Navvye/Olympiad-Problems-CSV huggingface dataset:
1999 ELMO Problems
Prove that $$ 2^6 \frac{abcd+1}{(a+b+c+d)^2} \le a^2+b^2+c^2+d^2+\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}+\frac{1}{d^2} $$ for $a,b,c,d > 0$.
1999 ELMO Problems
Let $a_1, a_2, a_3, \cdots$ be an infinite sequence of real numbers.  Prove that there exists a increasing sequence $j_1, j_2, j_3, \cdots$ of positive integers such that the sequence $a_{j_1}, a_{j_2}, a_{j_3}, \cdots$ is either nondecreasing or nonincreasing.
2019 ELMO Shortlist
Let $a$, $b$, $c$ be positive reals such that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1$. Show that $$a^abc+b^bca+c^cab\ge 27bc+27ca+27ab.$$



Proposed by Milan Haiman
2019 ELMO Shortlist
Find all functions $f:\mathbb Z\to \mathbb Z$ such that for all surjective functions $g:\mathbb Z\to \mathbb Z$, $f+g$ is also surjective. (A function $g$ is surjective over $\mathbb Z$ if for all integers $y$, there exists an integer $x$ such that $g(x)=y$.)



Proposed by Sean Li
2019 ELMO Shortlist
Find all nondecreasing functions $f:\mathbb R\to \mathbb R$ such that, for all $x,y\in \mathbb R$, $$f(f(x))+f(y)=f(x+f(y))+1.$$

Proposed by Carl Schildkraut
2019 ELMO Shortlist
Elmo and Elmo's clone are playing a game. Initially, $n\geq 3$ points are given on a circle. On a player's turn, that player must draw a triangle using three unused points as vertices, without creating any crossing edges. The first player who cannot move loses. If Elmo's clone goes first and players alternate turns, who wins? (Your answer may be in terms of $n$.)



Proposed by Milan Haiman
2019 ELMO Shortlist
Adithya and Bill are playing a game on a connected graph with $n > 2$ vertices, two of which are labeled $A$ and $B$, so that $A$ and $B$ are distinct and non-adjacent and known to both players. Adithya starts on vertex $A$ and Bill starts on $B$. Each turn, both players move simultaneously: Bill moves to an adjacent vertex, while Adithya may either move to an adjacent vertex or stay at his current vertex. Adithya loses if he is on the same vertex as Bill, and wins if he reaches $B$ alone. Adithya cannot see where Bill is, but Bill can see where Adithya is. Given that Adithya has a winning strategy, what is the maximum possible number of edges the graph may have? (Your answer may be in terms of $n$.)



Proposed by Steven Liu
2019 ELMO Shortlist
In the game of Ring Mafia, there are $2019$ counters arranged in a circle. $673$ of these counters are mafia, and the remaining $1346$ counters are town. Two players, Tony and Madeline, take turns with Tony going first. Tony does not know which counters are mafia but Madeline does.



On Tony’s turn, he selects any subset of the counters (possibly the empty set) and removes all counters in that set. On Madeline’s turn, she selects a town counter which is adjacent to a mafia counter and removes it. Whenever counters are removed, the remaining counters are brought closer together without changing their order so that they still form a circle. The game ends when either all mafia counters have been removed, or all town counters have been removed.



Is there a strategy for Tony that guarantees, no matter where the mafia counters are placed and what Madeline does, that at least one town counter remains at the end of the game?



Proposed by Andrew Gu
2019 ELMO Shortlist
Given a permutation of $1,2,3,\dots,n$, with consecutive elements $a,b,c$ (in that order), we may perform either of the moves:



If $a$ is the median of $a$, $b$, and $c$, we may replace $a,b,c$ with $b,c,a$ (in that order)

If $c$ is the median of $a$, $b$, and $c$, we may replace $a,b,c$ with $c,a,b$ (in that order)



What is the least number of sets in a partition of all $n!$ permutations, such that any two permutations in the same set are obtainable from each other by a sequence of moves?



Proposed by Milan Haiman
2019 ELMO Shortlist
Let $ABC$ be an acute triangle with orthocenter $H$ and circumcircle $\Gamma$. Let $BH$ intersect $AC$ at $E$, and let $CH$ intersect $AB$ at $F$. Let $AH$ intersect $\Gamma$ again at $P \neq A$. Let $PE$ intersect $\Gamma$ again at $Q \neq P$. Prove that $BQ$ bisects segment $\overline{EF}$.



Proposed by Luke Robitaille
2019 ELMO Shortlist
Let $\triangle ABC$ be an acute triangle with incenter $I$ and circumcenter $O$. The incircle touches sides $BC,CA,$ and $AB$ at $D,E,$ and $F$ respectively, and $A'$ is the reflection of $A$ over $O$. The circumcircles of $ABC$ and $A'EF$ meet at $G$, and the circumcircles of $AMG$ and $A'EF$ meet at a point $H\neq G$, where $M$ is the midpoint of $EF$. Prove that if $GH$ and $EF$ meet at $T$, then $DT\perp EF$.



Proposed by Ankit Bisain
2019 ELMO Shortlist
Let triangle $ABC$ have altitudes $BE$ and $CF$ which meet at $H$. The reflection of $A$ over $BC$ is $A'$. Let $(ABC)$ meet $(AA'E)$ at $P$ and $(AA'F)$ at $Q$. Let $BC$ meet $PQ$ at $R$. Prove that $EF \parallel HR$.



Proposed by Daniel Hu
2019 ELMO Shortlist
Given a triangle $ABC$ for which $\angle BAC \neq 90^{\circ}$, let $B_1, C_1$ be variable points on $AB,AC$, respectively. Let $B_2,C_2$ be the points on line $BC$ such that a spiral similarity centered at $A$ maps $B_1C_1$ to $C_2B_2$. Denote the circumcircle of $AB_1C_1$ by $\omega$. Show that if $B_1B_2$ and $C_1C_2$ concur on $\omega$ at a point distinct from $B_1$ and $C_1$, then $\omega$ passes through a fixed point other than $A$.



Proposed by Max Jiang
2019 ELMO Shortlist
Let $ABC$ be an acute scalene triangle and let $P$ be a point in the plane. For any point $Q\neq A,B,C$, define $T_A$ to be the unique point such that $\triangle T_ABP \sim \triangle T_AQC$ and $\triangle T_ABP, \triangle T_AQC$ are oriented in the same direction (clockwise or counterclockwise). Similarly define $T_B, T_C$.



a) Find all $P$ such that there exists a point $Q$ with $T_A,T_B,T_C$ all lying on the circumcircle of $\triangle ABC$. Call such a pair $(P,Q)$ a tasty pair with respect to $\triangle ABC$.



b) Keeping the notations from a), determine if there exists a tasty pair which is also tasty with respect to $\triangle T_AT_BT_C$.



Proposed by Vincent Huang
2019 ELMO Shortlist
Let $f:\mathbb N\to \mathbb N$. Show that $f(m)+n\mid f(n)+m$ for all positive integers $m\le n$ if and only if $f(m)+n\mid f(n)+m$ for all positive integers $m\ge n$.



Proposed by Carl Schildkraut
2019 ELMO Shortlist
A positive integer $b$ and a sequence $a_0,a_1,a_2,\dots$ of integers $0\le a_i<b$ is given. It is known that $a_0\neq 0$ and the sequence $\{a_i\}$ is eventually periodic but has infinitely many nonzero terms. Let $S$ be the set of positive integers $n$ so that $n\mid (a_0a_1\dots a_n)_b$. Given that $S$ is infinite, show that there are infinitely many primes that divide at least one element of $S$.



Proposed by Carl Schildkraut and Holden Mui
2019 ELMO Shortlist
Given an even positive integer $m$, find all positive integers $n$ for which there exists a bijection $f:[n]\to [n]$ so that, for all $x,y\in [n]$ for which $n\mid mx-y$, $$(n+1)\mid f(x)^m-f(y).$$

Note: For a positive integer $n$, we let $[n] = \{1,2,\dots, n\}$.



Proposed by Milan Haiman and Carl Schildkraut
2018 ELMO Shortlist
Let $f:\mathbb{R}\to\mathbb{R}$ be a bijective function. Does there always exist an infinite number of functions $g:\mathbb{R}\to\mathbb{R}$ such that $f(g(x))=g(f(x))$ for all $x\in\mathbb{R}$?



Proposed by Daniel Liu
2018 ELMO Shortlist
Let $a, b, c,x, y, z$ be positive reals such that $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=1$. Prove that $$a^x+b^y+c^z\ge \frac{4abcxyz}{(x+y+z-3)^2}.$$

Proposed by Daniel Liu
2018 ELMO Shortlist
Elmo calls a monic polynomial with real coefficients tasty if all of its coefficients are in the range $[-1,1]$. A monic polynomial  $P$ with real coefficients and complex roots $\chi_1,\cdots,\chi_m$ (counted with multiplicity) is given to Elmo, and he discovers that there does not exist a monic polynomial $Q$ with real coefficients such that $PQ$ is tasty. Find all possible values of $\max\left(|\chi_1|,\cdots,|\chi_m|\right)$.



Proposed by Carl Schildkraut
2018 ELMO Shortlist
We say that a positive integer $n$ is $m$-expressible if it is possible to get $n$ from some $m$ digits and the six operations $+,-,\times,\div$, exponentiation $^\wedge$, and concatenation $\oplus$. For example, $5625$ is $3$-expressible (in two ways): both $5\oplus (5^\wedge 4)$ and $(7\oplus 5)^\wedge 2$ yield $5625$.



Does there exist a positive integer $N$ such that all positive integers with $N$ digits are $(N-1)$-expressible?



Proposed by Krit Boonsiriseth
2018 ELMO Shortlist
Let $ABC$ be an acute triangle with orthocenter $H$, and let $P$ be a point on the nine-point circle of $ABC$. Lines $BH, CH$ meet the opposite sides $AC, AB$ at $E, F$, respectively. Suppose that the circumcircles $(EHP), (FHP)$ intersect lines $CH, BH$ a second time at $Q,R$, respectively. Show that as $P$ varies along the nine-point circle of $ABC$, the line $QR$ passes through a fixed point.



Proposed by Brandon Wang
2018 ELMO Shortlist
Let $ABCDEF$ be a hexagon inscribed in a circle $\Omega$ such that triangles $ACE$ and $BDF$ have the same orthocenter. Suppose that segments $BD$ and $DF$ intersect $CE$ at $X$ and $Y$, respectively. Show that there is a point common to $\Omega$, the circumcircle of $DXY$, and the line through $A$ perpendicular to $CE$.



Proposed by Michael Ren and Vincent Huang
2018 ELMO Shortlist
Let scalene triangle $ABC$ have altitudes $AD, BE, CF$ and circumcenter $O$. The circumcircles of $\triangle ABC$ and $\triangle ADO$ meet at $P \ne A$. The circumcircle of $\triangle ABC$ meets lines $PE$ at $X \ne P$ and $PF$ at $Y \ne P$. Prove that $XY \parallel BC$.



Proposed by Daniel Hu
2018 ELMO Shortlist
Determine all nonempty finite sets of positive integers $\{a_1, \dots, a_n\}$ such that $a_1 \cdots a_n$ divides $(x + a_1) \cdots (x + a_n)$ for every positive integer $x$.



Proposed by Ankan Bhattacharya
2018 ELMO Shortlist
Call a number $n$ good if it can be expressed as $2^x+y^2$ for where $x$ and $y$ are nonnegative integers.

(a) Prove that there exist infinitely many sets of $4$ consecutive good numbers.

(b) Find all sets of $5$ consecutive good numbers.



Proposed by Michael Ma
2018 ELMO Shortlist
Say a positive integer $n>1$ is $d$-coverable if for each non-empty subset $S\subseteq \{0, 1, \ldots, n-1\}$, there exists a polynomial  $P$ with integer coefficients and degree at most $d$ such that $S$ is exactly the set of residues modulo $n$ that $P$ attains as it ranges over the integers. For each $n$, find the smallest $d$ such that $n$ is $d$-coverable, or prove no such $d$ exists.



Proposed by Carl Schildkraut
2017 ELMO Shortlist
Let $0<k<\frac{1}{2}$ be a real number and let $a_0, b_0$ be arbitrary real numbers in $(0,1)$. The sequences $(a_n)_{n\ge 0}$ and $(b_n)_{n\ge 0}$ are then defined recursively by



$$a_{n+1} = \dfrac{a_n+1}{2} \text{ and } b_{n+1} = b_n^k$$

for $n\ge 0$. Prove that $a_n<b_n$ for all sufficiently large $n$.



Proposed by Michael Ma
2017 ELMO Shortlist
Let $m$ and $n$ be fixed distinct positive integers. A wren is on an infinite board indexed by $\mathbb Z^2$, and from a square $(x,y)$ may move to any of the eight squares $(x\pm m, y\pm n)$ or $(x\pm n, y \pm m)$. For each $\{m,n\}$, determine the smallest number $k$ of moves required to travel from $(0,0)$ to $(1,0)$, or prove that no such $k$ exists.



Proposed by Michael Ren
2017 ELMO Shortlist
Consider a finite binary string $b$ with at least $2017$ ones. Show that one can insert some plus signs in between pairs of digits such that the resulting sum, when performed in base $2$, is equal to a power of two.



Proposed by David Stoner
2017 ELMO Shortlist
There are $n$ MOPpers $p_1,...,p_n$ designing a carpool system to attend their morning class. Each $p_i$'s car fits $\chi (p_i)$ people ($\chi : \{p_1,...,p_n\} \to \{1,2,...,n\}$). A $c$-fair carpool system is an assignment of one or more drivers on each of several days, such that each MOPper drives $c$ times, and all cars are full on each day. (More precisely, it is a sequence of sets $(S_1, ...,S_m)$ such that $|\{k: p_i\in S_k\}|=c$ and $\sum_{x\in S_j} \chi(x) = n$ for all $i,j$. )

Suppose it turns out that a $2$-fair carpool system is possible but not a $1$-fair carpool system. Must $n$ be even?



Proposed by Nathan Ramesh and Palmer Mebane
2017 ELMO Shortlist
Let $ABC$ be a scalene triangle with $\angle A = 60^{\circ}$. Let $E$ and $F$ be the feet of the angle bisectors of $\angle ABC$ and $\angle ACB$, respectively, and let $I$ be the incenter of $\triangle ABC$. Let $P,Q$ be distinct points such that $\triangle PEF$ and $\triangle QEF$ are equilateral. If $O$ is the circumcenter of of $\triangle APQ$, show that $\overline{OI}\perp \overline{BC}$.



Proposed by Vincent Huang
2017 ELMO Shortlist
Call the ordered pair of distinct circles $(\omega, \gamma)$ scribable if there exists a triangle with circumcircle $\omega$ and incircle $\gamma$. Prove that among $n$ distinct circles there are at most $(n/2)^2$ scribable pairs.



Proposed by Daniel Liu
2017 ELMO Shortlist
Let $ABC$ be an acute triangle with incenter $I$ and circumcircle $\omega$. Suppose a circle $\omega_B$ is tangent to $BA,BC$, and internally tangent to $\omega$ at $B_1$, while a circle $\omega_C$ is tangent to $CA, CB$, and internally tangent to $\omega$ at $C_1$. If $B_2, C_2$ are the points opposite to $B,C$ on $\omega$, respectively, and $X$ denotes the intersection of $B_1C_2, B_2C_1$, prove that $XA=XI$.



Proposed by Vincent Huang and Nathan Weckwerth
2017 ELMO Shortlist
For each integer $C>1$ decide whether there exist pairwise distinct positive integers $a_1,a_2,a_3,...$ such that for every $k\ge 1$, $a_{k+1}^k$ divides $C^ka_1a_2...a_k$.



Proposed by Daniel Liu
2014 ELMO Shortlist
In a non-obtuse triangle $ABC$, prove that

$$ \frac{\sin A \sin B}{\sin C} + \frac{\sin B \sin C}{\sin A} + \frac{\sin C \sin A}{ \sin B} \ge \frac 52. $$Proposed by Ryan Alweiss
2014 ELMO Shortlist
Given positive reals $a,b,c,p,q$ satisfying $abc=1$ and $p \geq q$, prove that $$ p \left(a^2+b^2+c^2\right) + q\left( \frac{1}{a} + \frac{1}{b} + \frac{1}{c}\right) \geq (p+q) (a+b+c). $$Proposed by AJ Dennis
2014 ELMO Shortlist
Let $a,b,c,d,e,f$ be positive real numbers.  Given that $def+de+ef+fd=4$, show that $$ ((a+b)de+(b+c)ef+(c+a)fd)^2 \geq\ 12(abde+bcef+cafd).  $$Proposed by Allen Liu
2014 ELMO Shortlist
Let $\mathbb R^\ast$ denote the set of nonzero reals.  Find all functions $f: \mathbb R^\ast \to \mathbb R^\ast$ satisfying $$ f(x^2+y)+1=f(x^2+1)+\frac{f(xy)}{f(x)} $$ for all $x,y \in \mathbb R^\ast$ with $x^2+y\neq 0$.



Proposed by Ryan Alweiss
2014 ELMO Shortlist
Let $a,b,c$ be positive reals such that $a+b+c=ab+bc+ca$. Prove that $$ (a+b)^{ab-bc}(b+c)^{bc-ca}(c+a)^{ca-ab} \ge a^{ca}b^{ab}c^{bc}. $$Proposed by Sammy Luo
2014 ELMO Shortlist
Find all positive integers $n$ with $n \ge 2$ such that the polynomial $$ P(a_1, a_2, ..., a_n) = a_1^n+a_2^n + ... + a_n^n - n a_1 a_2 ... a_n $$ in the $n$ variables $a_1$, $a_2$, $\dots$, $a_n$ is irreducible over the real numbers, i.e. it cannot be factored as the product of two nonconstant polynomials with real coefficients.



Proposed by Yang Liu
2014 ELMO Shortlist
Let $a, b, c$ be positive reals with $a^{2014}+b^{2014}+c^{2014}+abc=4$. Prove that

$$ \frac{a^{2013}+b^{2013}-c}{c^{2013}} + \frac{b^{2013}+c^{2013}-a}{a^{2013}} + \frac{c^{2013}+a^{2013}-b}{b^{2013}} \ge a^{2012}+b^{2012}+c^{2012}. $$Proposed by David Stoner
2014 ELMO Shortlist
Let $a$, $b$, $c$ be positive reals.  Prove that $$ \sqrt{\frac{a^2(bc+a^2)}{b^2+c^2}}+\sqrt{\frac{b^2(ca+b^2)}{c^2+a^2}}+\sqrt{\frac{c^2(ab+c^2)}{a^2+b^2}}\ge a+b+c. $$Proposed by Robin Park
2014 ELMO Shortlist
You have some cyan, magenta, and yellow beads on a non-reorientable circle, and you can perform only the following operations:

1. Move a cyan bead right (clockwise) past a yellow bead, and turn the yellow bead magenta.

2. Move a magenta bead left of a cyan bead, and insert a yellow bead left of where the magenta bead ends up.

3. Do either of the above, switching the roles of the words ``magenta'' and ``left'' with those of ``yellow'' and ``right'', respectively.

4. Pick any two disjoint consecutive pairs of beads, each either yellow-magenta or magenta-yellow, appearing somewhere in the circle, and swap the orders of each pair.

5. Remove four consecutive beads of one color.





Starting with the circle: ``yellow, yellow, magenta, magenta, cyan, cyan, cyan'', determine whether or not you can reach

a) ``yellow, magenta, yellow, magenta, cyan, cyan, cyan'',

b) ``cyan, yellow, cyan, magenta, cyan'',

c) ``magenta, magenta, cyan, cyan, cyan'',

d) ``yellow, cyan, cyan, cyan''.



Proposed by Sammy Luo
2014 ELMO Shortlist
Let $r$ and $b$ be positive integers. The game of Monis, a variant of Tetris, consists of a single column of red and blue blocks.  If two blocks of the same color ever touch each other, they both vanish immediately.  A red block falls onto the top of the column exactly once every $r$ years, while a blue block falls exactly once every $b$ years.



(a) Suppose that $r$ and $b$ are odd, and moreover the cycles are offset in such a way that no two blocks ever fall at exactly the same time.  Consider a period of $rb$ years in which the column is initially empty.  Determine, in terms of $r$ and $b$, the number of blocks in the column at the end.



(b) Now suppose $r$ and $b$ are relatively prime and $r+b$ is odd.  At time $t=0$, the column is initially empty.  Suppose a red block falls at times $t = r, 2r, \dots, (b-1)r$ years, while a blue block falls at times $t = b, 2b, \dots, (r-1)b$ years.  Prove that at time $t=rb$, the number of blocks in the column is $\left\lvert 1+2(r-1)(b+r)-8S \right\rvert$, where $$ S = \left\lfloor \frac{2r}{r+b} \right\rfloor

     + \left\lfloor \frac{4r}{r+b} \right\rfloor

     + ...

     + \left\lfloor \frac{(r+b-1)r}{r+b} \right\rfloor

     . $$

Proposed by Sammy Luo
2014 ELMO Shortlist
Let $n$ be a positive integer. For any $k$, denote by $a_k$ the number of permutations of $\{1,2,\dots,n\}$ with exactly $k$ disjoint cycles. (For example, if $n=3$ then $a_2=3$ since $(1)(23)$, $(2)(31)$, $(3)(12)$ are the only such permutations.) Evaluate

$$ a_n n^n + a_{n-1} n^{n-1} + \dots + a_1 n. $$Proposed by Sammy Luo
2014 ELMO Shortlist
Let $f_0$ be the function from $\mathbb{Z}^2$ to $\{0,1\}$ such that $f_0(0,0)=1$ and $f_0(x,y)=0$ otherwise. For each positive integer $m$, let $f_m(x,y)$ be the remainder when $$ f_{m-1}(x,y) + \sum_{j=-1}^{1} \sum_{k=-1}^{1} f_{m-1}(x+j,y+k) $$ is divided by $2$.

Finally, for each nonnegative integer $n$, let $a_n$ denote the number of pairs $(x,y)$ such that $f_n(x,y) = 1$.

Find a closed form for $a_n$.



Proposed by Bobby Shen
2014 ELMO Shortlist
Let $ABC$ be a triangle with symmedian point $K$. Select a point $A_1$ on line $BC$ such that the lines $AB$, $AC$, $A_1K$ and $BC$ are the sides of a cyclic quadrilateral.  Define $B_1$ and $C_1$ similarly.  Prove that $A_1$, $B_1$, and $C_1$ are collinear.



Proposed by Sammy Luo
2014 ELMO Shortlist
$ABCD$ is a cyclic quadrilateral inscribed in the circle $\omega$. Let $AB \cap CD = E$, $AD \cap BC = F$. Let $\omega_1, \omega_2$ be the circumcircles of $AEF, CEF$, respectively. Let $\omega \cap \omega_1 = G$, $\omega \cap \omega_2 = H$. Show that $AC, BD, GH$ are concurrent.



Proposed by Yang Liu
2014 ELMO Shortlist
Let $A_1A_2A_3 \cdots A_{2013}$ be a cyclic $2013$-gon. Prove that for every point $P$ not the circumcenter of the $2013$-gon, there exists a point $Q\neq P$ such that $\frac{A_iP}{A_iQ}$ is constant for $i \in \{1, 2, 3, \cdots, 2013\}$.



Proposed by Robin Park
2014 ELMO Shortlist
Let $ABCD$ be a quadrilateral inscribed in circle $\omega$. Define $E = AA \cap CD$, $F = AA \cap BC$, $G = BE \cap \omega$, $H = BE \cap AD$, $I = DF \cap \omega$, and $J = DF \cap AB$. Prove that $GI$, $HJ$, and the $B$-symmedian are concurrent.



Proposed by Robin Park
2014 ELMO Shortlist
Let $P$ be a point in the interior of an acute triangle $ABC$, and let $Q$ be its isogonal conjugate.  Denote by $\omega_P$ and $\omega_Q$ the circumcircles of triangles $BPC$ and $BQC$, respectively.  Suppose the circle with diameter $\overline{AP}$ intersects $\omega_P$ again at $M$, and line $AM$ intersects $\omega_P$ again at $X$.  Similarly, suppose the circle with diameter $\overline{AQ}$ intersects $\omega_Q$ again at $N$, and line $AN$ intersects $\omega_Q$ again at $Y$.



Prove that lines $MN$ and $XY$ are parallel.



(Here, the points $P$ and $Q$ are isogonal conjugates with respect to $\triangle ABC$ if the internal angle bisectors of $\angle BAC$, $\angle CBA$, and $\angle ACB$ also bisect the angles $\angle PAQ$, $\angle PBQ$, and $\angle PCQ$, respectively.  For example, the orthocenter is the isogonal conjugate of the circumcenter.)



Proposed by Sammy Luo
2014 ELMO Shortlist
Let $ABCD$ be a cyclic quadrilateral with center $O$.

Suppose the circumcircles of triangles $AOB$ and $COD$ meet again at $G$, while the circumcircles of triangles $AOD$ and $BOC$ meet again at $H$.

Let $\omega_1$ denote the circle passing through $G$ as well as the feet of the perpendiculars from $G$ to $AB$ and $CD$.

Define $\omega_2$ analogously as the circle passing through $H$ and the feet of the perpendiculars from $H$ to $BC$ and $DA$.

Show that the midpoint of $GH$ lies on the radical axis of $\omega_1$ and $\omega_2$.



Proposed by Yang Liu
2014 ELMO Shortlist
Let $ABC$ be a triangle with circumcenter $O$. Let $P$ be a point inside $ABC$, so let the points $D, E, F$ be on $BC, AC, AB$ respectively so that the Miquel point of $DEF$ with respect to $ABC$ is $P$. Let the reflections of $D, E, F$ over the midpoints of the sides that they lie on be $R, S, T$. Let the Miquel point of $RST$ with respect to the triangle $ABC$ be $Q$. Show that $OP = OQ$.



Proposed by Yang Liu
2014 ELMO Shortlist
In triangle $ABC$ with incenter $I$ and circumcenter $O$, let $A',B',C'$ be the points of tangency of its circumcircle with its $A,B,C$-mixtilinear circles, respectively.  Let $\omega_A$ be the circle through $A'$ that is tangent to $AI$ at $I$, and define $\omega_B, \omega_C$ similarly. Prove that $\omega_A,\omega_B,\omega_C$ have a common point $X$ other than $I$, and that $\angle AXO = \angle OXA'$.



Proposed by Sammy Luo
2014 ELMO Shortlist
Let $P$ be a point inside a triangle $ABC$ such that $\angle PAC= \angle PCB$. Let the projections of $P$ onto $BC$, $CA$, and $AB$ be $X,Y,Z$ respectively. Let $O$ be the circumcenter of $\triangle XYZ$, $H$ be the foot of the altitude from $B$ to $AC$, $N$ be the midpoint of $AC$, and $T$ be the point such that $TYPO$ is a parallelogram.  Show that $\triangle THN$ is similar to $\triangle PBC$.



Proposed by Sammy Luo
2014 ELMO Shortlist
We are given triangles $ABC$ and $DEF$ such that $D\in BC, E\in CA, F\in AB$, $AD\perp EF, BE\perp FD, CF\perp DE$.  Let the circumcenter of $DEF$ be $O$, and let the circumcircle of $DEF$ intersect $BC,CA,AB$ again at $R,S,T$ respectively. Prove that the perpendiculars to $BC,CA,AB$ through $D,E,F$ respectively intersect at a point $X$, and the lines $AR,BS,CT$ intersect at a point $Y$, such that $O,X,Y$ are collinear.



Proposed by Sammy Luo
2014 ELMO Shortlist
Let $AB=AC$ in $\triangle ABC$, and let $D$ be a point on segment $AB$. The tangent at $D$ to the circumcircle $\omega$ of $BCD$ hits $AC$ at $E$. The other tangent from $E$ to $\omega$ touches it at $F$, and $G=BF \cap CD$, $H=AG \cap BC$. Prove that $BH=2HC$.



Proposed by David Stoner
2014 ELMO Shortlist
Let $ABC$ be a nondegenerate acute triangle with circumcircle $\omega$ and let its incircle $\gamma$ touch $AB, AC, BC$ at $X, Y, Z$ respectively. Let $XY$ hit arcs $AB, AC$ of $\omega$ at $M, N$ respectively, and let $P \neq X, Q \neq Y$ be the points on $\gamma$ such that $MP=MX,  NQ=NY$. If $I$ is the center of $\gamma$, prove that $P, I, Q$ are collinear if and only if $\angle BAC=90^\circ$.



Proposed by David Stoner
2014 ELMO Shortlist
Does there exist a strictly increasing infinite sequence of perfect squares $a_1, a_2, a_3, ...$ such that for all $k\in \mathbb{Z}^+$ we have that $13^k | a_k+1$?



Proposed by Jesse Zhang