Navvye/Olympiad-Problems-CSV huggingface dataset:
1989 ITAMO
Given a real number $\alpha$, a function $f$ is defined on pairs of nonnegative integers by

$f(0,0) = 1, f(m,0) = f(0,m) = 0$ for $m > 0$,

$f(m,n) = \alpha f(m,n-1)+(1- \alpha)f(m -1,n-1)$ for $m,n > 0$.

Find the values of $\alpha$ such that $| f(m,n)| < 1989$ holds for any integers $m,n \ge 0$.
1988 ITAMO
Players $A$ and $B$ play the following game: $A$ tosses a coin $n$ times, and $B$ does $n+1$ times. The player who obtains more ”heads” wins; or in the case of equal balances, $A$ is assigned victory. Find the values of $n$ for which this game is fair (i.e. both players have equal chances for victory).
1988 ITAMO
In a basketball tournament any two of the $n$ teams $S_1,S_2,...,S_n$ play one match (no draws).

Denote by $v_i$ and $p_i$ the number of victories and defeats of team $S_i$ ($i = 1,2,...,n$), respectively.

Prove that $v^2_1 +v^2_2 +...+v^2_n = p^2_1 +p^2_2 +...+p^2_n$
1988 ITAMO
A regular pentagon of side length $1$ is given. Determine the smallest $r$ for which the pentagon can be covered by five discs of radius $r$ and justify your answer.
1988 ITAMO
Show that all terms of the sequence $1,11,111,1111,...$ in base $9$ are triangular numbers, i.e. of the form $\frac{m(m+1)}{2} $for an integer $m$
1988 ITAMO
Given four non-coplanar points, is it always possible to find a plane such that the orthogonal projections of the points onto the plane are the vertices of a parallelogram? How many such planes are there in general?
1988 ITAMO
The edge lengths of the base of a tetrahedron are $a,b,c$, and the lateral edge lengths are $x,y,z$. If $d$ is the distance from the top vertex to the centroid of the base, prove that $x+y+z \le  a+b+c+3d$.
1988 ITAMO
Given $n \ge 3$ positive integers not exceeding $100$, let $d$ be their greatest common divisor. Show that there exist three of these numbers whose greatest common divisor is also equal to $d$.
1987 ITAMO
Show that $3x^5 +5x^3 -8x$ is divisible by $120$ for any integer $x$
1987 ITAMO
A tetrahedron has the property that the three segments connecting the pairs of midpoints of opposite edges are equal and mutually orthogonal. Prove that this tetrahedron is regular.
1987 ITAMO
Show how to construct (by a ruler and a compass) a right-angled triangle, given its inradius and circumradius.
1987 ITAMO
Given $I_0 = \{-1,1\}$, define $I_n$ recurrently as the set of solutions $x$ of the equations $x^2 -2xy+y^2- 4^n = 0$,

where $y$ ranges over all elements of $I_{n-1}$. Determine the union of the sets $I_n$ over all nonnegative integers $n$.
1987 ITAMO
Let $a_1,a_2,...$ and $b_1,b_2,..$. be two arbitrary infinite sequences of natural numbers.

Prove that there exist different indices $r$ and $s$ such that $a_r \ge a_s$ and $b_r \ge b_s$.
1987 ITAMO
There are three balls of distinct colors in a bag. We repeatedly draw out the balls one by one, the balls are put back into the bag after each drawing. What is the probability that, after $n$ drawings,

(a) exactly one color occured?

(b) exactly two oclors occured?

(c) all three colors occured?
1987 ITAMO
A square paper of side $n$ is divided into $n^2$ unit square cells. A maze is drawn on the paper with unit walls between some cells in such a way that one can reach every cell from every other cell not crossing any wall. Find, in terms of $n$, the largest possible total length of the walls.
1986 ITAMO
Two circles $\alpha$ and $\beta$ intersect at points $P$ and $Q$. The lines connecting a point $R$ on $\beta$ with $P$ and $Q$ intersect $\alpha$ again at $S$ and $T$ respectively. Prove that $ST$ is parallel to the line tangent to $\beta$ at $R$.
1986 ITAMO
Determine the general term of the sequence ($a_n$) given by $a_0 =\alpha > 0$ and $a_{n+1} =\frac{a_n}{1+a_n}$

.
1986 ITAMO
Two numbers are randomly selected from interval $I = [0, 1]$.

Given $\alpha \in I$, what is the probability that the smaller of the two numbers does not exceed $\alpha$?



reason for this postposted for the post collection, hide shall be deleted the next days
1986 ITAMO
Prove that a circle centered at point $(\sqrt{2},\sqrt{3})$ in the cartesian plane passes through at most one point with integer coordinates.



reason for this postposted for the post collection, hide shall be deleted the next days
1986 ITAMO
Given an acute triangle $T$ with sides $a,b,c$, find the tetrahedra with base $T$ whose all faces are acute triangles of the same area.
1986 ITAMO
Show that for any positive integer $n$ there exists an integer $m > 1$ such that $(\sqrt2-1)^n=\sqrt{m}-\sqrt{m-1}$.
1986 ITAMO
On a long enough highway, a passenger in a bus observes the traffic. He notes that, during an hour, the bus going with a constant velocity overpasses $a$ cars and gets overpassed by $b$ cars, while $c$ cars pass in the opposite direction. Assuming that the traffic is the same in both directions, is it possible to determine the number of cars that pass along the highway per hour? (You may assume that the velocity of a car can take only two values.)
1985 ITAMO
Let $x_1 = 97$, and for $n > 1$ let $x_n = \frac{n}{x_{n - 1}}$.  Calculate the product $x_1 x_2 \dotsm x_8$.
1985 ITAMO
When a right triangle is rotated about one leg, the volume of the cone produced is $800 \pi$ $\text{cm}^3$.  When the triangle is rotated about the other leg, the volume of the cone produced is $1920 \pi$ $\text{cm}^3$.  What is the length (in cm) of the hypotenuse of the triangle?
1985 ITAMO
Find $c$ if $a$, $b$, and $c$ are positive integers which satisfy $c=(a + bi)^3 - 107i$, where $i^2 = -1$.
1985 ITAMO
A small square is constructed inside a square of area 1 by dividing each side of the unit square into $n$ equal parts, and then connecting the vertices to the division points closest to the opposite vertices.  Find the value of $n$ if the the area of the small square is exactly 1/1985.

[asy]

size(200);

pair A=(0,1), B=(1,1), C=(1,0), D=origin;

draw(A--B--C--D--A--(1,1/6));

draw(C--(0,5/6)^^B--(1/6,0)^^D--(5/6,1));

pair point=( 0.5 , 0.5 );

//label("$A$", A, dir(point--A));

//label("$B$", B, dir(point--B));

//label("$C$", C, dir(point--C));

//label("$D$", D, dir(point--D));

label("$1/n$", (11/12,1), N, fontsize(9));[/asy]
1985 ITAMO
A sequence of integers $a_1$, $a_2$, $a_3$, $\ldots$ is chosen so that $a_n = a_{n - 1} - a_{n - 2}$ for each $n \ge 3$.  What is the sum of the first 2001 terms of this sequence if the sum of the first 1492 terms is 1985, and the sum of the first 1985 terms is 1492?
1985 ITAMO
As shown in the figure, triangle $ABC$ is divided into six smaller triangles by lines drawn from the vertices through a common interior point.  The areas of four of these triangles are as indicated.  Find the area of triangle $ABC$.



[asy]

size(200);

pair A=origin, B=(14,0), C=(9,12), D=foot(A, B,C), E=foot(B, A, C), F=foot(C, A, B), H=orthocenter(A, B, C);

draw(F--C--A--B--C^^A--D^^B--E);

label("$A$", A, SW);

label("$B$", B, SE);

label("$C$", C, N);

label("84", centroid(H, C, E), fontsize(9.5));

label("35", centroid(H, B, D), fontsize(9.5));

label("30", centroid(H, F, B), fontsize(9.5));

label("40", centroid(H, A, F), fontsize(9.5));[/asy]
1985 ITAMO
Assume that $a$, $b$, $c$, and $d$ are positive integers such that $a^5 = b^4$, $c^3 = d^2$, and $c - a = 19$.  Determine $d - b$.
1985 ITAMO
The sum of the following seven numbers is exactly 19:

$$a_1=2.56,\qquad a_2=2.61,\qquad a_3=2.65,\qquad a_4=2.71,$$$$a_5=2.79,\qquad a_6=2.82,\qquad a_7=2.86.$$It is desired to replace each $a_i$ by an integer approximation $A_i$, $1 \le i \le 7$, so that the sum of the $A_i$'s is also 19 and so that $M$, the maximum of the "errors" $|A_i - a_i|$, is as small as possible.  For this minimum $M$, what is $100M$?
1985 ITAMO
In a circle, parallel chords of lengths 2, 3, and 4 determine central angles of $\alpha$, $\beta$, and $\alpha + \beta$ radians, respectively, where $\alpha + \beta < \pi$.  If $\cos \alpha$, which is a positive rational number, is expressed as a fraction in lowest terms, what is the sum of its numerator and denominator?
1985 ITAMO
How many of the first 1000 positive integers can be expressed in the form

$$ \lfloor 2x \rfloor + \lfloor 4x \rfloor + \lfloor 6x \rfloor + \lfloor 8x \rfloor, $$

where $x$ is a real number, and $\lfloor z \rfloor$ denotes the greatest integer less than or equal to $z$?
1985 ITAMO
An ellipse has foci at $(9,20)$ and $(49,55)$ in the $xy$-plane and is tangent to the $x$-axis.  What is the length of its major axis?
1985 ITAMO
Let $A$, $B$, $C$, and $D$ be the vertices of a regular tetrahedron, each of whose edges measures 1 meter.  A bug, starting from vertex $A$, observes the following rule: at each vertex it chooses one of the three edges meeting at that vertex, each edge being equally likely to be chosen, and crawls along that edge to the vertex at its opposite end.  Let $p = n/729$ be the probability that the bug is at vertex $A$ when it has crawled exactly 7 meters.  Find the value of $n$.
1985 ITAMO
The numbers in the sequence 101, 104, 109, 116, $\dots$ are of the form $a_n = 100 + n^2$, where $n = 1$, 2, 3, $\dots$.  For each $n$, let $d_n$ be the greatest common divisor of $a_n$ and $a_{n + 1}$.  Find the maximum value of $d_n$ as $n$ ranges through the positive integers.
1985 ITAMO
In a tournament each player played exactly one game against each of the other players.  In each game the winner was awarded 1 point, the loser got 0 points, and each of the two players earned 1/2 point if the game was a tie.  After the completion of the tournament, it was found that exactly half of the points earned by each player were earned against the ten players with the least number of points.  (In particular, each of the ten lowest scoring players earned half of her/his points against the other nine of the ten).  What was the total number of players in the tournament?
1985 ITAMO
Three 12 cm $\times$ 12 cm squares are each cut into two pieces $A$ and $B$, as shown in the first figure below, by joining the midpoints of two adjacent sides.  These six pieces are then attached to a regular hexagon, as shown in the second figure, so as to fold into a polyhedron.  What is the volume (in $\text{cm}^3$) of this polyhedron?



[asy]

defaultpen(fontsize(10));

size(250);

draw(shift(0, sqrt(3)+1)*scale(2)*rotate(45)*polygon(4));

draw(shift(-sqrt(3)*(sqrt(3)+1)/2, -(sqrt(3)+1)/2)*scale(2)*rotate(165)*polygon(4));

draw(shift(sqrt(3)*(sqrt(3)+1)/2, -(sqrt(3)+1)/2)*scale(2)*rotate(285)*polygon(4));

filldraw(scale(2)*polygon(6), white, black);

pair X=(2,0)+sqrt(2)*dir(75), Y=(-2,0)+sqrt(2)*dir(105), Z=(2*dir(300))+sqrt(2)*dir(225);

pair[] roots={2*dir(0), 2*dir(60), 2*dir(120), 2*dir(180), 2*dir(240), 2*dir(300)};

draw(roots[0]--X--roots[1]);

label("$B$", centroid(roots[0],X,roots[1]));

draw(roots[2]--Y--roots[3]);

label("$B$", centroid(roots[2],Y,roots[3]));

draw(roots[4]--Z--roots[5]);

label("$B$", centroid(roots[4],Z,roots[5]));

label("$A$", (1+sqrt(3))*dir(90));

label("$A$", (1+sqrt(3))*dir(210));

label("$A$", (1+sqrt(3))*dir(330));

draw(shift(-10,0)*scale(2)*polygon(4));

draw((sqrt(2)-10,0)--(-10,sqrt(2)));

label("$A$", (-10,0));

label("$B$", centroid((sqrt(2)-10,0),(-10,sqrt(2)),(sqrt(2)-10, sqrt(2))));[/asy]
2021 Japan MO Finals
Find all functions from positive integers to themselves, such that for any positive integers $m,n$ the two conditions below are equivalent:

・$n$ divides $m$.

・$f(n)$ divides $f(m)-n$.
2021 Japan MO Finals
Let $n\geq 2$ be an integer. Players $A$ and $B$ play a game using $n\times 2021$ grid of square unit cells. Firstly, $A$ paints each cell either black of white. $B$ places a piece in one of the cells in the uppermost row, and designates one of the cells in the lowermost row as the goal. Then, $A$ repeats the following operation $n-1$ times:

・When the cell with the piece is painted white, $A$ moves the piece to the cell one below.

・Otherwise, $A$ moves the piece to the next cell on the left or right, and then to the cell one below.

Find the minimum possible value of $n$ such that $A$ can always move a piece to the goal, regardless of $B$'s choice.
2021 Japan MO Finals
Points $D,E$ on the side $AB,AC$ of an acute-angled triangle $ABC$ respectively satisfy $BD=CE$. Furthermore, points $P$ on the segmet $DE$ and $Q$ on the arc $BC$ of the circle $ABC$ not containing $A$ satisfy $BP:PC=EQ:QD$. Points $A,B,C,D,E,P,Q$ are pairwise distinct.

Prove that $\angle BPC=\angle BAC+\angle EQD$ holds.
2021 Japan MO Finals
Let $a_1,a_2,\dots,a_{2021}$ be $2021$ integers which satisfy

$$ a_{n+5}+a_n>a_{n+2}+a_{n+3}$$for all integers $n=1,2,\dots,2016$. Find the minimum possible value of the difference between the maximum value and the minimum value among $a_1,a_2,\dots,a_{2021}$.
2021 Japan MO Finals
Let $n$ be a positive integer. Find all integers $k$ among $1,2,\dots,2n^2$ which satisfy the following condition:

・There is a $2n\times 2n$ grid of square unit cells. When $k$ different cells are painted black while the other cells are painted white, the minimum possible number of $2\times 2$ squares that contain both black and white cells is $2n-1$.
2020 Japan MO Finals
Find all pairs of positive integers $(m,n)$ such that $\frac{n^2+1}{2m}$ and $\sqrt{2^{n-1}+m+4}$ are both integers.
2020 Japan MO Finals
Triangle $ABC$ satisfies $BC<AB$ and $BC<AC$. Points $D,E$ lie on segments $AB,AC$ respectively, satisfying $BD=CE=BC$. Lines $BE$ and $CD$ meet at point $P$, circumcircles of triangle $ABE$ and $ACD$ meet at point $Q$ other than $A$. Prove that lines $PQ$ and $BC$ are perpendicular.
2020 Japan MO Finals
Find all functions $f:\mathbb{Z}^{+}\to\mathbb{Z}^{+}$ such that $$m^{2}+f(n)^{2}+(m-f(n))^{2}\geq f(m)^{2}+n^{2}$$for all pairs of positive integers $(m,n)$.
2020 Japan MO Finals
Let $n\geq 2$ be an integer. $3n$ distinct points are plotted on the circle, where A and B perform the following operation : Firstly, $A$ picks exactly 2 points which haven't been connected yet and connects them by a segment. Secondly, B picks exactly 1 point with no piece and place a piece. Prove that, after consecutive $n$ operations, despite how B acts, A can make the number of segments no less than $\displaystyle \frac{n-1}{6}$, which connect a point with a piece and a point with no piece.
2020 Japan MO Finals
Find all infinite sequences of positive integers $\{a_{n}\}_{n\geq 1}$ satisfying the following condition : there exists a positive constant $c$ such that $\gcd(a_{m}+n,a_{n}+m)>c(m+n)$ holds for all pairs of positive integers $(m,n)$.
2019 Japan MO Finals
Find all triples $(a,\ b,\ c)$ of positive integers such that



$$a^2+b+3=(b^2-c^2)^2.$$
2019 Japan MO Finals
Let $n\geq 3$ be an odd number. We will play a game using a $n$ by $n$ grid. The game is comprised of $n^2$ turns, in every turn, we will perform the following operation sequentially.



$\bullet$ We will choose a square with an unwritten integer, and write down an integer among 1 through $n^2$. We can write down any integer only at once through the game.



$\bullet$ For each row, colum including the square, if the sum of integers is a multiple of $n$, then we will get 1 point (both of each sum is a multiple of $n$, we will get 2 points).}



Determine the maximum possible value of the points as the total sum that we can obtain by the end of the game.
2019 Japan MO Finals
Find all functions $f:\mathbb R^{+} \rightarrow \mathbb R^{+}$ such that



$$f\left(\frac{f(y)}{f(x)}+1\right)=f\left(x+\frac{y}{x}+1\right)-f(x)$$

for all $x,\ y\in\mathbb{R^{+}}$.
2019 Japan MO Finals
Let $ABC$ be a triangle with its inceter $I$, incircle $w$, and let $M$ be a midpoint of the side $BC$. A line through the point $A$ perpendicular to the line $BC$ and a line through the point $M$ perpendicular to the line $AI$ meet at $K$. Show that a circle with line segment $AK$ as the diameter touches $w$.
2019 Japan MO Finals
Let $S$ be a set which is comprised of positive integers. We call $S$ a $\it{beautiful\ number}$ when the element belonging to $S$ of which any two distinct elements $x,\ y,\ z$, at least of them  will be a divisor of $x+y+z$. Show that there exists an integer $N$ satisfying the following condition, and also determine the smallest $N$ as such :



For any set S of $\it{beautiful\ number}$, there exists, $n_{s}\geq 2$  being an integer, the number of the element belonging to $S$ which is not a multiple of $n_s$, is less than or equal to $N$.
2018 Japan MO Finals
Positive integers between $1$ to $100$ inclusive are written on a blackboard, each exactly once. One operation involves choosing two numbers $a$ and $b$ on the blackboard and erasing them, then writing the greatest common divisor of $a^2+b^2+2$ and $a^2b^2+3$. After a number of operations, there is only one positive integer left on the blackboard. Prove this number cannot be a perfect square.
2018 Japan MO Finals
Given a scalene triangle $\triangle ABC$, $D,E$ lie on segments $AB,AC$ respectively such that $CA=CD, BA=BE$. Let $\omega$ be the circumcircle of $\triangle ADE$. $P$ is the reflection of $A$ across $BC$, and $PD,PE$ meets $\omega$ again at $X,Y$ respectively. Prove that $BX$ and $CY$ intersect on $\omega$.
2018 Japan MO Finals
Let $S=\{1,2,\ldots ,999\}$. Consider a function $f: S\to S$, such that for any $n\in S$,

$$f^{n+f(n)+1}(n)=f^{nf(n)}(n)=n.$$Prove that there exists $a\in S$, such that $f(a)=a$. Here $f^k(n) = \underbrace{f(f(\ldots f}_{k}(n)\ldots))$.
2018 Japan MO Finals
Let $n$ be an odd positive integer, and consider an infinite square grid. Prove that it is impossible to fill in one of $1,2$ or $3$ in every cell, which simultaneously satisfies the following conditions:

(1) Any two cells which share a common side does not have the same number filled in them.

(2) For any $1\times 3$ or $3\times 1$ subgrid, the numbers filled does not contain $1,2,3$ in that order be it reading from top to bottom, bottom to top, or left to right, or right to left.

(3) The sum of numbers of any $n\times n$ subgrid is the same.
2018 Japan MO Finals
Let $T$ be a positive integer. Find all functions $f: \mathbb {Z}^+ \times \mathbb {Z}^+ \to \mathbb {Z}^+$, such that there exists integers $C_0,C_1,\ldots ,C_T$ satisfying:

(1) For any positive integer $n$, the number of positive integer pairs $(k,l)$ such that $f(k,l)=n$ is exactly $n$.

(2) For any $t=0,1,\ldots ,T,$ as well as for any positive integer pair $(k,l)$, the equality $f(k+t,l+T-t)-f(k,l)=C_t$ holds.
2017 Japan MO Finals
Let $a,b,c$ be positive integers. Prove that $lcm(a,b) \neq lcm(a+c,b+c)$.
2017 Japan MO Finals
Let $N$ be a positive integer. There are positive integers $a_{1}, a_{2},\cdots, a_{N}$ and none of them are multiples of $2^{N+1}$. For each integer $n\geq N+1$, set $a_{n}$ as below:



If the remainder of $a_{k}$ divided by $2^{n}$ is the smallest amongst the remainders of $a_{1},\cdots, a_{n-1}$ divided by $2^{n}$, set $a_{n}=2a_{k}$. If there are several integers $k$ which satisfy the above condition, put the biggest one.



Prove the existence of a positive integer $M$ which satisfies $a_{n}=a_{M}$ for $n\geq M$.
2017 Japan MO Finals
Let $ABC$ be an acute-angled triangle with the circumcenter $O$. Let $D,E$ and $F$ be the feet of the altitudes from $A,B$ and $C$, respectively, and let $M$ be the midpoint of $BC$. $AD$ and $EF$ meet at $X$, $AO$ and $BC$ meet at $Y$, and let $Z$ be the midpoint of $XY$. Prove that $A,Z,M$ are collinear.
2017 Japan MO Finals
Let $n \geq 3$ be an integer. There are $n$ people, and a meeting which at least $3$ people attend is held everyday. Each attendant shake hands with the rest attendants at every meeting. After the $n$th meeting, every pair of the $n$ people shook hands exactly once. Prove that every meeting was attended by the same number of attendants.
2017 Japan MO Finals
Let $x_{1}, x_{2},\cdots,x_{1000}$ be integers, and $\displaystyle \sum_{i=1}^{1000} x_{i}^{k}$ are all multiples of $2017$ for any positive integers $k \leq 672$. Prove that $x_{1}, x_{2},\cdots,x_{1000}$ are all multiples of $2017$.

Note: $2017$ is a prime number.
2016 Japan MO Finals
Let $p$ be an odd prime number. For positive integer $k$ satisfying $1\le k\le p-1$, the number of divisors of $kp+1$ between $k$ and $p$ exclusive is $a_k$. Find the value of $a_1+a_2+\ldots + a_{p-1}$.
2016 Japan MO Finals
Let $ABCD$ be a concyclic quadrilateral such that $AB:AD=CD:CB.$ The line $AD$ intersects the line $BC$ at $X$, and the line $AB$ intersects the line $CD$ at $Y$. Let $E,\ F,\ G$ and $H$ are the midpoints of the edges $AB,\ BC,\ CD$ and $DA$ respectively. The bisector of angle $AXB$ intersects the segment $EG$ at $S$, and that of angle $AYD$ intersects the segment $FH$ at $T$. Prove that the lines $ST$ and $BD$ are pararell.
2016 Japan MO Finals
Let $n$ be a positive integer. In JMO kingdom there are $2^n$ citizens and a king. In terms of currency, the kingdom uses paper bills with value $\$2^n$ and coins with value $\$2^a(a=0,1\ldots ,n-1)$. Every citizen has infinitely many paper bills. Let the total number of coins in the kingdom be $S$. One fine day, the king decided to implement a policy which is to be carried out every night:



Each citizen must decide on a finite amount of money based on the coins that he currently has, and he must pass that amount to either another citizen or the king;

Each citizen must pass exactly $1 more than the amount he received from other citizens.





Find the minimum value of $S$ such that the king will be able to collect money every night eternally.
2016 Japan MO Finals
Find all functions $f: \mathbb{R}\rightarrow \mathbb{R}$ such that



$$f(yf(x)-x)=f(x)f(y)+2x$$

for all $x,\ y\in{\mathbb{R}}$.
2016 Japan MO Finals
$m,n$ are positive integers such that $m\ge 2$, $n< \frac{3}{2}(m-1)$. In a country there are $m$ cities and $n$ roads, each road connect two different cities, and there can be multiple roads between two cities. Prove that there exist a way to separate the cities into two groups $\alpha$ and $\beta$, where all roads connecting a city in $\alpha$ to a city in $\beta$ is converted to a highway, and satisfies the following conditions:



Both groups have at least one city, and

for each city, the number of highways coming out from that city does not exceed $1$.
2015 Japan MO Finals
Find all positive integers $n$ such that $\frac {10^{n}}{n^3+n^2+n+1}$ is an integer.
2015 Japan MO Finals
Scalene triangle $ABC$ has circumcircle $\Gamma$ and incenter $I$. The incircle of triangle $ABC$ touches side $AB,AC$ at $D,E$ respectively. Circumcircle of triangle $BEI$ intersects $\Gamma$ again at $P$ distinct from $B$, circumcircle of triangle $CDI$ intersects $\Gamma$ again at $Q$ distinct from $C$. Prove that the $4$ points $D,E,P,Q$ are concyclic.
2015 Japan MO Finals
Let $a$ be a fixed positive integer. For a given positive integer $n$, consider the following assertion.



In an infinite two-dimensional grid of squares, $n$ different cells are colored black. Let $K$ denote the number of $a$ by $a$ squares in the grid containing exactly $a$ black cells. Then over all possible choices of the $n$ black cells, the maximum possible $K$ is $a(n+1-a)$.



Prove that there exists a positive integer $N$ such that for all $n\ge N$, this assertion is true.



(link is  for anyone who wants to correct my translation)
2014 Japan MO Finals
Let $O$ be the circumcenter of triangle $ABC$, and let $l$ be the line passing through the midpoint of segment $BC$ which is also perpendicular to the bisector of angle $ \angle BAC $. Suppose that the midpoint of segment $AO$ lies on $l$. Find $ \angle BAC $.
2014 Japan MO Finals
Find all ordered triplets of positive integers $(a,\ b,\ c)$ such that $2^a+3^b+1=6^c$.
2014 Japan MO Finals
In a school, there are $n$ students and some of them are friends each other. (Friendship is mutual.) Define $ a, b $ the minimum value which satisfies the following conditions:

(1) We can divide students into $ a $ teams such that two students in the same team are always friends.

(2) We can divide students into $ b $ teams such that two students in the same team are never friends.

Find the maximum value of $ N = a+b $ in terms of $n$.