Navvye/Olympiad-Problems-CSV huggingface dataset:
2014 Japan MO Finals
Let $ \Gamma $ be the circumcircle of triangle $ABC$, and let $l$ be the tangent line of $\Gamma $ passing $A$. Let $ D, E $ be the points each on side $AB, AC$ such that $ BD : DA= AE : EC $. Line $ DE $ meets $\Gamma $ at points $ F, G $. The line parallel to $AC$ passing $ D $ meets $l$ at $H$, the line parallel to $AB$ passing $E$ meets $l$ at $I$. Prove that there exists a circle passing four points $ F, G, H, I $ and tangent to line $ BC$.
2014 Japan MO Finals
Find the maximum value of real number $k$ such that

$$\frac{a}{1+9bc+k(b-c)^2}+\frac{b}{1+9ca+k(c-a)^2}+\frac{c}{1+9ab+k(a-b)^2}\geq \frac{1}{2}$$

holds for all non-negative real numbers $a,\ b,\ c$ satisfying $a+b+c=1$.
2013 Japan MO Finals
Let $n,\ k$ be positive integers with $n\geq k$. There are $n$ persons, each person belongs to exactly one of group $1$, group $2,\ \cdots$, group $k$ and more than or equal to one person belong to any  groups. Show that $n^2$ sweets can be delivered to $n$ persons in such way that all of  the following condition are satisfied.



$\bullet$ At least one sweet are delivered to each person.



$\bullet$ $a_i$ sweet are delivered to each person belonging to group $i\ (1\leq i\leq k).$



$\bullet$ If $1\leq i<j\leq k$, then $a_i>a_j.$
2013 Japan MO Finals
Find all functions $f:\mathbb{Z}\rightarrow\mathbb{R}$ such that  the equality

$$f(m)+f(n)=f(mn)+f(m+n+mn) $$

holds for all $m,n\in\mathbb{Z}.$
2013 Japan MO Finals
Let $n\geq 2$ be a positive integer. Find the minimum value of positive integer $m$ for which there exist positive integers $a_1,\ a_2,\ \cdots, a_n$ such that :



$\bullet\ a_1<a_2<\cdots <a_n=m$



$\bullet \ \frac{a_1^2+a_2^2}{2},\ \frac{a_2^2+a_3^2}{2},\ \cdots,\ \frac{a_{n-1}^2+a_n^2}{2}$ are all square numbers.
2013 Japan MO Finals
Given an acute-angled triangle ABC, let $H$ be the orthocenter. A cirlcle passing through the points $B,\ C$ and a cirlcle with a diameter $AH$ intersect at two distinct points $X,\ Y$. Let $D$ be the foot of the perpendicular drawn from $A$ to line $BC$, and let $K$ be the foot of the perpendicular drawn from $D$ to line $XY$. Show that $\angle{BKD}=\angle{CKD}$.
2013 Japan MO Finals
Let $n$ be a positive integer. Given are points $P_1,\ P_2,\ \cdots,\ P_{4n}$ of which any three points are not collinear. For $i=1,\ 2,\ \cdots,\ 4n$, rotating half-line $P_iP_{i-1}$ clockwise in $90^\circ$ about the pivot  $P_i$ gives half-line $P_iP_{i+1}.$ Find the maximum value of the number of the pairs of $(i,\ j)$ such that line segments $P_iP_{i+1}$ and $P_jP_{j+1}$ intersect at except endpoints.

Note that : $P_0=P_{4n},\ P_{4n+1}=P_1$ and $1\leq i<j\leq 4n.$
2012 Japan MO Finals
Given a triangle $ABC$, the tangent of the circumcircle at $A$ intersects with the line $BC$ at $P$. Let $Q,\ R$ be the points of symmetry for $P$ across the lines $AB,\ AC$ respectively. Prove that the line $BC$ intersects orthogonally with the line $QR$.
2012 Japan MO Finals
Find all functions $ f : \mathbb{R} \mapsto \mathbb{R}$ such that $f(f(x+y)f(x-y))=x^2-yf(y)$ for all $ \ x,y \in \mathbb{R}.$
2012 Japan MO Finals
Let $p$ be prime. Find all possible integers $n$ such that for all integers $x$, if $x^n-1$ is divisible by $p$, then $x^n-1$ is divisible by $p^2$ as well.
2012 Japan MO Finals
Given two triangles $PAB$ and $PCD$ such that $PA=PB,\ PC=PD,\ P,\ A,\ C$ and $B,\ P,\ D$ are collinear in this order respectively.

The circle $S_1$ passing through $A,\ C$ intersects with the circle $S_2$ passing through $B,\ D$ at distinct points $X,\ Y$.

Prove that the circumcenter of the triangle $PXY$ is the midpoint of the centers of $S_1,\ S_2$.
2012 Japan MO Finals
Given is a piece at the origin of the coordinate plane. Two persons $A,\ B$ act as following.

First, $A$ marks on a lattice point, on which the piece cannot be anymore put. Then $B$ moves the piece from the point $(x,\ y)$ to the point $(x+1,\ y)$ or $(x,\ y+1)$, a number of $m$ times $(1\leq m\leq k)$. Note that we may not move the piece to a marked point. If $A$ wins when $B$ can't move any pieces, then find all possible integers $k$ such that $A$ will win in a finite number of moves, regardless of how $B$ moves the piece.
2011 Japan MO Finals
Given an acute triangle $ABC$ with the midpoint $M$ of $BC$. Draw the perpendicular $HP$ from the orthocenter $H$ of $ABC$ to $AM$.

Show that $AM\cdot PM=BM^2$.
2011 Japan MO Finals
Find all of quintuple of positive integers $(a,n,p,q,r)$ such that $a^n-1=(a^p-1)(a^q-1)(a^r-1) $.
2011 Japan MO Finals
Person $A$ writes down  non negative integers in each $N$ grid running in a line horizontally. When $A$ says one non negative integer,

Person $B$ replaces some number in $N$ grid by the number that $A$ said. Repeat this procedure, when these numbers are arranged in the order of monotone increasing in the wider sense, the procedure is over. Is it possible that $B$ can finish in regard less of $A$?
2011 Japan MO Finals
Find all functions $ f : \mathbb{R} \to\mathbb{R}$ such that $f(f(x)-f(y))=f(f(x))-2x^2f(y)+f(y^2)$ for all $x,y \in \mathbb{R}.$
2011 Japan MO Finals
Given 4 points on a plane. Suppose radii of 4 incircles of the triangles, which can be formed by any 3 points taken from the 4 points, are equal.

Prove that all of the triangles are congruent.
2010 Japan MO Finals
Given an acute-angled triangle $ ABC$ such that $ AB\neq AC$. Draw the perpendicular $ AH$ from $ A$ to $ BC$. Suppose that if we take points $ P,\ Q$ in such a way that three points $ A,\ B,\ P$ and three points $ A,\ C,\ Q$ are collinear in this order respectively, then we have four points $ B,\ C,\ P,\ Q$ are concyclic and $ HP = HQ$. Prove that $ H$ is the circumcenter of $ \triangle{APQ}$.
2010 Japan MO Finals
Let $ k$ be positive integer and $ m$ be odd number. Prove that there exists positive integer $ n$ such that $ n^n - m$ is divisible by $ 2^k$.
2010 Japan MO Finals
There are $2010$ islands and $2009$ bridges connecting them. Suppose that any bridges are connected by one bridge or not the endpoints are connected to 2 distinct islands and we can travel a few times by crossing bridges from each island to any other islands.

Now a letter from each island was sent to some island, note that, some letter may sent to same island, then the following fact was proved that:

In case of connecting island $A$ and island $B$ by bridge, the habitant of island $A$ and that of island $B$ are mutually connected by bridge or the same island (itself).

Prove that at least one of the following statements (1) or (2) hold.



(1) There exists island for which a letter was sent to the same island.



(2) There exist 2 islands, connecting bridge, whose letter are exchanged each other.
2010 Japan MO Finals
Let $ x,\ y,\ z$ be positive real numbers.



Prove that

$$ \frac {1 + yz + zx}{(1 + x + y)^2} + \frac {1 + zx + xy}{(1 + y + z)^2} + \frac {1 + xy + yz}{(1 + z + x)^2}\geq 1$$
2010 Japan MO Finals
Given a convex $2010$ polygonal whose any 3 diagonals have no intersection points except vertices. Consider  closed broken lines which have $2010$ diagonals (not including edges) and they pass through each vertex exactly one time. Find the possible maximum value of the number of self-crossing. Note that we call closed broken lines such that broken line $P_1P_2\cdots P_nP_{n+1}$ has the property $P_1=P_{n+1}.$
2009 Japan MO Finals
Find all positive integers $ n$ such that $ 8^n + n$ is divisible by $ 2^n + n$.
2009 Japan MO Finals
Let $ N$ be postive integer. Some integers are written in a black board and those satisfy the following conditions.



1. Any numbers written are integers which are from 1 to $ N$.



2. More than one integer which is from 1 to $ N$ is written.



3. The sum of numbers written is even.



If we mark $ X$ to some numbers written and mark $ Y$  to all remaining numbers, then prove that we can set the sum of numbers marked $ X$ are equal to that of numbers marked $ Y$.
2009 Japan MO Finals
Let $ k\geq 2$ be  integer, $ n_1,\ n_2,\ n_3$ be positive integers and $ a_1,\ a_2,\ a_3$ be integers from $ 1,\ 2,\ \cdots ,\ k - 1$.

Let $ b_i = a_i\sum_{j = 0}^{n_i} k^{j}\ (i = 1,\ 2,\ 3)$. Find all possible pairs of integers $ (n_1,\ n_2,\ n_3)$ such that $ b_1b_2 = b_3$.
2009 Japan MO Finals
Let $ \Gamma$ be a circumcircle. A circle with center $ O$ touches to line segment $ BC$ at $ P$ and touches the arc $ BC$ of $ \Gamma$ which doesn't have $ A$ at $ Q$. If $ \angle {BAO} = \angle {CAO}$, then prove that $ \angle {PAO} = \angle {QAO}$.
2009 Japan MO Finals
Find all functions $ f$, defined on the non negative real numbers and taking non negative real numbers such that $ f(x^2)+f(y)=f(x^2+y+xf(4y))$ for any non negative real numbers $ x,\ y$.
2008 Japan MO Finals
Let $ P(x)$ be a polynomial with integer coefficients such that $ P(n^2) = 0$ for some non zero integers $ n.$ Prove that $ P(a^2)\neq 1$ for all non zero rational numbers $ a\neq 0.$
2008 Japan MO Finals
There are 2008 red cards and 2008 white cards. 2008  players sit down in circular toward the inside of the circle in situation that 2 red cards and 2 white cards from each card are delivered to each person. Each person conducts the following procedure in one turn as follows.



$ (*)$ If you have more than one red card, then you will pass one red card to the left-neighbouring player.



If you have no red card, then you will pass one white card to the left -neighbouring player.

Find the maximum value of the number of turn required for the state such that all person will have one red card and one white card first.
2008 Japan MO Finals
Given an acute-angled triangle $ ABC$ with circumcenter $ O.$ The circle passing through two points $ A,\ O$ intersects with the line $ AB$ and $ AC$ at $ P,\ Q$ other than $ A$ respectively. If the lengths of the line segments $ PQ,\ BC$ are equal, then find the angle $ \leq 90^\circ$ that the lines $ PQ$ and $ BC$ make.
2008 Japan MO Finals
Find all functions $ f : \mathbb{R} \to \mathbb{R}$ such that $$ f(x+y)f(f(x)-y)=xf(x)-yf(y)$$for all $ \ x,y \in \mathbb{R}.$
2008 Japan MO Finals
Can it be existed postive integers $ n$ such that there are integers $ b$ and non zero integers $ a_i\ (i=1,\ 2,\ \cdots n)$ for rational numbers $ r$ which satisfies $ r=b+\sum_{i=1}^n \frac{1}{a_i}?$
2007 Japan MO Finals
Let $n$ be positive integers. Two persons play a game in which they are calling a integer $m\ (1\leq m\leq n)$ alternately. Note that you may not call the number which have already said. The game is over when no one can call numbers, if the sum of the numbers that the lead have said  is divisible by 3, then the lead wins, otherwise the the second move wins. Find $n$ for which there exists the way of forestalling.
2007 Japan MO Finals
Find all functions $f,$ defined on the positive real numbers and taking real numbers such that

$$f(x)+f(y)\leq \frac{f(x+y)}{2},\ \ \frac{f(x)}{x}+\frac{f(y)}{y}\geq \frac{f(x+y)}{x+y}$$ for all $x,\ y>0.$
2007 Japan MO Finals
Let $\Gamma$ be the circumcircle of triangle $ABC.$ Denote the circle which touches to the sides $AB, \ AC$ and touches to $\Gamma$ internally at $P$ by $\Gamma_{A},$ and the circle which touches to the sides $AB, \ BC$ and touches to $\Gamma$ internally at $Q$ by $\Gamma_{B},$ and the circle which touches to the sides $AC, \ BC$ and touches to $\Gamma$ internally at $R$ by $\Gamma_{C}.$ Prove that the lines $AP,\ BQ,\ CR$ are concurrent.
2007 Japan MO Finals
On a plane, call the band with width $d$ be the set of all points which are away the distance of less than or equal to $\frac{d}{2}$ from a line.Given four points $A,\ B,\ C,\ D$ on a plane.If you take three points among them, there exists the band with width $1$ containing them. Prove that there exist the band with width $\sqrt{2}$ containing all four points .
2007 Japan MO Finals
For real positive numbers $x,$ the set $A(x)$ is defined by

$$A(x)=\{[nx]\ | \ n\in{ \mathbb{N}}\},$$ where $[r]$ denotes the greatest integer not exceeding real numbers $r.$Find all irrational numbers $\alpha >1$ satisfying the following condition.

Condition: If  positive real number $\beta$ satisfies $A(\alpha)\supset A(\beta),$ then $\frac{\beta}{\alpha}$ is integer.
2006 Japan MO Finals
Given five distinct points $A,M,B,C,D$ in this order on the circumference of the circle $O$ such that $MA=MB.$

Let $P,Q$ be the intersection points of the line $AC$ and $MD$, and that of the line $BD$ and $MC,$ respectively.

If two intersection points of the line $PQ$ and the circumference of the circle $O$ are $X,\ Y,$ then prove that $MX=MY.$
2006 Japan MO Finals
Determine all integers $k$ for which there exist infinitely the pairs of integers $(a,b,c)$ satisfying the following equation.



$$ (a^2-k)(b^2-k)=c^2-k.  $$
2006 Japan MO Finals
Find all functions $f,$ defined on real numbers and taking real values such that $\left\{f(x)\right\}^2+2yf(x)+f(y)=f(y+f(x))$ for all real numbers $x,y.$
2006 Japan MO Finals
Let $m,\ n$ be integers such that $2\leq m\leq n$ and let $a,\ a'$ be integers which are less than or equal to $m$ and let $b,\ b'$ be integers which are less than or equal to $n$ such that $(a,b)\neq (a'\ b').$ Given a town of the rectangular shaped chessboard which is made up of $m's$ road running north and south which is called Line and $n's$ road running west and east which is called Street. Denote the intersection point of the $a$ th Line from the west and $b$ th Street from the north by $A$, and $a'$ th Line from the west and $b'$ th Street from the north by $B,$ including the edge for both cases.Find all pair of $(m,n,a,b,a', b')$ such that by passing through each crossroads of the town exactly one time, you can reach the point $B$ from the point $A$ including in the start point and goal one.
2006 Japan MO Finals
For any positive real numbers $x_1,\ x_2,\ x_3,\ y_1,\ y_2,\ y_3,\ z_1,\ z_2,\ z_3,$ find the maximum value of real number $A$ such that if $$ M = (x_1^3+x_2^3+x_3^3+1)(y_1^3+y_2^3+y_3^3+1)(z_1^3+z_2^3+z_3^3+1)  $$ and $$ N = A(x_1+y_1+z_1)(x_2+y_2+z_2)(x_3+y_3+z_3),  $$ then $M \geq N$ always holds, then find the condition that the equality holds.
2005 Japan MO Finals
Double-faced coins are arranged with all the heads facing upward in the shape of $17\times17.$At one operation, you turn over $5$ consecutive coins in longitudinal or $5$ ones in transversal or $5$ ones in oblique at the same time.Now can you make all those reverses face upward when you repeat this operation?
2005 Japan MO Finals
Let $P(x,y),Q(x,y)$ be two-variable polynomials with the coefficients of integer.Supposed that when $a_n,b_n$ are determined for certain integers $a_0,\ b_0$ by $a_{n+1}=P(a_n,b_n),\ \ b_{n+1}=Q(a_n,b_n)\ \ (n=0,1,2,\cdots)$ there existed positive integer $k$ such that $(a_1,b_1)\neq (a_0,b_0)$ and $(a_k,b_k)=(a_0,b_0)$.Prove that the number of  the lattice points on the segment with end points of $(a_n,b_n)$ and $(a_{n+1},b_{n+1})$ is indepedent of $n$.
2005 Japan MO Finals
Let $a,b,c$ be positive real numbers such that $a+b+c=1.$Prove the following inequality.



$$ a\sqrt[3]{1+b-c}+b\sqrt[3]{1+c-a}+c\sqrt[3]{1+a-b}\leqq 1 $$
2005 Japan MO Finals
Given two points $A$ and $B$ on a circle $\varGamma$. Let the tangents to this circle $\varGamma$ at the points $A$ and $B$ meet at a point $X$. Let $C$ and $D$ be two points on the circle $\varGamma$ such that the points $C$, $D$, $X$ are collinear in this order and such that the lines $CA$ and $BD$ are perpendicular.



Let the line $CA$ intersect the line $BD$ at a point $F$.

Let the line $CD$ intersect the line $AB$ at a point $G$.

Let $H$ be the point of intersection of the segment $BD$ and the perpendicular bisector of the segment $GX$.



Prove that the four points $X$, $F$, $G$, $H$ lie on one circle.
2005 Japan MO Finals
You are the boss. You have ten men and there are ten tasks. Your men have two parameters to each task, one is enthusiasm, the other is ability.Now you are to assign the tasks to your men one by one.When man $A$ has more enthusiasm about task $v$ than about task $u,$ and man $A$ has better ability in task $v$ than man $B$ does, though if you assign task $u$ to man $A$ and task $v$ to man $B,$ man $A$ feel unsatisfied.Or, if there is a more efficient way than yours that you can assign each task to men with better ability, you will be brought to account by your employer.Prove that there is a way to assign tasks without any dissatisfaction or disadvantage.
2004 Japan MO Finals
Find all functions $f : \mathbb{R} \mapsto \mathbb{R}$ such that $f(xf(x)+f(y)) =(f(x))^2+y$ for all $\ x,y \in \mathbb{R}.$
2004 Japan MO Finals
Given two planes $\pi _1,\ \pi _2$ intersecting orthogonally in space. Let $A,B$ be two distinct points on the line of intersection of $\pi _1$ and $\pi _2,$ and $C$ be the point which is on  $\pi _2$ but not on $\pi_1.$ Denote by $P$ the intersection point of the bisector of $\angle {BCA}$ and $AB$, and denote $S$ by the circumference on $\pi _1$ with a diameter $AB.$ For an arbiterary plane $\pi _3$ which contains $CP,$ if $D,E$ are the intersection points of $\pi_3$ and $S,$ then prove that $CP$ is the bisector of $\angle {DCE}.$
2004 Japan MO Finals
For positive real numbers $a,b,c$ satisfying $a+b+c=1,$



Prove that we have $\frac{1+a}{1-a}+\frac{1+b}{1-b}+\frac{1+c}{1-c}\leqq 2\left(\frac{b}{a}+\frac{c}{b}+\frac{a}{c}\right).$ Note that you don't need to state for the condition

for which the equality holds.
2004 Japan MO Finals
In a land any towns are connected by roads with exactly other three towns. Last year we made a trip starting from a town coming back to the town by visiting exactly one time all towns in land. This year we are plan to make a trip in the same way as last year's trip. Note that you can't take such order completely same as last year's one or trace the order only reversely. Prove that this is possible.
2003 Japan MO Finals
A point $P$ lies in $\triangle ABC$. The lines $BP,CP$ meet $AC,AB$ at $Q,R$ respectively. Given that $AR=RB=CP, CQ=PQ$, find $\angle BRC$.
2003 Japan MO Finals
We have two distinct positive integers $a,b$ with $a|b$. Each of $a,b$ consists of $2n$ decimal digits. The first $n$ digits of $a$ are identical to the last $n$ digits of $b$, and vice versa. Determine $a,b$.
2003 Japan MO Finals
Find the greatest real number $k$ such that, for any positive $a,b,c$ with $a^{2}>bc$, $(a^{2}-bc)^{2}>k(b^{2}-ca)(c^{2}-ab)$.
2003 Japan MO Finals
Let $p,q\geq 2$ be coprime integers. A list of integers $(r,a_{1},a_{2},...,a_{n})$ with $|a_{i}|\geq 2$ for all $i$ is said to be an expansion of $p/q$ if

$\frac{p}{q}=r+\frac{1}{a_{1}+\frac{1}{a_{2}+\frac{1}{...+\frac{1}{a_{n}}}}}$.

Now define the weight of an expansion $(r,a_{1},a_{2},...,a_{n})$ to be the product

$(|a_{1}|-1)(|a_{2}|-1)...(|a_{n}|-1)$.

Show that the sum of the weights of all expansions of $p/q$ is $q$.
2003 Japan MO Finals
Find the greatest possible integer $n$ such that one can place $n$ points in a plane with no three on a line, and color each of them either red, green, or yellow so that:

(i) inside each triangle with all vertices red there is a green point.

(ii) inside each triangle with all vertices green there is a yellow point.

(iii) inside each triangle with all vertices yellow there is a red point.
2002 Japan MO Finals
Distinct points $ A,M,B$ with $ AM = MB$ are given on circle $ (C_0)$ in this order. Let $ P$ be a point on the arc $ AB$ not containing $ M$. Circle $ (C_1)$ is internally tangent to $ (C_0)$ at $ P$ and tangent to $ AB$ at $ Q$. Prove that the product $ MP\cdot MQ$ is independent of the position of $ P$.
2002 Japan MO Finals
There are $ n \ge 3$ coins on a circle. Consider a coin and the two coins adjacent to it; if there are an odd number of heads among the three, we call it good. An operation consists of turning over all good coins simultaneously. Initially, exactly one of the $ n$ coins is a head. The operation is repeatedly performed.

(a) Prove that if $ n$ is odd, the coins will never be all-tails.

(b) For which values of $ n$ is it possible to make the coins all-tails after several operations? Find, in terms of $ n$, the number of operations needed for this to occur.
2002 Japan MO Finals
Denote by $ S(n)$ the sum of decimal digits of a positive integer $ n$. Show that there exist $ 2002$ distinct positive integers $ n_1, n_2, \cdots, n_{2002}$ such that $ n_1 + S(n_1) = n_2 + S(n_2) = \cdots = n_{2002} + S(n_{2002})$.
2002 Japan MO Finals
Let $n\geq 3$ be natural numbers, and let $a_1,\ a_2,\ \cdots,\ a_n,\ \ b_1,\ b_2,\ \cdots,\ b_n$ be positive numbers such that $a_1+a_2+\cdots +a_n=1,\ b_1^2+b_2^2+\cdots +b_n^2=1.$ Prove that $a_1(b_1+a_2)+a_2(b_2+a_3)+\cdots +a_n(b_n+a_1)<1.$
2002 Japan MO Finals
Let $ S$ be a set of 2002 points in the coordinate plane, no two of which have the same $ x-$ or $ y-$coordinate. For any two points $ P,Q \in S$, consider the rectangle with one diagonal $ PQ$ and the sides parallel to the axes. Denote by $ W_{PQ}$ the number of points of $ S$ lying in the interior of this rectangle. Determine the maximum $ N$ such that, no matter how the points of $ S$ are distributed, there always exist points $ P,Q$ in $ S$ with $ W_{PQ}\ge N$.
2001 Japan MO Finals
Each square of an $m\times n$ chessboard is painted black or white in such a way that for every black square, the number of black squares adjacent to it is odd (two squares are adjacent if they share one edge). Prove that the number of black squares is even.
2001 Japan MO Finals
An integer $n>0$ is written in decimal system as $\overline{a_ma_{m-1}\ldots a_1}$. Find all $n$ such that

$$n=(a_m+1)(a_{m-1}+1)\cdots (a_1+1)$$
2001 Japan MO Finals
Three nonnegative real numbers satisfy $a,b,c$ satisfy $a^2\le b^2+c^2, b^2\le c^2+a^2$ and $c^2\le a^2+b^2$. Prove the inequality

$$(a+b+c)(a^2+b^2+c^2)(a^3+b^3+c^3)\ge 4(a^6+b^6+c^6).$$

When does equality hold?
2001 Japan MO Finals
Let $p$ be a prime number and $m$ a positive integer. Show that there exists a positive integer $n$ such that the decimal representation of $p^n$ contains a string of $m$ consecutive zeros.
2001 Japan MO Finals
Suppose that $ABC$ and $PQR$ are triangles such that $A,P$ are the midpoints of $QR,BC$ respectively, and $QR,BC$ are the bisectors of $\angle BAC,\angle QPR$. Prove that $AB+AC=PQ+PR$.
2000 Japan MO Finals
Consider the points $O(0,0)$ and $A(0,1/ 2)$ on the coordinate plane. Prove that there is no finite sequence of rational points $P_1,P_2,\ldots ,P_n$ in the plane such that

$$OP_1=P_1P_2=\ldots =P_{n-1}P_n=P_nA=1 $$
2000 Japan MO Finals
Let $3n$ cards, denoted by distinct letters $a_1,a_2,\ldots ,a_{3n}$, be put in line in this order from left to right. After each shuffle, the sequence $a_1,a_2,\ldots ,a_{3n}$ is replaced by the sequence $a_3,a_6,\ldots ,a_{3n},a_2,a_5,\ldots ,a_{3n-1},a_1,a_4,\ldots ,a_{3n-2}$. Is it possible to replace the sequence of cards $1,2,\ldots ,192$ by the reverse sequence $192,191,\ldots ,1$ by a finite number of shuffles?
2000 Japan MO Finals
Given five points $A,B,C,D,E$ in a plane, no three of which are collinear, prove the inequality

$$AB+BC+CA+DE\le AD+AE+BD+BE+CD+CE $$
2000 Japan MO Finals
Given a natural number $n\ge 3$, prove that there exists a set $A_n$ with the following two properties:

1) $A_n $ consists of $n$ distinct natural numbers

2) For any $a\in A$, the remainder of the product of all elements of $A_n\backslash \{a\}$ divided by $a$ is $1$.
2000 Japan MO Finals
Finitely many lines are given in a plane. We call an intersection point a point that belongs to at least two of the given lines, and a good intersection point a point that belongs to exactly two lines. Assuming there at least two intersection points, find the minimum number of good intersection points.
1999 Japan MO Finals
One can place a stone on each of the squares of a $1999\times 1999$ board. Find the minimum number of stones that must be placed so that, for any blank square on the board, the total number of stones placed in the corresponding row and column is at least $1999$.
1999 Japan MO Finals
Let $f(x)=x^3+17$. Prove that for every integer $n\ge 2$ there exists a natural number $x$ for which $f(x)$ is divisible by $3^n$ but not by $3^{n+1}$.
1999 Japan MO Finals
From a group of $2n+1$ weights, if we remove any weight, the remaining $2n$, can be divided in two groups of $n$ elements, such that they have the same total weight. Prove all weights are equal.