Difference between revisions of "2014 USAJMO Problems"
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[[2014 USAJMO Problems/Problem 4|Solution]] | [[2014 USAJMO Problems/Problem 4|Solution]] | ||
===Problem 5=== | ===Problem 5=== | ||
+ | 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 USAJMO Problems/Problem 5|Solution]] | [[2014 USAJMO Problems/Problem 5|Solution]] | ||
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===Problem 6=== | ===Problem 6=== | ||
[[2014 USAJMO Problems/Problem 6|Solution]] | [[2014 USAJMO Problems/Problem 6|Solution]] |
Revision as of 17:42, 30 April 2014
Contents
Day 1
Problem 1
Let ,
,
be real numbers greater than or equal to
. Prove that
Solution
Problem 2
Let be a non-equilateral, acute triangle with
, and let
and
denote the circumcenter and orthocenter of
, respectively.
(a) Prove that line intersects both segments
and
.
(b) Line intersects segments
and
at
and
, respectively. Denote by
and
the respective areas of triangle
and quadrilateral
. Determine the range of possible values for
.
Problem 3
Let be the set of integers. Find all functions
such that
for all
with
.
Day 2
Problem 4
Problem 5
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.