Difference between revisions of "2014 USAMO Problems"
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===Problem 1=== | ===Problem 1=== | ||
− | 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 < | + | 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=== | ||
− | 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>. | + | 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=== | ||
+ | |||
+ | 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>. | ||
+ | |||
[[2014 USAMO Problems/Problem 3|Solution]] | [[2014 USAMO Problems/Problem 3|Solution]] | ||
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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. | ||
+ | |||
[[2014 USAMO Problems/Problem 4|Solution]] | [[2014 USAMO Problems/Problem 4|Solution]] | ||
===Problem 5=== | ===Problem 5=== | ||
+ | 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>. | ||
+ | |||
[[2014 USAMO Problems/Problem 5|Solution]] | [[2014 USAMO Problems/Problem 5|Solution]] | ||
+ | |||
===Problem 6=== | ===Problem 6=== | ||
+ | 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> | ||
+ | |||
[[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 02:14, 14 February 2024
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.
Problem 5
Let be a triangle with orthocenter
and let
be the second intersection of the circumcircle of triangle
with the internal bisector of the angle
. Let
be the circumcenter of triangle
and
the orthocenter of triangle
. Prove that the length of segment
is equal to the circumradius of triangle
.
Problem 6
Prove that there is a constant with the following property: If
are positive integers such that
for all
, then
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.