Navvye/Olympiad-Problems-CSV huggingface dataset:
2020 Korea - Final Round
Let $ABCD$ be an isosceles trapezoid such that $AB \parallel CD$ and $\overline{AD}=\overline{BC}, \overline{AB}>\overline{CD}$. Let $E$ be a point such that $\overline{EC}=\overline{AC}$ and $EC \perp BC$, and $\angle ACE<90^{\circ}$. Let $\Gamma$ be a circle with center $D$ and radius $DA$, and $\Omega$ be the circumcircle of triangle $AEB$. Suppose that $\Gamma$ meets $\Omega$ again at $F(\neq A)$, and let $G$ be a point on $\Gamma$ such that $\overline{BF}=\overline{BG}$.

Prove that the lines $EG, BD$ meet on $\Omega$.
2020 Korea - Final Round
There are $2020$ groups, each of which consists of a boy and a girl, such that each student is contained in exactly one group. Suppose that the students shook hands so that the following conditions are satisfied:



boys didn't shake hands with boys, and girls didn't shake hands with girls;

in each group, the boy and girl had shake hands exactly once;

any boy and girl, each in different groups, didn't shake hands more than once;

for every four students in two different groups, there are at least three handshakes.



Prove that one can pick $4038$ students and arrange them at a circular table so that every two adjacent students had shake hands.
2020 Korea - Final Round
Find all $f: \mathbb{Q}_{+} \rightarrow \mathbb{R}$ such that $$ f(x)+f(y)+f(z)=1 $$holds for every positive rationals $x, y, z$ satisfying $x+y+z+1=4xyz$.
2020 Korea - Final Round
Do there exist two positive reals $\alpha, \beta$ such that each positive integer appears exactly once in the following sequence? $$ 2020, [\alpha], [\beta], 4040, [2\alpha], [2\beta], 6060, [3\alpha], [3\beta], \cdots $$If so, determine all such pairs; if not, prove that it is impossible.
2020 Korea - Final Round
Let $ABC$ be an acute triangle such that $\overline{AB}=\overline{AC}$. Let $M, L, N$ be the midpoints of segment $BC, AM, AC$, respectively. The circumcircle of triangle $AMC$, denoted by $\Omega$, meets segment $AB$ at $P(\neq A)$, and the segment $BL$ at $Q$. Let $O$ be the circumcenter of triangle $BQC$. Suppose that the lines $AC$ and $PQ$ meet at $X$, $OB$ and $LN$ meet at $Y$, and $BQ$ and $CO$ meets at $Z$. Prove that the points $X, Y, Z$ are collinear.
2020 Korea - Final Round
Find all positive integers $n$ such that $6(n^4-1)$ is a square of an integer.
2019 Korea - Final Round
There are $n$ cards such that for each $i=1,2, \cdots n$, there are exactly one card labeled $i$. Initially the cards are piled with increasing order from top to bottom. There are two operations:



$A$ : One can take the top card of the pile and move it to the bottom;



$B$ : One can remove the top card from the pile.



The operation $ABBABBABBABB \cdots $ is repeated until only one card gets left. Let $L(n)$ be the labeled number on the final pile. Find all integers $k$ such that $L(3k)=k$.
2019 Korea - Final Round
For a rectangle $ABCD$ which is not a square, there is $O$ such that $O$ is on the perpendicular bisector of $BD$ and $O$ is in the interior of $\triangle BCD$. Denote by $E$ and $F$ the second intersections of the circle centered at $O$ passing through $B, D$ and $AB, AD$. $BF$ and $DE$ meets at $G$, and $X, Y, Z$ are the foots of the perpendiculars from $G$ to $AB, BD, DA$. $L, M, N$ are the foots of the perpendiculars from $O$ to $CD, BD, BC$. $XY$ and $ML$ meets at $P$, $YZ$ and $MN$ meets at $Q$. Prove that $BP$ and $DQ$ are parallel.
2019 Korea - Final Round
Prove that there exist infinitely many positive integers $k$ such that the sequence $\{x_n\}$ satisfying



$$ x_1=1, x_2=k+2, x_{n+2}-(k+1)x_{n+1}+x_n=0(n \ge 0)$$

does not contain any prime number.
2019 Korea - Final Round
Let triangle $ABC$ be an acute scalene triangle with orthocenter $H$. The foot of perpendicular from $A$ to $BC$ is $O$, and denote $K,L$ by the midpoints of $AB, AC$, respectively. For a point $D(\neq O,B,C)$ on segment $BC$, let $E,F$ be the orthocenters of triangles $ABD, ACD$, respectively, and denote $M,N$ by the midpoints of $DE,DF$. The perpendicular line from $M$ to $KH$ cuts the perpendicular line from $N$ to $LH$ at $P$. If $Q$ is the midpoint of $EF$, and $S$ is the orthocenter of triangle $HPQ$, then prove that as $D$ varies on $BC$, the ratio $\frac{OS}{OH}$, $\frac{OQ}{OP}$ remains constant.
2019 Korea - Final Round
Find all pairs $(p,q)$ such that the equation $$x^4+2px^2+qx+p^2-36=0$$has exactly $4$ integer roots(counting multiplicity).
2019 Korea - Final Round
A sequence $\{x_n \}=x_0, x_1, x_2, \cdots $ satisfies $x_0=a(1\le a \le 2019, a \in \mathbb{R})$, and $$x_{n+1}=\begin{cases}1+1009x_n &\ (x_n \le 2) \\ 2021-x_n &\ (2<x_n \le 1010) \\ 3031-2x_n &\ (1010<x\le 1011) \\ 2020-x_n &\ (1011<x_n) \end{cases}$$for each non-negative integer $n$. If there exist some integer $k>1$ such that $x_k=a$, call such minimum $k$ a  fundamental period of $\{x_n \}$. Find all integers which can be a fundamental period of some seqeunce; and for such minimal odd period $k(>1)$, find all values of $x_0=a$ such that the fundamental period of $\{x_n \}$ equals $k$.
2018 Korea - Final Round
Find all integers of the form $\frac{m-6n}{m+2n}$ where $m,n$ are nonzero rational numbers satisfying $m^3=(27n^2+1)(m+2n)$.
2018 Korea - Final Round
Triangle $ABC$ satisfies $\angle ABC < \angle BCA < \angle CAB < 90^{\circ}$. $O$ is the circumcenter of triangle $ABC$, and $K$ is the reflection of $O$ in $BC$. $D,E$ is the foot of perpendicular line from $K$ to line $AB$, $AC$, respectively. Line $DE$ meets $BC$ at $P$, and a circle with diameter $AK$ meets the circumcircle of triangle $ABC$ at $Q(\neq A)$. If   $PQ$ cuts the perpendicular bisector of $BC$ at $S$, then prove that $S$ lies on the circle with diameter $AK$.
2018 Korea - Final Round
For 31 years, n (>6) tennis players have records of wins. It turns out that for every two players, there is a third player who has won over them before. Prove that for every integer $k,l$ such that $2^{2^k+1}-1>n, 1<l<2k+1$, there exist $l$ players ($A_1, A_2, ... , A_l$) such that every player $A_{i+1}$ won over $A_i$. ($A_{l+1}$ is same as $A_1$)
2018 Korea - Final Round
Triangle $ABC$ satisfies $\angle C=90^{\circ}$. A circle passing $A,B$ meets segment $AC$ at $G(\neq A,C)$ and it meets segment $BC$ at point $D(\neq B)$. Segment $AD$ cuts segment $BG$ at $H$, and let $l$, the perpendicular bisector of segment $AD$, cuts the perpendicular bisector of segment $AB$ at point $E$. A line passing $D$ is perpendicular to $DE$ and cuts $l$ at point $F$. If the circumcircle of triangle $CFH$ cuts $AC$, $BC$ at $P(\neq C),Q(\neq C)$ respectively, then prove that $PQ$ is perpendicular to $FH$.
2018 Korea - Final Round
Determine whether or not two polynomials $P, Q$ with degree no less than 2018 and with integer coefficients exist such that $$P(Q(x))=3Q(P(x))+1$$for all real numbers $x$.
2018 Korea - Final Round
Twenty ants live on the faces of an icosahedron, one ant on each side, where the icosahedron have each side with length 1. Each ant moves in a counterclockwise direction on each face, along the side/edges. The speed of each ant must be no less than 1 always. Also, if two ants meet, they should meet at the vertex of the icosahedron. If five ants meet at the same time at a vertex, we call that a collision. Can the ants move forever, in a way that no collision occurs?
2017 Korea - Final Round
A acute triangle $\triangle ABC$ has circumcenter $O$. The circumcircle of $OAB$, called $O_1$, and the circumcircle of $OAC$, called $O_2$, meets $BC$ again at $D ( \not=B )$ and $E ( \not= C )$ respectively. The perpendicular bisector of $BC$ hits $AC$ again at $F$. Prove that the circumcenter of $\triangle ADE$ lies on $AC$ if and only if the centers of $O_1, O_2$ and $F$ are colinear.
2017 Korea - Final Round
For a positive integer $n$, $(a_0, a_1, \cdots , a_n)$ is a $n+1$-tuple with integer entries.

For all $k=0, 1, \cdots , n$, we denote $b_k$ as the number of $k$s in $(a_0, a_1, \cdots ,a_n)$.

For all $k = 0,1, \cdots , n$, we denote $c_k$ as the number of $k$s in $(b_0, b_1, \cdots ,b_n)$.

Find all $(a_0, a_1, \cdots ,a_n)$ which satisfies $a_0 = c_0$, $a_1=c_1$, $\cdots$, $a_n=c_n$.
2017 Korea - Final Round
For a positive integer $n$, denote $c_n=2017^n$. A function $f: \mathbb{N} \rightarrow \mathbb{R}$ satisfies the following two conditions.



1. For all positive integers $m, n$, $f(m+n) \le 2017 \cdot f(m) \cdot f(n+325)$.



2. For all positive integer $n$, we have $0<f(c_{n+1})<f(c_n)^{2017}$.



Prove that there exists a sequence $a_1, a_2, \cdots $ which satisfies the following.

For all $n, k$ which satisfies $a_k<n$, we have $f(n)^{c_k} < f(c_k)^n$.
2017 Korea - Final Round
For a positive integer $n \ge 2$, define a sequence $a_1, a_2, \cdots ,a_n$ as the following.



$$ a_1 = \frac{n(2n-1)(2n+1)}{3}$$$$a_k = \frac{(n+k-1)(n-k+1)}{2(k-1)(2k+1)}a_{k-1}, \text{      } (k=2,3, \cdots n)$$

(a) Show that $a_1, a_2, \cdots a_n$ are all integers.



(b) Prove that there are exactly one number out of $a_1, a_2, \cdots a_n$ which is not a multiple of $2n-1$ and exactly one number out of $a_1, a_2, \cdots a_n$ which is not a multiple of $2n+1$ if and only if $2n-1$ and $2n+1$ are all primes.
2017 Korea - Final Round
Let there be cyclic quadrilateral $ABCD$ with $L$ as the midpoint of $AB$ and $M$ as the midpoint of $CD$. Let $AC \cap BD = E$, and let rays $AB$ and $DC$ meet again at $F$. Let $LM \cap DE = P$. Let $Q$ be the foot of the perpendicular from $P$ to $EM$. If the orthocenter of $\triangle FLM$ is $E$, prove the following equality.



$$\frac{EP^2}{EQ} = \frac{1}{2} \left( \frac{BD^2}{DF} - \frac{BC^2}{CF} \right)$$
2017 Korea - Final Round
A room has $2017$ boxes in a circle. A set of boxes is friendly if there are at least two boxes in the set, and for each boxes in the set, if we go clockwise starting from the box, we would pass either $0$ or odd number of boxes before encountering a new box in the set. $30$ students enter the room and picks a set of boxes so that the set is friendly, and each students puts a letter inside all of the boxes that he/she chose. If the set of the boxes which have $30$ letters inside is not friendly, show that there exists two students $A, B$ and boxes $a, b$ satisfying the following condition.



(i). $A$ chose $a$ but not $b$, and $B$ chose $b$ but not $a$.



(ii). Starting from $a$ and going clockwise to $b$, the number of boxes that we pass through, not including $a$ and $b$, is not an odd number, and none of $A$ or $B$ chose such boxes that we passed.
2016 Korea - Final Round
In a acute triangle $\triangle ABC$, denote $D, E$ as the foot of the perpendicular from $B$ to $AC$ and $C$ to $AB$.

Denote the reflection of $E$ with respect to $AC, BC$ as $S, T$.

The circumcircle of $\triangle CST$ hits $AC$ at point $X (\not= C)$.

Denote the circumcenter of $\triangle CST$ as $O$. Prove that $XO \perp DE$.
2016 Korea - Final Round
Two integers $n, k$ satisfies $n \ge 2$ and $k \ge \frac{5}{2}n-1$.

Prove that whichever $k$ lattice points with $x$ and $y$ coordinate no less than $1$ and no more than $n$ we pick, there must be a circle passing through at least four of these points.
2016 Korea - Final Round
Prove that for all rationals $x,y$, $x-\frac{1}{x}+y-\frac{1}{y}=4$ is not true.
2016 Korea - Final Round
If $x,y,z$ satisfies $x^2+y^2+z^2=1$, find the maximum possible value of $$(x^2-yz)(y^2-zx)(z^2-xy)$$
2016 Korea - Final Round
An acute triangle $\triangle ABC$ has incenter $I$, and the incircle hits $BC, CA, AB$ at $D, E, F$.

Lines $BI, CI, BC, DI$ hits $EF$ at $K, L, M, Q$ and the line connecting the midpoint of segment $CL$ and $M$ hits the line segment $CK$ at $P$. Prove that $$PQ=\frac{AB \cdot KQ}{BI}$$
2016 Korea - Final Round
Let $U$ be a set of $m$ triangles. Prove that there exists a subset $W$ of $U$ which satisfies the following.



(i). The number of triangles in $W$ is at least $0.45m^{\frac{4}{5}}$



(ii) There are no points $A, B, C, D, E, F$ such that triangles $ABC$, $BCD$, $CDE$, $DEF$, $EFA$, $FAB$ are all in $W$.
2015 Korea - Final Round
Find all functions $f: R \rightarrow R$ such that

$f(x^{2015} + (f(y))^{2015}) = (f(x))^{2015} + y^{2015}$ holds for all reals $x, y$
2015 Korea - Final Round
In a triangle $\triangle ABC$ with incenter $I$, the incircle meets lines $BC, CA, AB$ at $D, E, F$ respectively.

Define the circumcenter of $\triangle IAB$ and $\triangle IAC$ $O_1$ and $O_2$ respectively.

Let the two intersections of the circumcircle of $\triangle ABC$ and line $EF$ be $P, Q$.

Prove that the circumcenter of $\triangle DPQ$ lies on the line $O_1O_2$.
2015 Korea - Final Round
There are at least $3$ subway stations in a city.

In this city, there exists a route that passes through more than $L$ subway stations, without revisiting.

Subways run both ways, which means that if you can go from subway station A to B, you can also go from B to A.

Prove that at least one of the two holds.



$\text{(i)}$. There exists three subway stations $A$, $B$, $C$ such that there does not exist a route from $A$ to $B$ which doesn't pass through $C$.



$\text{(ii)}$. There is a cycle passing through at least $\lfloor \sqrt{2L} \rfloor$ stations, without revisiting a same station more than once.
2015 Korea - Final Round
$\triangle ABC$ is an acute triangle and its orthocenter is $H$.

The circumcircle of $\triangle ABH$ intersects line $BC$ at $D$.

Lines $DH$ and $AC$ meets at $P$, and the circumcenter of $\triangle ADP$ is $Q$.

Prove that the circumcenter of $\triangle ABH$ lies on the circumcircle of $\triangle BDQ$.
2015 Korea - Final Round
For a fixed positive integer $k$, there are two sequences $A_n$ and $B_n$.

They are defined inductively, by the following recurrences.

$A_1 = k$, $A_2 = k$, $A_{n+2} = A_{n}A_{n+1}$

$B_1 = 1$, $B_2 = k$, $B_{n+2} = \frac{B^3_{n+1}+1}{B_{n}}$

Prove that for all positive integers $n$, $A_{2n}B_{n+3}$ is an integer.
2015 Korea - Final Round
There are $2015$ distinct circles in a plane, with radius $1$.

Prove that you can select $27$ circles, which form a set $C$, which satisfy the following.



For two arbitrary circles in $C$, they intersect with each other or

For two arbitrary circles in $C$, they don't intersect with each other.
2014 Korea - Final Round
Suppose $x$, $y$, $z$ are positive numbers such that $x+y+z=1$. Prove that

$$

\frac{(1+xy+yz+zx)(1+3x^3 + 3y^3 + 3z^3)}{9(x+y)(y+z)(z+x)}

\ge 

\left(

	\frac{x \sqrt{1+x} }{\sqrt[4]{3+9x^2}}

	+ \frac{y \sqrt{1+y} }{\sqrt[4]{3+9y^2}}

	+ \frac{z \sqrt{1+z}}{\sqrt[4]{3+9z^2}}

\right)^2. $$
2014 Korea - Final Round
Let $ABC$ be a isosceles triangle with $ AC = BC > AB$. Let $ E, F $ be the midpoints of segments $ AC, AB$, and let $l$ be the perpendicular bisector of $AC$. Let $ l $ meets $ AB$ at $K$, the line through $B$ parallel to $KC$ meets $AC$ at point $L$, and line $FL$ meets $ l$ at $W$. Let $ P $ be a point on segment $BF$. Let $H$ be the orthocenter of triangle $ACP$ and line $BH$ and $CP$ meet at point $J$. Line $FJ$ meets $l$ at $M$. Prove that $ AW = PW $ if and only if $B$ lies on the circumcircle of $EFM$.
2014 Korea - Final Round
There are $n$ students sitting on a round table. You collect all of $ n $ name tags and give them back arbitrarily. Each student gets one of $n$ name tags. Now $n$ students repeat following operation:

The students who have their own name tags exit the table. The other students give their name tags to the student who is sitting right to him.

Find the number of ways giving name tags such that there exist a student who don't exit the table after 4 operations.
2014 Korea - Final Round
Let $ ABC $ be a isosceles triangle with $ AC=BC$. Let $ D $ a point on a line $ BA $ such that $ A $ lies between $ B, D $. Let $O_1 $ be the circumcircle of triangle $ DAC $. $ O_1 $ meets $ BC $ at point $ E $. Let $ F $ be the point on $ BC $ such that $ FD $ is tangent to circle $O_1 $, and let $O_2 $ be the circumcircle of $ DBF$. Two circles $O_1 , O_2 $ meet at point $ G ( \ne D) $. Let $ O $ be the circumcenter of triangle $ BEG$. Prove that the line $FG$ is tangent to circle $O$ if and only if $ DG \bot FO$.
2014 Korea - Final Round
Let $p>5$ be a prime. Suppose that there exist integer $k$ such that $ k^2 + 5 $ is divisible by $p$. Prove that there exist two positive integers $m,n$ satisfying $ p^2 = m^2 + 5n^2 $.
2014 Korea - Final Round
In an island there are $n$ castles, and each castle is in country $A$ or $B$. There is one commander per castle, and each commander belongs to the same country as the castle he's initially in. There are some (two-way) roads between castles (there may be roads between castles of different countries), and call two castles adjacent if there is a road between them.

Prove that the following two statements are equivalent:

(1) If some commanders from country $B$ move to attack an adjacent castle in country $A$, some commanders from country $A$ could appropriately move in defense to adjacent castles in country $A$ so that in every castle of country $A$, the number of country $A$'s commanders defending that castle is not less than the number of country $B$'s commanders attacking that castle. (Each commander can defend or attack only one castle at a time.)

(2) For any arbitrary set $X$ of castles in country $A$, the number of country $A$'s castles that are in $X$ or adjacent to at least one of the castle in $X$ is not less than the number of country $B$'s castles that are adjacent to at least one of the castles in $X$.
2013 Korea - Final Round
For a triangle $ \triangle ABC  (\angle B > \angle C) $, $ D $ is a point on $ AC $ satisfying $ \angle ABD = \angle C $. Let $ I $ be the incenter of $ \triangle ABC $, and circumcircle of $ \triangle CDI $ meets $ AI $ at $ E ( \ne I )$. The line passing $ E $ and parallel to $ AB $ meets the line $ BD $ at $ P $. Let $ J $ be the incenter of $ \triangle ABD $, and $ A' $ be the point such that $ AI = IA' $. Let $ Q $ be the intersection point of $ JP $ and $ A'C $. Prove that $ QJ = QA' $.
2013 Korea - Final Round
Find all functions $ f : \mathbb{R}\to\mathbb{R}$ satisfying following conditions.

(a) $ f(x) \ge 0 $ for all $ x \in \mathbb{R} $.

(b) For $ a, b, c, d \in \mathbb{R} $ with $ ab + bc + cd = 0 $, equality $ f(a-b) + f(c-d) = f(a) + f(b+c) + f(d) $ holds.
2013 Korea - Final Round
For a positive integer $n \ge 2 $, define set $ T = \{ (i,j) | 1 \le i < j \le n , i | j \} $. For nonnegative real numbers $ x_1 , x_2 , \cdots , x_n $ with $ x_1 + x_2 + \cdots + x_n = 1 $, find the maximum value of

$$ \sum_{(i,j) \in T} x_i x_j $$

in terms of $n$.
2013 Korea - Final Round
For a triangle $ ABC $, let $ B_1 ,C_1 $ be the excenters of $ B, C $. Line $B_1 C_1 $ meets with the circumcircle of $ \triangle ABC $ at point $ D (\ne A) $. $ E $ is the point which satisfies $ B_1 E \bot CA $ and $ C_1 E \bot AB $. Let $ w $ be the circumcircle of $ \triangle ADE $. The tangent to the circle $ w $ at $ D $ meets $ AE $ at $ F $. $ G , H $ are the points on $ AE, w $ such that $ DGH \bot AE $. The circumcircle of $ \triangle HGF $ meets $ w $ at point $ I ( \ne H ) $, and $ J $ be the foot of perpendicular from $ D $ to $ AH $. Prove that $ AI $ passes the midpoint of $ DJ $.
2013 Korea - Final Round
Two coprime positive integers $ a, b $ are given. Integer sequence $ \{ a_n \}, \{b_n \} $ satisties

$$ (a+b \sqrt2 )^{2n} = a_n + b_n \sqrt2 $$

Find all prime numbers $ p $ such that there exist positive integer $ n \le p $ satisfying $ p | b_n $.
2013 Korea - Final Round
For any permutation $ f : \{ 1, 2, \cdots , n \} \to \{1, 2, \cdots , n \} $, and define

$$ A = \{ i | i > f(i) \} $$

$$ B = \{ (i, j) | i<j \le f(j) < f(i) \ or \ f(j) < f(i) < i < j \} $$

$$ C = \{ (i, j) | i<j \le f(i) < f(j) \  or \  f(i) < f(j) < i < j \} $$

$$ D = \{ (i, j) | i< j \  and  \ f(i) > f(j)\} $$

Prove that $ |A| + 2|B| + |C| = |D| $.
2012 Korea - Final Round
Let $ x, y, z $ be positive real numbers. Prove that

$$ \frac{2x^2 + xy}{(y+ \sqrt{zx} + z )^2} + \frac{2y^2 + yz}{(z+ \sqrt{xy} + x )^2} + \frac{2z^2 + zx}{(x+ \sqrt{yz} +y )^2} \ge 1 $$
2012 Korea - Final Round
For a triangle $ ABC $ which $ \angle B \ne 90^{\circ} $ and $ AB \ne AC $, define $ P_{ABC} $ as follows ;



Let $ I $ be the incenter of triangle $ABC$, and let  $ D, E, F $ be the intersection points with the incircle and segments $ BC, CA, AB $. Two lines $ AB $ and $ DI $ meet at  $ S $ and let $ T $ be the intersection point of line $ DE $ and the line which is perpendicular with $ DF $ at $ F $. The line $ ST $ intersects line $ EF $ at $ R$. Now define $ P_{ABC} $ be one of the intersection points of the incircle and the circle with diameter $ IR $, which is located in other side with $ A $ about $ IR $.



Now think of an isosceles triangle $ XYZ $ such that $ XZ = YZ > XY $. Let $ W $ be the point on the side $ YZ $ such that $ WY < XY $ and Let $ K = P_{YXW} $ and $ L = P_{ZXW} $. Prove that $ 2 KL \le XY $.
2012 Korea - Final Round
$ A_1 , A_2 , \cdots , A_n $ are given subsets. Let $ S = \left\{ 1, 2, \cdots , n \right\} $. For any $ X \subset S $, let

$$ N(X)= \left\{ i \in S-X \ | \  \forall j \in X, \  A_i \cap A_j \ne \emptyset \right\} $$

Let $ m $ be an integer such that $ 3 \le m \le n-2 $. Prove that there exist $ X \subset S $ such that $ |X|=m $ and $ |N(X)| \ne 1 $.
2012 Korea - Final Round
Let $ABC$ be an acute triangle. Let $ H $ be the foot of perpendicular from $ A $ to $ BC $. $ D, E $ are the points on $ AB, AC $ and let $ F, G $ be the foot of perpendicular from $ D, E $ to $ BC $. Assume that $ DG \cap EF $ is on $ AH $. Let $ P $ be the foot of perpendicular from $ E $ to $ DH $. Prove that $ \angle APE = \angle CPE $.
2012 Korea - Final Round
Let $n$ be a given positive integer. Prove that there exist infinitely many integer triples $(x,y,z)$ such that

$$nx^2+y^3=z^4,\ \gcd (x,y)=\gcd (y,z)=\gcd (z,x)=1.$$
2012 Korea - Final Round
Let $M$ be the set of positive integers which do not have a prime divisor greater than 3. For any infinite family of subsets of $M$,  say $A_1,A_2,\ldots $, prove that there exist $i\ne j$ such that for each $x\in A_i$  there exists some $y\in A_j $ such that $y\mid x$.
2011 Korea - Final Round
Prove that there is no positive integers $x,y,z$ satisfying

$$ x^2 y^4 - x^4 y^2 + 4x^2 y^2 z^2 +x^2 z^4 -y^2 z^4 =0 $$
2011 Korea - Final Round
$ABC$ is an acute triangle. $P$(different from $B,C$) is a point on side $BC$.

$H$ is an orthocenter, and $D$ is a foot of perpendicular from $H$ to $AP$.

The circumcircle of the triangle $ABD$ and $ACD$ is $O _1$ and $O_2$, respectively.

A line $l$ parallel to $BC$ passes $D$ and meet $O_1$ and $O_2$ again at $X$ and $Y$, respectively.

$l$ meets $AB$ at $E$, and $AC$ at $F$. Two lines $XB$ and $YC$ intersect at $Z$.

Prove that $ZE=ZF$ is a necessary and sufficient condition for $BP=CP$.
2011 Korea - Final Round
There are $n$ boys $a_1, a_2, \ldots, a_n$ and $n$ girls $b_1, b_2, \ldots, b_n $. Some pairs of them are connected. Any two boys or two girls are not connected, and $a_i$ and $b_i$ are not connected for all $i \in \{ 1,2,\ldots,n\}$. Now all boys and girls are divided into several groups satisfying two conditions:

(i) Every groups contains an equal number of boys and girls.

(ii) There is no connected pair in the same group.

Assume that the number of connected pairs is $m$. Show that we can make the number of groups not larger than $\max\left \{2, \dfrac{2m}{n} +1\right \}$.
2011 Korea - Final Round
Find the maximal value of the following expression, if $a,b,c$ are nonnegative and $a+b+c=1$.

$$ \frac{1}{a^2 -4a+9} + \frac {1}{b^2 -4b+9} + \frac{1}{c^2 -4c+9} $$
2011 Korea - Final Round
$ABC$ is a triangle such that $AC<AB<BC$ and $D$ is a point on side $AB$ satisfying $AC=AD$.

The circumcircle of $ABC$ meets with the bisector of angle $A$ again at $E$ and meets with $CD$ again at $F$.

$K$ is an intersection point of $BC$ and $DE$. Prove that $CK=AC$ is a necessary and sufficient condition for $DK \cdot EF = AC \cdot DF$.