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Difference between revisions of "2014 USAMO Problems"

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===Problem 1===
 
===Problem 1===
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Let <math>a,b,c,d</math> be real numbers such that <math>b-d \ge 5</math> and all zeros <math>x_1, x_2, x_3,</math> and <math>x_4</math> of the polynomial <math>P(x)=x^4+ax^3+bx^2+cx+d</math> are real. Find the smallest value the product <math>(x_1^2+1)(x_2^2+1)(x_3^2+1)(x_4^2+1)</math> can take.
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Let <math>a,b,c,d</math> be real numbers such that <math>b-d \ge 5</math> and all zeros <math>x_1, x_2, x_3,</math> and <math>x_4</math> of the polynomial <math>P(x)=x^4+ax^3+bx^2+cx+d</math> are real. Find the smallest value the product <cmath>(x_1^2+1)(x_2^2+1)(x_3^2+1)(x_4^2+1)</cmath> can take.
  
 
[[2014 USAMO Problems/Problem 1|Solution]]
 
[[2014 USAMO Problems/Problem 1|Solution]]
 
===Problem 2===
 
===Problem 2===
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Let <math>\mathbb{Z}</math> be the set of integers. Find all functions <math>f : \mathbb{Z} \rightarrow \mathbb{Z}</math> such that <cmath>xf(2f(y)-x)+y^2f(2x-f(y))=\frac{f(x)^2}{x}+f(yf(y))</cmath> for all <math>x, y \in \mathbb{Z}</math> with <math>x \neq 0</math>.
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Let <math>\mathbb{Z}</math> be the set of integers. Find all functions <math>f : \mathbb{Z} \rightarrow \mathbb{Z}</math> such that <cmath>xf(2f(y)-x)+y^2f(2x-f(y))=\frac{(f(x))^2}{x}+f(yf(y))</cmath> for all <math>x, y \in \mathbb{Z}</math> with <math>x \neq 0</math>.
  
 
[[2014 USAMO Problems/Problem 2|Solution]]
 
[[2014 USAMO Problems/Problem 2|Solution]]
  
 
===Problem 3===
 
===Problem 3===
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Prove that there exists an infinite set of points <cmath>\ldots,\,\,\,\,P_{-3},\,\,\,\,P_{-2},\,\,\,\,P_{-1},\,\,\,\,P_0,\,\,\,\,P_1,\,\,\,\,P_2,\,\,\,\,P_3,\,\,\,\,\ldots</cmath> in the plane with the following property: For any three distinct integers <math>a,b,</math> and <math>c</math>, points <math>P_a</math>, <math>P_b</math>, and <math>P_c</math> are collinear if and only if <math>a+b+c=2014</math>.
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[[2014 USAMO Problems/Problem 3|Solution]]
 
[[2014 USAMO Problems/Problem 3|Solution]]
  
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===Problem 4===
 
===Problem 4===
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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===
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Let <math>ABC</math> be a triangle with orthocenter <math>H</math> and let <math>P</math> be the second intersection of the circumcircle of triangle <math>AHC</math> with the internal bisector of the angle <math>\angle BAC</math>.  Let <math>X</math> be the circumcenter of triangle <math>APB</math> and <math>Y</math> the orthocenter of triangle <math>APC</math>.  Prove that the length of segment <math>XY</math> is equal to the circumradius of triangle <math>ABC</math>.
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[[2014 USAMO Problems/Problem 5|Solution]]
 
[[2014 USAMO Problems/Problem 5|Solution]]
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===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]]
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{{USAMO newbox|year=2014|before=[[2013 USAMO Problems]]|after=[[2015 USAMO Problems]]}}
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{{MAA Notice}}

Latest revision as of 01:14, 14 February 2024

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

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 $\angle BAC$. Let $X$ be the circumcenter of triangle $APB$ and $Y$ the orthocenter of triangle $APC$. Prove that the length of segment $XY$ is equal to the circumradius of triangle $ABC$.

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


2014 USAMO (Problems • Resources)
Preceded by
2013 USAMO Problems 		Followed by
2015 USAMO Problems
1 • 2 • 3 • 4 • 5 • 6
All USAMO Problems and Solutions


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