Then the players alternately take turns with
A moving first. In his move, A may choose two adjacent hexagons in the grid which are empty
and place a counter in both of them. In his move, B may choose any counter on the board
and remove it. If at any time there are k consecutive grid cells in a line all of which contain
a counter, A wins. Find the minimum value of k for which A cannot win in a finite number
of moves, or prove that no such minimum value exists.

5 Let ABC be a triangle with orthocenter H and let P be the second intersection of the
circumcircle of triangle AHC with the internal bisector of the angle BAC. Let X be the
circumcenter of triangle AP B and Y the orthocenter of triangle AP C. Prove that the length
of segment XY is equal to the circumradius of triangle ABC.

6 Prove that there is a constant c > 0 with the following property: If a, b, n are positive integers
such that gcd(a + i, b + j) > 1 for all i, j {0, 1, . . . n}, then
n
min{a, b} > cn n 2 .

The problems on this page are copyrighted by the Mathematical Association of


Americas American Mathematics Competitions.

This file was downloaded from the AoPS Math Olympiad Resources Page Page 2
http://www.artofproblemsolving.com/

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