Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "2014 USAJMO Problems"

(→‎Problem 3)
 
(17 intermediate revisions by 7 users not shown)
Line 2: Line 2:
  
 
===Problem 1===
 
===Problem 1===
−
Let <math>a</math>, <math>b</math>, <math>c</math> be real numbers greater than or equal to <math>1</math>. Prove that <cmath>\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 </cmath>
+
Let <math>a</math>, <math>b</math>, <math>c</math> be real numbers greater than or equal to <math>1</math>. Prove that <cmath>\min{\left (\frac{10a^2-5a+1}{b^2-5b+10},\frac{10b^2-5b+1}{c^2-5c+10},\frac{10c^2-5c+1}{a^2-5a+10}\right )}\leq abc. </cmath>
 
[[2014 USAJMO Problems/Problem 1|Solution]]
 
[[2014 USAJMO Problems/Problem 1|Solution]]
  
 
===Problem 2===
 
===Problem 2===
 +
Let <math>\triangle{ABC}</math> be a non-equilateral, acute triangle with <math>\angle A=60^\circ</math>, and let <math>O</math> and <math>H</math> denote the circumcenter and orthocenter of <math>\triangle{ABC}</math>, respectively.
 +
 +
(a) Prove that line <math>OH</math> intersects both segments <math>AB</math> and <math>AC</math>.
 +
 +
(b) Line <math>OH</math> intersects segments <math>AB</math> and <math>AC</math> at <math>P</math> and <math>Q</math>, respectively. Denote by <math>s</math> and <math>t</math> the respective areas of triangle <math>APQ</math> and quadrilateral <math>BPQC</math>. Determine the range of possible values for <math>s/t</math>.
 +
 
[[2014 USAJMO Problems/Problem 2|Solution]]
 
[[2014 USAJMO Problems/Problem 2|Solution]]
 +
 
===Problem 3===
 
===Problem 3===
 
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>.
Line 15: Line 22:
  
 
===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.
 +
 
[[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.
 +
 
[[2014 USAJMO Problems/Problem 5|Solution]]
 
[[2014 USAJMO Problems/Problem 5|Solution]]
 +
 
===Problem 6===
 
===Problem 6===
 +
Let <math>ABC</math> be a triangle with incenter <math>I</math>, incircle <math>\gamma</math> and circumcircle <math>\Gamma</math>.  Let <math>M,N,P</math> be the midpoints of sides <math>\overline{BC}</math>, <math>\overline{CA}</math>, <math>\overline{AB}</math> and let <math>E,F</math> be the tangency points of <math>\gamma</math> with <math>\overline{CA}</math> and <math>\overline{AB}</math>, respectively.  Let <math>U,V</math> be the intersections of line <math>EF</math> with line <math>MN</math> and line <math>MP</math>, respectively, and let <math>X</math> be the midpoint of arc <math>BAC</math> of <math>\Gamma</math>.
 +
 +
(a) Prove that <math>I</math> lies on ray <math>CV</math>.
 +
 +
(b) Prove that line <math>XI</math> bisects <math>\overline{UV}</math>.
 +
 
[[2014 USAJMO Problems/Problem 6|Solution]]
 
[[2014 USAJMO Problems/Problem 6|Solution]]
 +
 +
== See Also ==
 +
*[[USAJMO Problems and Solutions]]
 +
 +
{{USAJMO box|year=2014|before=[[2013 USAJMO Problems]]|after=[[2015 USAJMO Problems]]}}
 +
{{MAA Notice}}

Latest revision as of 15:42, 5 August 2023

Day 1

Problem 1

Let $a$, $b$, $c$ be real numbers greater than or equal to $1$. Prove that \[\min{\left (\frac{10a^2-5a+1}{b^2-5b+10},\frac{10b^2-5b+1}{c^2-5c+10},\frac{10c^2-5c+1}{a^2-5a+10}\right )}\leq abc.\] Solution

Problem 2

Let $\triangle{ABC}$ be a non-equilateral, acute triangle with $\angle A=60^\circ$, and let $O$ and $H$ denote the circumcenter and orthocenter of $\triangle{ABC}$, respectively.

(a) Prove that line $OH$ intersects both segments $AB$ and $AC$.

(b) Line $OH$ intersects segments $AB$ and $AC$ at $P$ and $Q$, respectively. Denote by $s$ and $t$ the respective areas of triangle $APQ$ and quadrilateral $BPQC$. Determine the range of possible values for $s/t$.

Solution

Problem 3

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

Day 2

Problem 4

Let $b\geq 2$ be an integer, and let $s_b(n)$ denote the sum of the digits of $n$ when it is written in base $b$. Show that there are infinitely many positive integers that cannot be represented in the form $n+s_b(n)$, where $n$ is a positive integer.

Solution

Problem 5

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 6

Let $ABC$ be a triangle with incenter $I$, incircle $\gamma$ and circumcircle $\Gamma$. Let $M,N,P$ be the midpoints of sides $\overline{BC}$, $\overline{CA}$, $\overline{AB}$ and let $E,F$ be the tangency points of $\gamma$ with $\overline{CA}$ and $\overline{AB}$, respectively. Let $U,V$ be the intersections of line $EF$ with line $MN$ and line $MP$, respectively, and let $X$ be the midpoint of arc $BAC$ of $\Gamma$.

(a) Prove that $I$ lies on ray $CV$.

(b) Prove that line $XI$ bisects $\overline{UV}$.

Solution

See Also

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

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2014_USAJMO_Problems&oldid=196449"