Navvye/Olympiad-Problems-CSV huggingface dataset:
2018 ELMO Problems
Let $ABC$ be a scalene triangle with orthocenter $H$ and circumcenter $O$. Let $P$ be the midpoint of $\overline{AH}$ and let $T$ be on line $BC$ with $\angle TAO=90^{\circ}$. Let $X$ be the foot of the altitude from $O$ onto line $PT$. Prove that the midpoint of $\overline{PX}$ lies on the nine-point circle* of $\triangle ABC$.



*The nine-point circle of $\triangle ABC$ is the unique circle passing through the following nine points: the midpoint of the sides, the feet of the altitudes, and the midpoints of $\overline{AH}$, $\overline{BH}$, and $\overline{CH}$.



Proposed by Zack Chroman
2018 ELMO Problems
Let $a_1,a_2,\dots,a_m$ be a finite sequence of positive integers. Prove that there exist nonnegative integers $b,c,$ and $N$ such that $$\left\lfloor \sum_{i=1}^m \sqrt{n+a_i} \right\rfloor =\left\lfloor \sqrt{bn+c} \right\rfloor$$holds for all integers $n>N.$



Proposed by Carl Schildkraut
2018 ELMO Problems
A windmill is a closed line segment of unit length with a distinguished endpoint, the pivot. Let $S$ be a finite set of $n$ points such that the distance between any two points of $S$ is greater than $c$. A configuration of $n$ windmills is admissible if no two windmills intersect and each point of $S$ is used exactly once as a pivot.



An admissible configuration of windmills is initially given to Geoff in the plane. In one operation Geoff can rotate any windmill around its pivot, either clockwise or counterclockwise and by any amount, as long as no two windmills intersect during the process. Show that Geoff can reach any other admissible configuration in finitely many operations, where



(i) $c = \sqrt 3$,



(ii) $c = \sqrt 2$.



Proposed by Michael Ren
2017 ELMO Problems
Let $a_1,a_2,\dots, a_n$ be positive integers with product $P,$ where $n$ is an odd positive integer. Prove that $$\gcd(a_1^n+P,a_2^n+P,\dots, a_n^n+P)\le 2\gcd(a_1,\dots, a_n)^n.$$

Proposed by Daniel Liu
2017 ELMO Problems
Let $ABC$ be a triangle with orthocenter $H,$ and let $M$ be the midpoint of $\overline{BC}.$ Suppose that $P$ and $Q$ are distinct points on the circle with diameter $\overline{AH},$ different from $A,$ such that $M$ lies on line $PQ.$ Prove that the orthocenter of $\triangle APQ$ lies on the circumcircle of $\triangle ABC.$



Proposed by Michael Ren
2017 ELMO Problems
nic$\kappa$y is drawing kappas in the cells of a square grid. However, he does not want to draw kappas in three consecutive cells (horizontally, vertically, or diagonally). Find all real numbers $d>0$ such that for every positive integer $n,$ nic$\kappa$y can label at least $dn^2$ cells of an $n\times n$ square.



Proposed by Mihir Singhal and Michael Kural
2017 ELMO Problems
An integer $n>2$ is called tasty if for every ordered pair of positive integers $(a,b)$ with $a+b=n,$ at least one of $\frac{a}{b}$ and $\frac{b}{a}$ is a terminating decimal. Do there exist infinitely many tasty integers?



Proposed by Vincent Huang
2017 ELMO Problems
The edges of $K_{2017}$ are each labeled with $1,2,$ or $3$ such that any triangle has sum of labels at least $5.$ Determine the minimum possible average of all $\dbinom{2017}{2}$ labels.



(Here $K_{2017}$ is defined as the complete graph on 2017 vertices, with an edge between every pair of vertices.)



Proposed by Michael Ma
2017 ELMO Problems
Find all functions $f:\mathbb{R}\to \mathbb{R}$ such that for all real numbers $a,b,$ and $c$:



(i) If $a+b+c\ge 0$ then $f(a^3)+f(b^3)+f(c^3)\ge 3f(abc).$



(ii) If $a+b+c\le 0$ then $f(a^3)+f(b^3)+f(c^3)\le 3f(abc).$



Proposed by Ashwin Sah
2016 ELMO Problems
Cookie Monster says a positive integer $n$ is $crunchy$ if there exist $2n$ real numbers $x_1,x_2,\ldots,x_{2n}$, not all equal, such that the sum of any $n$ of the $x_i$'s is equal to the product of the other $n$ of the $x_i$'s. Help Cookie Monster determine all crunchy integers.



Yannick Yao
2016 ELMO Problems
Oscar is drawing diagrams with trash can lids and sticks. He draws a triangle $ABC$ and a point $D$ such that $DB$ and $DC$ are tangent to the circumcircle of $ABC$. Let $B'$ be the reflection of $B$ over $AC$ and $C'$ be the reflection of $C$ over $AB$. If $O$ is the circumcenter of $DB'C'$, help Oscar prove that $AO$ is perpendicular to $BC$.



James Lin
2016 ELMO Problems
In a Cartesian coordinate plane, call a rectangle $standard$ if all of its sides are parallel to the $x$- and $y$- axes, and call a set of points $nice$ if no two of them have the same $x$- or $y$- coordinate. First, Bert chooses a nice set $B$ of $2016$ points in the coordinate plane. To mess with Bert, Ernie then chooses a set $E$ of $n$ points in the coordinate plane such that $B\cup E$ is a nice set with $2016+n$ points. Bert returns and then miraculously notices that there does not exist a standard rectangle that contains at least two points in $B$ and no points in $E$ in its interior. For a given nice set $B$ that Bert chooses, define $f(B)$ as the smallest positive integer $n$ such that Ernie can find a nice set $E$ of size $n$ with the aforementioned properties. Help Bert determine the minimum and maximum possible values of $f(B)$.



Yannick Yao
2016 ELMO Problems
Big Bird has a polynomial $P$ with integer coefficients such that $n$ divides $P(2^n)$ for every positive integer $n$. Prove that Big Bird's polynomial must be the zero polynomial.



Ashwin Sah
2016 ELMO Problems
Elmo is drawing with colored chalk on a sidewalk outside. He first marks a set $S$ of $n>1$ collinear points. Then, for every unordered pair of points $\{X,Y\}$ in $S$, Elmo draws the circle with diameter $XY$ so that each pair of circles which intersect at two distinct points are drawn in different colors. Count von Count then wishes to count the number of colors Elmo used. In terms of $n$, what is the minimum number of colors Elmo could have used?



Michael Ren
2016 ELMO Problems
Elmo is now learning olympiad geometry. In triangle $ABC$ with $AB\neq AC$, let its incircle be tangent to sides $BC$, $CA$, and $AB$ at $D$, $E$, and $F$, respectively. The internal angle bisector of $\angle BAC$ intersects lines $DE$ and $DF$ at $X$ and $Y$, respectively. Let $S$ and $T$ be distinct points on side $BC$ such that $\angle XSY=\angle XTY=90^\circ$. Finally, let $\gamma$ be the circumcircle of $\triangle AST$.



(a) Help Elmo show that $\gamma$ is tangent to the circumcircle of $\triangle ABC$.



(b) Help Elmo show that $\gamma$ is tangent to the incircle of $\triangle ABC$.



James Lin
2015 ELMO Problems
Define the sequence $a_1 = 2$ and $a_n = 2^{a_{n-1}} + 2$ for all integers $n \ge 2$. Prove that $a_{n-1}$ divides $a_n$ for all integers $n \ge 2$.



Proposed by Sam Korsky
2015 ELMO Problems
Let $m$, $n$, and $x$ be positive integers. Prove that $$ \sum_{i = 1}^n \min\left(\left\lfloor \frac{x}{i} \right\rfloor, m \right) = \sum_{i = 1}^m \min\left(\left\lfloor \frac{x}{i} \right\rfloor, n \right). $$

Proposed by Yang Liu
2015 ELMO Problems
Let $\omega$ be a circle and $C$ a point outside it; distinct points $A$ and $B$ are selected on $\omega$ so that $\overline{CA}$ and $\overline{CB}$ are tangent to $\omega$.  Let $X$ be the reflection of $A$ across the point $B$, and denote by $\gamma$ the circumcircle of triangle $BXC$.  Suppose $\gamma$ and $\omega$ meet at $D \neq B$ and line $CD$ intersects $\omega$ at $E \neq D$.  Prove that line $EX$ is tangent to the circle $\gamma$.



Proposed by David Stoner
2015 ELMO Problems
Let $a > 1$ be a positive integer. Prove that for some nonnegative integer $n$, the number $2^{2^n}+a$ is not prime.



Proposed by Jack Gurev
2015 ELMO Problems
Let $m, n, k > 1$ be positive integers.  For a set $S$ of positive integers, define $S(i,j)$ for $i<j$ to be the number of elements in $S$ strictly between $i$ and $j$.  We say two sets $(X,Y)$ are a fat pair if $$ X(i,j)\equiv Y(i,j) \pmod{n} $$ for every $i,j \in X \cap Y$.  (In particular, if $\left\lvert X \cap Y \right\rvert < 2$ then $(X,Y)$ is fat.)



If there are $m$ distinct sets of $k$ positive integers such that no two form a fat pair, show that $m<n^{k-1}$.



Proposed by Allen Liu
2014 ELMO Problems
Find all triples $(f,g,h)$ of injective functions from the set of real numbers to itself satisfying

\begin{align*}

  f(x+f(y)) &= g(x) + h(y) \\
  g(x+g(y)) &= h(x) + f(y) \\
  h(x+h(y)) &= f(x) + g(y) 

\end{align*}

for all real numbers $x$ and $y$. (We say a function $F$ is injective if $F(a)\neq F(b)$ for any distinct real numbers $a$ and $b$.)



Proposed by Evan Chen
2014 ELMO Problems
Define a beautiful number to be an integer of the form $a^n$, where $a\in\{3,4,5,6\}$ and $n$ is a positive integer.

Prove that each integer greater than $2$ can be expressed as the sum of pairwise distinct beautiful numbers.



Proposed by Matthew Babbitt
2014 ELMO Problems
We say a finite set $S$ of points in the plane is very if for every point $X$ in $S$, there exists an inversion with center $X$ mapping every point in $S$ other than $X$ to another point in $S$ (possibly the same point).



(a) Fix an integer $n$. Prove that if $n \ge 2$, then any line segment $\overline{AB}$ contains a unique very set $S$ of size $n$ such that $A, B \in S$.



(b) Find the largest possible size of a very set not contained in any line.



(Here, an inversion with center $O$ and radius $r$ sends every point $P$ other than $O$ to the point $P'$ along ray $OP$ such that $OP\cdot OP' = r^2$.)



Proposed by Sammy Luo
2014 ELMO Problems
Let $n$ be a positive integer and let $a_1$, $a_2$, \dots, $a_n$ be real numbers strictly between $0$ and $1$.  For any subset $S$ of $\{1, 2, ..., n\}$, define $$ f(S) = \prod_{i \in S} a_i \cdot \prod_{j \not \in S} (1-a_j). $$ Suppose that $\sum_{|S| \text{ odd}} f(S) = \frac{1}{2}$.  Prove that $a_k = \frac{1}{2}$ for some $k$.  (Here the sum ranges over all subsets of $\{1, 2, ..., n\}$ with an odd number of elements.)



Proposed by Kevin Sun
2014 ELMO Problems
Let $ABC$ be a triangle with circumcenter $O$ and orthocenter $H$. Let $\omega_1$ and $\omega_2$ denote the circumcircles of triangles $BOC$ and $BHC$, respectively. Suppose the circle with diameter $\overline{AO}$ intersects $\omega_1$ again at $M$, and line $AM$ intersects $\omega_1$ again at $X$. Similarly, suppose the circle with diameter $\overline{AH}$ intersects $\omega_2$ again at $N$, and line $AN$ intersects $\omega_2$ again at $Y$. Prove that lines $MN$ and $XY$ are parallel.



Proposed by Sammy Luo
2014 ELMO Problems
A $2^{2014} + 1$ by $2^{2014} + 1$ grid has some black squares filled. The filled black squares form one or more snakes on the plane, each of whose heads splits at some points but never comes back together. In other words, for every positive integer $n$ greater than $2$, there do not exist pairwise distinct black squares $s_1$, $s_2$, \dots, $s_n$ such that $s_i$ and $s_{i+1}$ share an edge for $i=1,2, \dots, n$ (here $s_{n+1}=s_1$).



What is the maximum possible number of filled black squares?



Proposed by David Yang
2013 ELMO Problems
Let $a_1,a_2,...,a_9$ be nine real numbers, not necessarily distinct, with average $m$. Let $A$ denote the number of triples $1 \le i < j < k \le 9$ for which $a_i + a_j + a_k \ge 3m$.  What is the minimum possible value of $A$?



Proposed by Ray Li
2013 ELMO Problems
Let $a,b,c$ be positive reals satisfying $a+b+c = \sqrt[7]{a} + \sqrt[7]{b} + \sqrt[7]{c}$.  Prove that $a^a b^b c^c \ge 1$.



Proposed by Evan Chen
2013 ELMO Problems
Let $m_1,m_2,...,m_{2013} > 1$ be 2013 pairwise relatively prime positive integers and $A_1,A_2,...,A_{2013}$ be 2013 (possibly empty) sets with $A_i\subseteq \{1,2,...,m_i-1\}$ for $i=1,2,...,2013$.  Prove that there is a positive integer $N$ such that

$$ N \le \left( 2\left\lvert A_1 \right\rvert + 1 \right)\left( 2\left\lvert A_2 \right\rvert + 1 \right)\cdots\left( 2\left\lvert A_{2013} \right\rvert + 1 \right) $$

and for each $i = 1, 2, ..., 2013$, there does not exist $a \in A_i$ such that $m_i$ divides $N-a$.



Proposed by Victor Wang
2013 ELMO Problems
Triangle $ABC$ is inscribed in circle $\omega$.  A circle with chord $BC$ intersects segments $AB$ and $AC$ again at $S$ and $R$, respectively.  Segments $BR$ and $CS$ meet at $L$, and rays $LR$ and $LS$ intersect $\omega$ at $D$ and $E$, respectively.  The internal angle bisector of $\angle BDE$ meets line $ER$ at $K$.  Prove that if $BE = BR$, then $\angle ELK = \tfrac{1}{2} \angle BCD$.



Proposed by Evan Chen
2013 ELMO Problems
For what polynomials $P(n)$ with integer coefficients can a positive integer be assigned to every lattice point in $\mathbb{R}^3$ so that for every integer $n \ge 1$, the sum of the $n^3$ integers assigned to any $n \times n \times n$ grid of lattice points is divisible by $P(n)$?



Proposed by Andre Arslan
2013 ELMO Problems
Consider a function $f: \mathbb Z \to \mathbb Z$ such that for every integer $n \ge 0$, there are at most $0.001n^2$ pairs of integers $(x,y)$ for which $f(x+y) \neq f(x)+f(y)$ and $\max\{ \lvert x \rvert, \lvert y \rvert \} \le n$.  Is it possible that for some integer $n \ge 0$, there are more than $n$ integers $a$ such that $f(a) \neq a \cdot f(1)$ and $\lvert a \rvert \le n$?



Proposed by David Yang
2012 ELMO Problems
In acute triangle $ABC$, let $D,E,F$ denote the feet of the altitudes from $A,B,C$, respectively, and let $\omega$ be the circumcircle of $\triangle AEF$. Let $\omega_1$ and $\omega_2$ be the circles through $D$ tangent to $\omega$ at $E$ and $F$, respectively. Show that $\omega_1$ and $\omega_2$ meet at a point $P$ on $BC$ other than $D$.



Ray Li.
2012 ELMO Problems
Find all ordered pairs of positive integers $(m,n)$ for which there exists a set $C=\{c_1,\ldots,c_k\}$ ($k\ge1$) of colors and an assignment of colors to each of the $mn$ unit squares of a $m\times n$ grid such that for every color $c_i\in C$ and unit square $S$ of color $c_i$, exactly two direct (non-diagonal) neighbors of $S$ have color $c_i$.



David Yang.
2012 ELMO Problems
Let $f,g$ be polynomials with complex coefficients such that $\gcd(\deg f,\deg g)=1$. Suppose that there exist polynomials $P(x,y)$ and $Q(x,y)$ with complex coefficients such that $f(x)+g(y)=P(x,y)Q(x,y)$. Show that one of $P$ and $Q$ must be constant.



Victor Wang.
2012 ELMO Problems
Let $a_0,b_0$ be positive integers, and define $a_{i+1}=a_i+\lfloor\sqrt{b_i}\rfloor$ and $b_{i+1}=b_i+\lfloor\sqrt{a_i}\rfloor$ for all $i\ge0$. Show that there exists a positive integer $n$ such that $a_n=b_n$.



David Yang.
2012 ELMO Problems
Let $ABC$ be an acute triangle with $AB<AC$, and let $D$ and $E$ be points on side $BC$ such that $BD=CE$ and $D$ lies between $B$ and $E$. Suppose there exists a point $P$ inside $ABC$ such that $PD\parallel AE$ and $\angle PAB=\angle EAC$. Prove that $\angle PBA=\angle PCA$.



Calvin Deng.
2012 ELMO Problems
A diabolical combination lock has $n$ dials (each with $c$ possible states), where $n,c>1$. The dials are initially set to states $d_1, d_2, \ldots, d_n$, where $0\le d_i\le c-1$ for each $1\le i\le n$. Unfortunately, the actual states of the dials (the $d_i$'s) are concealed, and the initial settings of the dials are also unknown. On a given turn, one may advance each dial by an integer amount $c_i$ ($0\le c_i\le c-1$), so that every dial is now in a state $d_i '\equiv d_i+c_i \pmod{c}$ with $0\le d_i ' \le c-1$. After each turn, the lock opens if and only if all of the dials are set to the zero state; otherwise, the lock selects a random integer $k$ and cyclically shifts the $d_i$'s by $k$ (so that for every $i$, $d_i$ is replaced by $d_{i-k}$, where indices are taken modulo $n$).



Show that the lock can always be opened, regardless of the choices of the initial configuration and the choices of $k$ (which may vary from turn to turn), if and only if $n$ and $c$ are powers of the same prime.



Bobby Shen.
2011 ELMO Problems
Let $ABCD$ be a convex quadrilateral. Let $E,F,G,H$ be points on segments $AB$, $BC$, $CD$, $DA$, respectively, and let $P$ be the intersection of $EG$ and $FH$. Given that quadrilaterals $HAEP$, $EBFP$, $FCGP$, $GDHP$ all have inscribed circles, prove that $ABCD$ also has an inscribed circle.



Evan O'Dorney.
2011 ELMO Problems
Wanda the Worm likes to eat Pascal's triangle. One day, she starts at the top of the triangle and eats $\textstyle\binom{0}{0}=1$. Each move, she travels to an adjacent positive integer and eats it, but she can never return to a spot that she has previously eaten. If Wanda can never eat numbers $a,b,c$ such that $a+b=c$, prove that it is possible for her to eat 100,000 numbers in the first 2011 rows given that she is not restricted to traveling only in the first 2011 rows.



(Here, the $n+1$st row of Pascal's triangle consists of entries of the form $\textstyle\binom{n}{k}$ for integers $0\le k\le n$. Thus, the entry $\textstyle\binom{n}{k}$ is considered adjacent to the entries $\textstyle\binom{n-1}{k-1}$, $\textstyle\binom{n-1}{k}$, $\textstyle\binom{n}{k-1}$, $\textstyle\binom{n}{k+1}$, $\textstyle\binom{n+1}{k}$, $\textstyle\binom{n+1}{k+1}$.)



Linus Hamilton.
2011 ELMO Problems
Determine whether there exist two reals $x,y$ and a sequence $\{a_n\}_{n=0}^{\infty}$ of nonzero reals such that $a_{n+2}=xa_{n+1}+ya_n$ for all $n\ge0$ and for every positive real number $r$, there exist positive integers $i,j$ such that $|a_i|<r<|a_j|$.



Alex Zhu.
2011 ELMO Problems
Find all functions $f:\mathbb{R}^+\to\mathbb{R}^+$ such that whenever $a>b>c>d>0$ and $ad=bc$,

$$f(a+d)+f(b-c)=f(a-d)+f(b+c).$$



Calvin Deng.
2011 ELMO Problems
Let $p>13$ be a prime of the form $2q+1$, where $q$ is prime. Find the number of ordered pairs of integers $(m,n)$ such that $0\le m<n<p-1$ and

$$3^m+(-12)^m\equiv 3^n+(-12)^n\pmod{p}.$$



Alex Zhu.



NoteThe original version asked for the number of solutions to $2^m+3^n\equiv 2^n+3^n\pmod{p}$ (still $0\le m<n<p-1$), where $p$ is a Fermat prime.
2011 ELMO Problems
Consider the infinite grid of lattice points in $\mathbb{Z}^3$. Little D and Big Z play a game, where Little D first loses a shoe on an unmunched point in the grid. Then, Big Z munches a shoe-free plane perpendicular to one of the coordinate axes. They continue to alternate turns in this fashion, with Little D's goal to lose a shoe on each of $n$ consecutive lattice points on a line parallel to one of the coordinate axes. Determine all $n$ for which Little D can accomplish his goal.



David Yang.
2010 ELMO Problems
Determine all (not necessarily finite) sets $S$ of points in the plane such that given any four distinct points in $S$, there is a circle passing through all four or a line passing through some three.



Carl Lian.
2010 ELMO Problems
Let $r$ and $s$ be positive integers. Define $a_0 = 0$, $a_1 = 1$, and $a_n = ra_{n-1} + sa_{n-2}$ for $n \geq 2$. Let $f_n = a_1a_2\cdots a_n$. Prove that $\displaystyle\frac{f_n}{f_kf_{n-k}}$ is an integer for all integers $n$ and $k$ such that $0 < k < n$.



Evan O' Dorney.
2010 ELMO Problems
Let $n > 1$ be a positive integer. A 2-dimensional grid, infinite in all directions, is given. Each 1 by 1 square in a given $n$ by $n$ square has a counter on it. A move consists of taking $n$ adjacent counters in a row or column and sliding them each by one space along that row or column. A returning sequence is a finite sequence of moves such that all counters again fill the original $n$ by $n$ square at the end of the sequence.



Assume that all counters are distinguishable except two, which are indistinguishable from each other. Prove that any distinguishable arrangement of counters in the $n$ by $n$ square can be reached by a returning sequence.

Assume all counters are distinguishable. Prove that there is no returning sequence that switches two counters and returns the rest to their original positions.





Mitchell Lee and Benjamin Gunby.
2010 ELMO Problems
Determine all strictly increasing functions $f: \mathbb{N}\to\mathbb{N}$ satisfying $nf(f(n))=f(n)^2$ for all positive integers $n$.



Carl Lian and Brian Hamrick.
2010 ELMO Problems
2010 MOPpers are assigned numbers 1 through 2010. Each one is given a red slip and a blue slip of paper. Two positive integers, A and B, each less than or equal to 2010 are chosen. On the red slip of paper, each MOPper writes the remainder when the product of A and his or her number is divided by 2011. On the blue slip of paper, he or she writes the remainder when the product of B and his or her number is divided by 2011. The MOPpers may then perform either of the following two operations:



Each MOPper gives his or her red slip to the MOPper whose number is written on his or her blue slip.

Each MOPper gives his or her blue slip to the MOPper whose number is written on his or her red slip.



Show that it is always possible to perform some number of these operations such that each MOPper is holding a red slip with his or her number written on it.



Brian Hamrick.
2010 ELMO Problems
Let $ABC$ be a triangle with circumcircle $\omega$, incenter $I$, and $A$-excenter $I_A$. Let the incircle and the $A$-excircle hit $BC$ at $D$ and $E$, respectively, and let $M$ be the midpoint of arc $BC$ without $A$. Consider the circle tangent to $BC$ at $D$ and arc $BAC$ at $T$. If $TI$ intersects $\omega$ again at $S$, prove that $SI_A$ and $ME$ meet on $\omega$.



Amol Aggarwal.
2009 ELMO Problems
Let $a,b,c$ be positive integers such that $a^2 - bc$ is a square. Prove that $2a + b + c$ is not prime.



Evan o'Dorney
2009 ELMO Problems
Let $ABC$ be a triangle such that $AB < AC$. Let $P$ lie on a line through $A$ parallel to line $BC$ such that $C$ and $P$ are on the same side of line $AB$. Let $M$ be the midpoint of segment $BC$.  Define $D$ on segment $BC$ such that $\angle BAD = \angle CAM$, and define $T$ on the extension of ray $CB$ beyond $B$ so that $\angle BAT = \angle CAP$.  Given that lines $PC$ and $AD$ intersect at $Q$, that lines $PD$ and $AB$ intersect at $R$, and that $S$ is the midpoint of segment $DT$, prove that if $A$,$P$,$Q$, and $R$ lie on a circle, then $Q$, $R$, and $S$ are collinear.



David Rush
2009 ELMO Problems
Let $a,b,c$ be nonnegative real numbers. Prove that $$ a(a - b)(a - 2b) + b(b - c)(b - 2c) + c(c - a)(c - 2a) \geq 0.$$Wenyu Cao
2009 ELMO Problems
Let $n$ be a positive integer. Given $n^2$ points in a unit square, prove that there exists a broken line of length $2n + 1$ that passes through all the points.



Allen Yuan
2009 ELMO Problems
Let $ABCDEFG$ be a regular heptagon with center $O$. Let $M$ be the centroid of $\triangle ABD$. Prove that $\cos^2(\angle GOM)$ is rational and determine its value.



Evan o'Dorney
2009 ELMO Problems
Let $p$ be an odd prime and $x$ be an integer such that $p \mid x^3 - 1$ but $p \nmid x - 1$. Prove that $$ p \mid (p - 1)!\left(x - \frac {x^2}{2} + \frac {x^3}{3} - \cdots - \frac {x^{p - 1}}{p - 1}\right).$$John Berman
2003 ELMO Problems
Let $ABCDEF$ be a convex equilateral hexagon with sides of length $1$.  Let $R_1$ be the area of the region contained within both $ACE$ and $BDF$, and let $R_2$ be the area of the region within the hexagon outside both triangles.  Prove that: $$ \min \{ [ACE], [BDF] \} + R_2 - R_1 \le \frac{3\sqrt{3}}{4}. $$
2003 ELMO Problems
In a set of $30$ MOPpers, prove that some two MOPpers have an even number of common friends.
2003 ELMO Problems
Let $k$ be a positive integer for which the equation $$ 2ab+2bc+2ca-a^2-b^2-c^2 = k $$ has some solution in positive integers $a,b,c$.  Prove that the equation has a solution for which $a$, $b$ and $c$ are the sides of a possibly degenerate triangle.
2003 ELMO Problems
Let $x,y,z \ge 1$ be real numbers such that $$ \frac{1}{x^2-1} + \frac{1}{y^2-1} + \frac{1}{z^2-1} = 1. $$ Prove that $$ \frac{1}{x+1} + \frac{1}{y+1} + \frac{1}{z+1} \le 1. $$
1999 ELMO Problems
In nonisosceles triangle $ABC$ the excenters of the triangle opposite $B$ and $C$ be $X_B$ and $X_C$, respectively.  Let the external angle bisector of $A$ intersect the circumcircle of $\triangle ABC$ again at $Q$.  Prove that $QX_B = QB = QC = QX_C$.
1999 ELMO Problems
Mr. Fat moves around on the lattice points according to the following rules: From point $(x,y)$ he may move to any of the points $(y,x)$, $(3x,-2y)$, $(-2x,3y)$, $(x+1,y+4)$ and $(x-1,y-4)$.  Show that if he starts at $(0,1)$ he can never get to $(0,0)$.