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Difference between revisions of "2014 USAMO Problems"

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[[2014 USAMO Problems/Problem 5|Solution]]
 
[[2014 USAMO Problems/Problem 5|Solution]]
 
===Problem 6===
 
===Problem 6===
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Prove that there is a constant <math>c>0</math> with the following property: If <math>a, b, n</math> are positive integers such that <math>\gcd(a+i, b+j)>1</math> for all <math>i, j\in\{0, 1, \ldots n\}</math>, then<cmath>\min\{a, b\}>c^n\cdot n^{\frac{n}{2}}.</cmath>
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[[2014 USAMO Problems/Problem 6|Solution]]
 
[[2014 USAMO Problems/Problem 6|Solution]]

Revision as of 16:45, 30 April 2014

Day 1

Problem 1

Let $a,b,c,d$ be real numbers such that $b-d \ge 5$ and all zeros $x_1, x_2, x_3,$ and $x_4$ of the polynomial $P(x)=x^4+ax^3+bx^2+cx+d$ are real. Find the smallest value the product $(x_1^2+1)(x_2^2+1)(x_3^2+1)(x_4^2+1)$ can take.

Solution

Problem 2

Let $\mathbb{Z}$ be the set of integers. Find all functions $f : \mathbb{Z} \rightarrow \mathbb{Z}$ such that \[xf(2f(y)-x)+y^2f(2x-f(y))=\frac{f(x)^2}{x}+f(yf(y))\] for all $x, y \in \mathbb{Z}$ with $x \neq 0$.

Solution

Problem 3

Prove that there exists an infinite set of points \[\ldots,\,\,\,\,P_{-3},\,\,\,\,P_{-2},\,\,\,\,P_{-1},\,\,\,\,P_0,\,\,\,\,P_1,\,\,\,\,P_2,\,\,\,\,P_3,\,\,\,\,\ldots\] in the plane with the following property: For any three distinct integers $a,b,$ and $c$, points $P_a$, $P_b$, and $P_c$ are collinear if and only if $a+b+c=2014$.

Solution

Day 2

Problem 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.

Solution

Problem 5

Solution

Problem 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 $i, j\in\{0, 1, \ldots n\}$, then\[\min\{a, b\}>c^n\cdot n^{\frac{n}{2}}.\]

Solution

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