Navvye/Olympiad-Problems-CSV huggingface dataset:
1987 India National Olympiad
Construct the $ \triangle ABC$, given $ h_a$, $ h_b$ (the altitudes from $ A$ and $ B$) and $ m_a$, the median from the vertex $ A$.
1987 India National Olympiad
Three congruent circles have a common point $ O$ and lie inside a given triangle. Each circle touches a pair of sides of the triangle. Prove that the incentre and the circumcentre of the triangle and the common point $ O$ are collinear.
1987 India National Olympiad
Prove that any triangle having two equal internal angle bisectors (each measured from a vertex to the opposite side) is isosceles.
1986 India National Olympiad
A person who left home between 4 p.m. and 5 p.m. returned between 5 p.m. and 6 p.m. and found that the hands of his watch had exactly exchanged place, when did he go out ?
1986 India National Olympiad
Solve

$$ \left\{ \begin{array}{l}

 \log_2 x+\log_4 y+\log_4 z=2 \\ 

 \log_3 y+\log_9 z+\log_9 x=2 \\ 

 \log_4 z+\log_{16} x+\log_{16} y=2 \\ 

 \end{array} \right.$$
1986 India National Olympiad
Two circles with radii a and b respectively touch each other externally. Let c be the radius of a circle that touches these two circles as well as a common tangent to the two circles. Prove that

$$ \frac{1}{\sqrt{c}}=\frac{1}{\sqrt{a}}+\frac{1}{\sqrt{b}}$$
1986 India National Olympiad
Find the least natural number whose last digit is 7 such that it becomes 5 times larger when this last digit is carried to the beginning of the number.
1986 India National Olympiad
If $ P(x)$ is a polynomial with integer coefficients and $ a$, $ b$, $ c$, three distinct integers, then show that it is impossible to have $ P(a)=b$, $ P(b)=c$, $ P(c)=a$.
1986 India National Olympiad
Construct a quadrilateral which is not a parallelogram, in which a pair of opposite angles and a pair of opposite sides are equal.
1986 India National Olympiad
If $ a$, $ b$, $ x$, $ y$ are integers greater than 1 such that $ a$ and $ b$ have no common factor except 1 and $ x^a = y^b$ show that $ x = n^b$, $ y = n^a$ for some integer $ n$ greater than 1.
1986 India National Olympiad
Suppose $ A_1,\dots, A_6$ are six sets each with four elements and $ B_1,\dots,B_n$ are $ n$ sets each with two elements, Let $ S = A_1 \cup A_2 \cup \cdots \cup A_6 = B_1 \cup \cdots \cup B_n$. Given that each elements of $ S$ belogs to exactly four of the $ A$'s and to exactly three of the $ B$'s, find $ n$.
1986 India National Olympiad
Show that among all quadrilaterals of a given perimeter the square has the largest area.
2019 Indonesia MO
Given that $n$ and $r$ are positive integers.

Suppose that

$$ 1 + 2 + \dots + (n - 1) = (n + 1) + (n + 2) + \dots + (n + r) $$Prove that $n$ is a composite number.
2019 Indonesia MO
Given $19$ red boxes and $200$ blue boxes filled with balls. None of which is empty.

Suppose that every red boxes have a maximum of $200$ balls and every blue boxes have a maximum of $19$ balls.

Suppose that the sum of all balls in the red boxes is less than the sum of all the balls in the blue boxes.

Prove that there exists a subset of the red boxes and a subset of the blue boxes such that their sum is the same.
2019 Indonesia MO
Given that $ABCD$ is a rectangle such that $AD > AB$, where $E$ is on $AD$ such that $BE \perp AC$. Let $M$ be the intersection of $AC$ and $BE$. Let the circumcircle of $\triangle ABE$ intersects $AC$ and $BC$ at $N$ and $F$.

Moreover, let the circumcircle of $\triangle DNE$ intersects $CD$ at $G$.

Suppose $FG$ intersects $AB$ at $P$.

Prove that $PM = PN$.
2019 Indonesia MO
Let us define a $\textit{triangle equivalence}$ a group of numbers that can be arranged as shown



$a+b=c$

$d+e+f=g+h$

$i+j+k+l=m+n+o$

and so on...



Where at the $j$-th row, the left hand side has $j+1$ terms and the right hand side has $j$ terms.



Now, we are given the first $N^2$ positive integers, where $N$ is a positive integer. Suppose we eliminate any one number that has the same parity with $N$.



Prove that the remaining $N^2-1$ numbers can be formed into a $\textit{triangle equivalence}$.



For example, if $10$ is eliminated from the first $16$ numbers, the remaining numbers can be arranged into a $\textit{triangle equivalence}$ as shown.



$1+3=4$

$2+5+8=6+9$

$7+11+12+14=13+15+16$
2019 Indonesia MO
Given that $a$ and $b$ are real numbers such that for infinitely many positive integers $m$ and $n$,

$$ \lfloor an + b \rfloor \ge \lfloor a + bn \rfloor $$$$ \lfloor a + bm \rfloor \ge \lfloor am + b \rfloor $$Prove that $a = b$.
2019 Indonesia MO
Given a circle with center $O$, such that $A$ is not on the circumcircle.

Let $B$ be the reflection of $A$ with respect to $O$. Now let $P$ be a point on the circumcircle. The line perpendicular to $AP$ through $P$ intersects the circle at $Q$.

Prove that $AP \times BQ$ remains constant as $P$ varies.
2019 Indonesia MO
Determine all solutions of

$$ x + y^2 = p^m $$$$ x^2 + y = p^n $$For $x,y,m,n$ positive integers and $p$ being a prime.
2019 Indonesia MO
Let $n > 1$ be a positive integer and $a_1, a_2, \dots, a_2n \in \{ -n, -n + 1, \dots, n - 1, n \}$. Suppose

$$ a_1 + a_2 + a_3 + \dots + a_{2n} = n + 1 $$Prove that some of $a_1, a_2, \dots, a_{2n}$ have sum 0.
2018 Indonesia MO
Let $a$ be a positive integer such that $\gcd(an+1, 2n+1) = 1$ for all integer $n$.



a) Prove that $\gcd(a-2, 2n+1) = 1$ for all integer $n$.

b) Find all possible $a$.
2018 Indonesia MO
Let $\Gamma_1, \Gamma_2$ be circles that touch at a point $A$, and $\Gamma_2$ is inside $\Gamma_1$. Let $B$ be on $\Gamma_2$, and let $AB$ intersect $\Gamma_1$ on $C$. Let $D$ be on $\Gamma_1$ and $P$ be on the line $CD$ (may be outside of the segment $CD$). $BP$ intersects $\Gamma_2$ at $Q$. Prove that $A,D,P,Q$ lie on a circle.
2018 Indonesia MO
Alzim and Badril are playing a game on a hexagonal lattice grid with 37 points (4 points a side), all of them uncolored. On his turn, Alzim colors one uncolored point with the color red, and Badril colors two uncolored points with the color blue. The game ends either when there is an equilateral triangle whose vertices are all red, or all points are colored. If the former happens, then Alzim wins, otherwise Badril wins. If Alzim starts the game, does Alzim have a strategy to guarantee victory?
2018 Indonesia MO
In a game, Andi and a computer take turns. At the beginning, the computer shows a polynomial $x^2 + mx + n$ where $m,n \in \mathbb{Z}$, such that it doesn't have real roots. Andi then begins the game. On his turn, Andi may change a polynomial in the form $x^2 + ax + b$ into either $x^2 + (a+b)x + b$ or $x^2 + ax + (a+b)$. However, Andi may only choose a polynomial that has real roots. On the computer's turn, it simply switches the coefficient of $x$ and the constant of the polynomial. Andi loses if he can't continue to play. Find all $(m,n)$ such that Andi always loses (in finitely many turns).
2018 Indonesia MO
Find all triples of reals $(x,y,z)$ satisfying:



$$\begin{cases}

\frac{1}{3} \min \{x,y\} + \frac{2}{3} \max \{x,y\} = 2017 \\
\frac{1}{3} \min \{y,z\} + \frac{2}{3} \max \{y,z\} = 2018 \\
\frac{1}{3} \min \{z,x\} + \frac{2}{3} \max \{z,x\} = 2019 \\
\end{cases}$$
2018 Indonesia MO
Find all prime numbers $p$ such that there exists a positive integer $n$ where $2^n p^2 + 1$ is a square number.
2018 Indonesia MO
Suppose there are three empty buckets and $n \ge 3$ marbles. Ani and Budi play a game. For the first turn, Ani distributes all the marbles into the buckets so that each bucket has at least one marble. Then Budi and Ani alternate turns, starting with Budi. On a turn, the current player may choose a bucket and take 1, 2, or 3 marbles from it. The player that takes the last marble wins. Find all $n$ such that Ani has a winning strategy, including what Ani's first move (distributing the marbles) should be for these $n$.
2018 Indonesia MO
Let $I, O$ be the incenter and circumcenter of the triangle $ABC$ respectively. Let the excircle $\omega_A$ of $ABC$ be tangent to the side $BC$ on $N$, and tangent to the extensions of the sides $AB, AC$ on $K, M$ respectively. If the midpoint of $KM$ lies on the circumcircle of $ABC$, prove that $O, I, N$ are collinear.
2017 Indonesia MO
$ABCD$ is a parallelogram. $g$ is a line passing $A$. Prove that the distance from $C$ to $g$ is either the sum or the difference of the distance from $B$ to $g$, and the distance from $D$ to $g$.
2017 Indonesia MO
Five people are gathered in a meeting. Some pairs of people shakes hands. An ordered triple of people $(A,B,C)$ is a trio if one of the following is true:



A shakes hands with B, and B shakes hands with C, or

A doesn't shake hands with B, and B doesn't shake hands with C.





If we consider $(A,B,C)$ and $(C,B,A)$ as the same trio, find the minimum possible number of trios.
2017 Indonesia MO
A positive integer $d$ is special if every integer can be represented as $a^2 + b^2 - dc^2$ for some integers $a, b, c$.



Find the smallest positive integer that is not special.

Prove 2017 is special.
2017 Indonesia MO
Determine all pairs of distinct real numbers $(x, y)$ such that both of the following are true:



$x^{100} - y^{100} = 2^{99} (x-y)$

$x^{200} - y^{200} = 2^{199} (x-y)$
2017 Indonesia MO
A polynomial $P$ has integral coefficients, and it has at least 9 different integral roots. Let $n$ be an integer such that $|P(n)| < 2017$. Prove that $P(n) = 0$.
2017 Indonesia MO
Find the number of positive integers $n$ not greater than 2017 such that $n$ divides $20^n + 17k$ for some positive integer $k$.
2017 Indonesia MO
Let $ABCD$ be a parallelogram. $E$ and $F$ are on $BC, CD$ respectively such that the triangles $ABE$ and $BCF$ have the same area. Let $BD$ intersect $AE, AF$ at $M, N$ respectively. Prove there exists a triangle whose side lengths are $BM, MN, ND$.
2017 Indonesia MO
A field is made of $2017 \times 2017$ unit squares. Luffy has $k$ gold detectors, which he places on some of the unit squares, then he leaves the area. Sanji then chooses a $1500 \times 1500$ area, then buries a gold coin on each unit square in this area and none other. When Luffy returns, a gold detector beeps if and only if there is a gold coin buried underneath the unit square it's on. It turns out that by an appropriate placement, Luffy will always be able to determine the $1500 \times 1500$ area containing the gold coins by observing the detectors, no matter how Sanji places the gold coins. Determine the minimum value of $k$ in which this is possible.
2016 Indonesia MO
Let $ABCD$ be a cyclic quadrilateral wih both diagonals perpendicular to each other and intersecting at point $O$. Let $E,F,G,H$ be the orthogonal projections of $O$ on sides $AB,BC,CD,DA$ respectively.

a. Prove that $\angle EFG + \angle  GHE = 180^o$

b. Prove that $OE$ bisects angle  $\angle FEH$ .
2016 Indonesia MO
Determine all triples of natural numbers $(a,b, c)$ with $b> 1$ such that $2^c + 2^{2016} = a^b$.
2016 Indonesia MO
There are $5$ boxes arranged in a circle. At first, there is one a box containing one ball, while the other boxes are empty. At each step, we can do one of the following two operations:

i. select one box that is not empty, remove one ball from the box and add one ball into both boxes next to the box,

ii. select an empty box next to a non-empty box, from the box the non-empty one moves one ball to the empty box.

Is it possible, that after a few steps, obtained conditions where each box contains exactly $17^{5^{2016}}$ balls?
2016 Indonesia MO
Given triangle $ABC$ such that angles $A$, $B$, $C$ satisfy

$$

\frac{\cos A}{20}+\frac{\cos B}{21}+\frac{\cos C}{29}=\frac{29}{420}

$$Prove that $ABC$ is right angled triangle
2016 Indonesia MO
Given positive integers $a,b,c,d$ such that $a\mid c^d$ and $b\mid d^c$. Prove that

$$

ab\mid (cd)^{max(a,b)}

$$
2016 Indonesia MO
For a quadrilateral $ABCD$, we call a square $amazing$ if all of its sides(extended if necessary) pass through distinct vertices of $ABCD$(no side passing through 2 vertices). Prove that for an arbitrary $ABCD$ such that its diagonals are not perpendicular, there exist at least 6 $amazing$ squares
2016 Indonesia MO
Suppose that $p> 2$ is a prime number. For each integer $k = 1, 2,..., p-1$, denote $r_k$ as the remainder of the division $k^p$ by $p^2$. Prove that $r_1+r_2+r_3+...+r_{p-1}=\frac{p^2(p-1)}{2}$
2016 Indonesia MO
Determine with proof, the number of permutations $a_1,a_2,a_3,...,a_{2016}$ of $1,2,3,...,2016$ such that the value of $|a_i-i|$ is fixed for all $i=1,2,3,...,2016$, and its value is an integer multiple of $3$.
2015 Indonesia MO
Albert, Bernard, and Cheryl are playing marbles. At the beginning, each of them brings 5 red marbles, 7 green marbles and 13 blue marbles and in the middle of the table, there is a box of infinitely many red, blue and green marbles. In each turn, each player may choose 2 marbles of different color and replace them with 2 marbles of the third color. After a finite number of steps, this conversation happens.



Albert : " I have only red marbles"

Bernard : "I have only blue marbles"

Cheryl: "I have only green marbles"



Which of the three are lying?
2015 Indonesia MO
For every natural number $a$ and $b$, define the notation $[a,b]$ as the least common multiple of $a $ and $b$ and the notation $(a,b)$ as the greatest common divisor of $a$ and $b$. Find all $n \in \mathbb{N}$ that satisfies

$$

4 \sum_{k=1}^{n} [n,k] = 1 + \sum_{k=1}^{n} (n,k) + 2n^2 \sum_{k=1}^{n} \frac{1}{(n,k)}

$$
2015 Indonesia MO
Given an acute triangle $ABC$. $\Gamma _{B}$ is a circle that passes through $AB$, tangent to $AC$ at $A$ and centered at $O_{B}$. Define $\Gamma_C$ and $O_C$ the same way. Let the altitudes of $\triangle ABC$ from $B$ and $C$ meets the circumcircle of $\triangle ABC$ at $X$ and $Y$, respectively. Prove that $A$, the midpoint of $XY$ and the midpoint of $O_{B}O_{C}$ is collinear.
2015 Indonesia MO
Let function pair $f,g : \mathbb{R^+} \rightarrow \mathbb{R^+}$ satisfies

$$

f(g(x)y + f(x)) = (y+2015)f(x)

$$for every $x,y \in \mathbb{R^+} $

a. Prove that $f(x) = 2015g(x)$ for every $x \in \mathbb{R^+}$

b. Give an example of function pair $(f,g)$ that satisfies the statement above and $f(x), g(x) \geq 1$ for every $x \in \mathbb{R^+}$
2015 Indonesia MO
Let $ABC$ be an acute angled triangle with circumcircle $O$.  Line $AO$ intersects the circumcircle of triangle  $ABC$ again at point $D$. Let $P$ be a point on the side $BC$. Line passing through $P$ perpendicular  to $AP$ intersects lines  $DB$ and $DC$ at $E$ and $F$ respectively . Line  passing through $D$ perpendicular to $BC$ intersects $EF$ at point $Q$. Prove that $EQ = FQ$ if and only if $BP = CP$.
2015 Indonesia MO
Let $a,b,c$ be positive real numbers. Prove that

$\sqrt{\frac{a}{b+c}+\frac{b}{c+a}}+\sqrt{\frac{b}{c+a}+\frac{c}{a+b}}+\sqrt{\frac{c}{a+b}+\frac{a}{b+c}}\ge 3$
2015 Indonesia MO
It is known that there are $3$ buildings in the same shape which are located in an equilateral triangle. Each building has a $2015$ floor with each floor having one window. In all three buildings, every $1$st floor is uninhabited, while each floor of others have exactly one occupant. All windows will be colored with one of red, green or blue. The residents of each floor of a building can see the color of the window in the other buildings of the the same floor and one floor just below it, but they cannot see the colors of the other windows of the two buildings. Besides that, sresidents cannot see the color of the window from any floor in the building itself. For example, resident of the $10$th floor can see the colors of the $9$th and $10$th floor windows for the other buildings (a total of $4$ windows) and he can't see the color of the other window. We want to color the windows so that each resident can see at lest $1$ window of each color. How many ways are there to color those windows?
2014 Indonesia MO
Is it possible to fill a $3 \times 3$ grid with each of the numbers $1,2,\ldots,9$ once each such that the sum of any two numbers sharing a side is prime?
2014 Indonesia MO
For some positive integers $m,n$, the system $x+y^2 = m$ and $x^2+y = n$ has exactly one integral solution $(x,y)$. Determine all possible values of $m-n$.
2014 Indonesia MO
Let $ABCD$ be a trapezoid (quadrilateral with one pair of parallel sides) such that $AB < CD$. Suppose that $AC$ and $BD$ meet at $E$ and $AD$ and $BC$ meet at $F$. Construct the parallelograms $AEDK$ and $BECL$. Prove that $EF$ passes through the midpoint of the segment $KL$.
2014 Indonesia MO
Determine all polynomials with integral coefficients $P(x)$ such that if $a,b,c$ are the sides of a right-angled triangle, then $P(a), P(b), P(c)$ are also the sides of a right-angled triangle. (Sides of a triangle are necessarily positive. Note that it's not necessary for the order of sides to be preserved; if $c$ is the hypotenuse of the first triangle, it's not necessary that $P(c)$ is the hypotenuse of the second triangle, and similar with the others.)
2014 Indonesia MO
A sequence of positive integers $a_1, a_2, \ldots$ satisfies $a_k + a_l = a_m + a_n$ for all positive integers $k,l,m,n$ satisfying $kl = mn$. Prove that if $p$ divides $q$ then $a_p \le a_q$.
2014 Indonesia MO
Let $ABC$ be a triangle. Suppose $D$ is on $BC$ such that $AD$ bisects $\angle BAC$. Suppose $M$ is on $AB$ such that $\angle MDA = \angle ABC$, and $N$ is on $AC$ such that $\angle NDA = \angle ACB$. If $AD$ and $MN$ intersect on $P$, prove that $AD^3 = AB \cdot AC \cdot AP$.
2014 Indonesia MO
Suppose that $k,m,n$ are positive integers with $k \le n$. Prove that:

$$\sum_{r=0}^m \dfrac{k \binom{m}{r} \binom{n}{k}}{(r+k) \binom{m+n}{r+k}} = 1$$
2014 Indonesia MO
A positive integer is called beautiful if it can be represented in the form $\dfrac{x^2+y^2}{x+y}$ for two distinct positive integers $x,y$. A positive integer that is not beautiful is ugly.



a) Prove that $2014$ is a product of a beautiful number and an ugly number.

b) Prove that the product of two ugly numbers is also ugly.
2013 Indonesia MO
In a $4 \times 6$ grid, all edges and diagonals are drawn (see attachment). Determine the number of parallelograms in the grid that uses only the line segments drawn and none of its four angles are right.
2013 Indonesia MO
Let $ABC$ be an acute triangle and $\omega$ be its circumcircle. The bisector of $\angle BAC$ intersects $\omega$ at [another point] $M$. Let $P$ be a point on $AM$ and inside $\triangle ABC$. Lines passing $P$ that are parallel to $AB$ and $AC$ intersects $BC$ on $E, F$ respectively. Lines $ME, MF$ intersects $\omega$ at points $K, L$ respectively. Prove that $AM, BL, CK$ are concurrent.
2013 Indonesia MO
Determine all positive real $M$ such that for any positive reals $a,b,c$, at least one of $a + \dfrac{M}{ab}, b + \dfrac{M}{bc}, c + \dfrac{M}{ca}$ is greater than or equal to $1+M$.
2013 Indonesia MO
Suppose $p > 3$ is a prime number and

$$S = \sum_{2 \le i < j < k \le p-1} ijk$$

Prove that $S+1$ is divisible by $p$.
2013 Indonesia MO
Let $P$ be a quadratic (polynomial of degree two) with a positive leading coefficient and negative discriminant. Prove that there exists three quadratics $P_1, P_2, P_3$ such that:

- $P(x) = P_1(x) + P_2(x) + P_3(x)$

- $P_1, P_2, P_3$ have positive leading coefficients and zero discriminants (and hence each has a double root)

- The roots of $P_1, P_2, P_3$ are different
2013 Indonesia MO
A positive integer $n$ is called "strong" if there exists a positive integer $x$ such that $x^{nx} + 1$ is divisible by $2^n$.



a. Prove that $2013$ is strong.

b. If $m$ is strong, determine the smallest $y$ (in terms of $m$) such that $y^{my} + 1$ is divisible by $2^m$.
2013 Indonesia MO
Let $ABCD$ be a parallelogram. Construct squares $ABC_1D_1, BCD_2A_2, CDA_3B_3, DAB_4C_4$ on the outer side of the parallelogram. Construct a square having $B_4D_1$ as one of its sides and it is on the outer side of $AB_4D_1$ and call its center $O_A$. Similarly do it for $C_1A_2, D_2B_3, A_3C_4$ to obtain $O_B, O_C, O_D$. Prove that $AO_A = BO_B = CO_C = DO_D$.
2013 Indonesia MO
Let $A$ be a set of positive integers. $A$ is called "balanced" if [and only if] the number of 3-element subsets of $A$ whose elements add up to a multiple of $3$ is equal to the number of 3-element subsets of $A$ whose elements add up to not a multiple of $3$.



a. Find a 9-element balanced set.

b. Prove that no set of $2013$ elements can be balanced.
2012 Indonesia MO
Show that for any positive integers $a$ and $b$, the number $$n=\mathrm{LCM}(a,b)+\mathrm{GCD}(a,b)-a-b$$ is an even non-negative integer.



Proposer: Nanang Susyanto
2012 Indonesia MO
Let $n\ge 3$ be an integer, and let $a_2,a_3,\ldots ,a_n$ be positive real numbers such that $a_{2}a_{3}\cdots a_{n}=1$. Prove that

$$(1 + a_2)^2 (1 + a_3)^3 \dotsm (1 + a_n)^n > n^n.$$



Proposed by Angelo Di Pasquale, Australia
2012 Indonesia MO
Given an acute triangle $ABC$ with $AB>AC$ that has circumcenter $O$. Line $BO$ and $CO$ meet the bisector of $\angle BAC$ at $P$ and $Q$, respectively. Moreover, line $BQ$ and $CP$ meet at $R$. Show that $AR$ is perpendicular to $BC$.



Proposer: Soewono and Fajar Yuliawan
2012 Indonesia MO
Given $2012$ distinct points $A_1,A_2,\dots,A_{2012}$ on the Cartesian plane. For any permutation $B_1,B_2,\dots,B_{2012}$ of $A_1,A_2,\dots,A_{2012}$ define the shadow of a point $P$ as follows: Point $P$ is rotated by $180^{\circ}$ around $B_1$ resulting $P_1$, point $P_1$ is rotated by $180^{\circ}$ around $B_2$ resulting $P_2$, ..., point $P_{2011}$ is rotated by $180^{\circ}$ around $B_{2012}$ resulting $P_{2012}$. Then, $P_{2012}$ is called the shadow of $P$ with respect to the permutation $B_1,B_2,\dots,B_{2012}$.

Let $N$ be the number of different shadows of $P$ up to all permutations of $A_1,A_2,\dots,A_{2012}$. Determine the maximum value of $N$.



Proposer: Hendrata Dharmawan
2012 Indonesia MO
Given positive integers $m$ and $n$. Let $P$ and $Q$ be two collections of $m \times n$ numbers of $0$ and $1$, arranged in $m$ rows and $n$ columns. An example of such collections for $m=3$ and $n=4$ is

$$\left[ \begin{array}{cccc}

1 & 1 & 1 & 0 \\
1 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 \end{array} \right].$$

Let those two collections satisfy the following properties:

(i) On each row of $P$, from left to right, the numbers are non-increasing,

(ii) On each column of $Q$, from top to bottom, the numbers are non-increasing,

(iii) The sum of numbers on the row in $P$ equals to the same row in $Q$,

(iv) The sum of numbers on the column in $P$ equals to the same column in $Q$.

Show that the number on row $i$ and column $j$ of $P$ equals to the number on row $i$ and column $j$ of $Q$ for $i=1,2,\dots,m$ and $j=1,2,\dots,n$.



Proposer: Stefanus Lie
2012 Indonesia MO
Let $\mathbb{R}^+$ be the set of all positive real numbers. Show that there is no function $f:\mathbb{R}^+ \to \mathbb{R}^+$ satisfying

$$f(x+y)=f(x)+f(y)+\dfrac{1}{2012}$$

for all positive real numbers $x$ and $y$.



Proposer: Fajar Yuliawan
2012 Indonesia MO
Let $n$ be a positive integer. Show that the equation $$\sqrt{x}+\sqrt{y}=\sqrt{n}$$ have solution of pairs of positive integers $(x,y)$ if and only if $n$ is divisible by some perfect square greater than $1$.



Proposer: Nanang Susyanto
2012 Indonesia MO
Given a triangle $ABC$, let the bisector of $\angle BAC$ meets the side $BC$ and circumcircle of triangle $ABC$ at $D$ and $E$, respectively. Let $M$ and $N$ be the midpoints of $BD$ and $CE$, respectively. Circumcircle of triangle $ABD$ meets $AN$ at $Q$. Circle passing through $A$ that is tangent to $BC$ at $D$ meets line $AM$ and side $AC$ respectively at $P$ and $R$. Show that the four points $B,P,Q,R$ lie on the same line.



Proposer: Fajar Yuliawan