Difference between revisions of "2014 USAJMO Problems"
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===Problem 4=== | ===Problem 4=== | ||
| + | Let <math>b\geq 2</math> be an integer, and let <math>s_b(n)</math> denote the sum of the digits of <math>n</math> when it is written in base <math>b</math>. Show that there are infinitely many positive integers that cannot be represented in the form <math>n+s_b(n)</math>, where <math>n</math> is a positive integer. | ||
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[[2014 USAJMO Problems/Problem 4|Solution]] | [[2014 USAJMO Problems/Problem 4|Solution]] | ||
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===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. | 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. | ||
Revision as of 16:42, 30 April 2014
Contents
Day 1
Problem 1
Let 
, 
, 
be real numbers greater than or equal to 
. Prove that ![\[\min{\left (\frac{10a^2-5a+1}{b^2-5b+1},\frac{10b^2-5b+1}{c^2-5c+10},\frac{10c^2-5c+1}{a^2-5a+10}\right )}\leq abc.\]](http://latex.artofproblemsolving.com/3/5/3/353bdb612bb60b2ed53c4b11766e658f325cd2f7.png)
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 ![\[xf(2f(y)-x)+y^2f(2x-f(y))=\frac{f(x)^2}{x}+f(yf(y))\]](http://latex.artofproblemsolving.com/9/f/d/9fdb11dbf102dfe0783b1d31466e276420e3913d.png)
for all 
with 
.
Day 2
Problem 4
Let 
be an integer, and let 
denote the sum of the digits of 
when it is written in base . Show that there are infinitely many positive integers that cannot be represented in the form 
, where is a positive integer.
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.
