Difference between revisions of "2014 USAMO Problems"
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===Problem 4=== | ===Problem 4=== | ||
+ | Let <math>k</math> be a positive integer. Two players <math>A</math> and <math>B</math> play a game on an infinite grid of regular hexagons. Initially all the grid cells are empty. Then the players alternately take turns with <math>A</math> moving first. In his move, <math>A</math> may choose two adjacent hexagons in the grid which are empty and place a counter in both of them. In his move, <math>B</math> may choose any counter on the board and remove it. If at any time there are <math>k</math> consecutive grid cells in a line all of which contain a counter, <math>A</math> wins. Find the minimum value of <math>k</math> for which <math>A</math> cannot win in a finite number of moves, or prove that no such minimum value exists. | ||
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[[2014 USAMO Problems/Problem 4|Solution]] | [[2014 USAMO Problems/Problem 4|Solution]] | ||
===Problem 5=== | ===Problem 5=== |
Revision as of 17:44, 30 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
.
Day 2
Problem 4
Let be a positive integer. Two players
and
play a game on an infinite grid of regular hexagons. Initially all the grid cells are empty. Then the players alternately take turns with
moving first. In his move,
may choose two adjacent hexagons in the grid which are empty and place a counter in both of them. In his move,
may choose any counter on the board and remove it. If at any time there are
consecutive grid cells in a line all of which contain a counter,
wins. Find the minimum value of
for which
cannot win in a finite number of moves, or prove that no such minimum value exists.