Navvye/Olympiad-Problems-CSV huggingface dataset:
2011 Indonesia MO
For a number $n$ in base $10$, let $f(n)$ be the sum of all numbers possible by removing some digits of $n$ (including none and all). For example, if $n = 1234$, $f(n) = 1234 + 123 + 124 + 134 + 234 + 12 + 13 + 14 + 23 + 24 + 34 + 1 + 2 + 3 + 4 = 1979$; this is formed by taking the sums of all numbers obtained when removing no digit from $n$ (1234), removing one digit from $n$ (123, 124, 134, 234), removing two digits from $n$ (12, 13, 14, 23, 24, 34), removing three digits from $n$ (1, 2, 3, 4), and removing all digits from $n$ (0). If $p$ is a 2011-digit integer, prove that $f(p)-p$ is divisible by $9$.



Remark: If a number appears twice or more, it is counted as many times as it appears. For example, with the number $101$, $1$ appears three times (by removing the first digit, giving $01$ which is equal to $1$, removing the first two digits, or removing the last two digits), so it is counted three times.
2011 Indonesia MO
For each positive integer $n$, let $s_n$ be the number of permutations $(a_1, a_2, \cdots, a_n)$ of $(1, 2, \cdots, n)$ such that $\dfrac{a_1}{1} + \dfrac{a_2}{2} + \cdots + \dfrac{a_n}{n}$ is a positive integer. Prove that $s_{2n} \ge n$ for all positive integer $n$.
2011 Indonesia MO
Given an acute triangle $ABC$, let $l_a$ be the line passing $A$ and perpendicular to $AB$, $l_b$ be the line passing $B$ and perpendicular to $BC$, and $l_c$ be the line passing $C$ and perpendicular to $CA$. Let $D$ be the intersection of $l_b$ and $l_c$, $E$ be the intersection of $l_c$ and $l_a$, and $F$ be the intersection of $l_a$ and $l_b$. Prove that the area of the triangle $DEF$ is at least three times of the area of $ABC$.
2011 Indonesia MO
An island has $10$ cities, where some of the possible pairs of cities are connected by roads. A tour route is a route starting from a city, passing exactly eight out of the other nine cities exactly once each, and returning to the starting city. (In other words, it is a loop that passes only nine cities instead of all ten cities.) For each city, there exists a tour route that doesn't pass the given city. Find the minimum number of roads on the island.
2011 Indonesia MO
[asy]

draw((0,1)--(4,1)--(4,2)--(0,2)--cycle);

draw((2,0)--(3,0)--(3,3)--(2,3)--cycle);

draw((1,1)--(1,2));

label("1",(0.5,1.5));

label("2",(1.5,1.5));

label("32",(2.5,1.5));

label("16",(3.5,1.5));

label("8",(2.5,0.5));

label("6",(2.5,2.5));

[/asy]

The image above is a net of a unit cube. Let $n$ be a positive integer, and let $2n$ such cubes are placed to build a $1 \times 2 \times n$ cuboid which is placed on a floor. Let $S$ be the sum of all numbers on the block visible (not facing the floor). Find the minimum value of $n$ such that there exists such cuboid and its placement on the floor so $S > 2011$.
2011 Indonesia MO
Let a sequence of integers $a_0, a_1, a_2, \cdots, a_{2010}$ such that $a_0 = 1$ and $2011$ divides $a_{k-1}a_k - k$ for all $k = 1, 2, \cdots, 2010$. Prove that $2011$ divides $a_{2010} + 1$.
2011 Indonesia MO
Let $a,b,c \in \mathbb{R}^+$ and $abc = 1$ such that $a^{2011} + b^{2011} + c^{2011} < \dfrac{1}{a^{2011}} + \dfrac{1}{b^{2011}} + \dfrac{1}{c^{2011}}$. Prove that $a + b + c < \dfrac{1}{a} + \dfrac{1}{b} + \dfrac{1}{c}$.
2011 Indonesia MO
Given a triangle $ABC$. Its incircle is tangent to $BC, CA, AB$ at $D, E, F$ respectively. Let $K, L$ be points on $CA, AB$ respectively such that $K \neq A \neq L, \angle EDK = \angle ADE, \angle FDL = \angle ADF$. Prove that the circumcircle of $AKL$ is tangent to the incircle of $ABC$.
2010 Indonesia MO
Let $a,b,c$ be three different positive integers. Show that the sequence

$$a+b+c,ab+bc+ca,3abc$$

could be neither an arithmetic nor geometric progression.



Fajar Yuliawan, Bandung
2010 Indonesia MO
Given an acute triangle $ABC$ with $AC>BC$ and the circumcenter of triangle $ABC$ is $O$. The altitude of triangle $ABC$ from $C$ intersects $AB$ and the circumcircle at $D$ and $E$, respectively. A line which passed through $O$ which is parallel to $AB$ intersects $AC$ at $F$. Show that the line $CO$, the line which passed through $F$ and perpendicular to $AC$, and the line which passed through $E$ and parallel with $DO$ are concurrent.



Fajar Yuliawan, Bandung
2010 Indonesia MO
A mathematical competition was attended by 120 participants from several contingents. At the closing ceremony, each participant gave 1 souvenir each to every other participants from the same contingent, and 1 souvenir to any person from every other contingents. It is known that there are 3840 souvenirs whom were exchanged.

Find the maximum possible contingents such that the above condition still holds?



Raymond Christopher Sitorus, Singapore
2010 Indonesia MO
Given that $m$ and $n$ are positive integers with property:

$$(mn)\mid(m^{2010}+n^{2010}+n)$$

Show that there exists a positive integer $k$ such that $n=k^{2010}$



Nanang Susyanto, Yogyakarta
2010 Indonesia MO
$m$ boys and $n$ girls ($m>n$) sat across a round table, supervised by a teacher, and they did a game, which went like this. At first, the teacher pointed a boy to start the game. The chosen boy put a coin on the table. Then, consecutively in a clockwise order, everyone did his turn. If the next person is a boy, he will put a coin to the existing pile of coins. If the next person is a girl, she will take a coin from the existing pile of coins. If there is no coin on the table, the game ends. Notice that depending on the chosen boy, the game could end early, or it could go for a full turn. If the teacher wants the game to go for at least a full turn, how many possible boys could be chosen?



Hendrata Dharmawan, Boston, USA
2010 Indonesia MO
Find all positive integers $n>1$ such that

$$\tau(n)+\phi(n)=n+1$$

Which in this case, $\tau(n)$ represents the amount of positive divisors of $n$, and $\phi(n)$ represents the amount of positive integers which are less than $n$ and relatively prime with $n$.



Raja Oktovin, Pekanbaru
2010 Indonesia MO
Given 2 positive reals $a$ and $b$. There exists 2 polynomials $F(x)=x^2+ax+b$ and $G(x)=x^2+bx+a$ such that all roots of polynomials $F(G(x))$ and $G(F(x))$ are real. Show that $a$ and $b$ are more than $6$.



Raja Oktovin, Pekanbaru
2010 Indonesia MO
Given an acute triangle $ABC$ with circumcenter $O$ and orthocenter $H$. Let $K$ be a point inside $ABC$ which is not $O$ nor $H$. Point $L$ and $M$ are located outside the triangle $ABC$ such that $AKCL$ and $AKBM$ are parallelogram. At last, let $BL$ and $CM$ intersects at $N$, and let $J$ be the midpoint of $HK$. Show that $KONJ$ is also a parallelogram.



Raja Oktovin, Pekanbaru
2009 Indonesia MO
Find all positive integers $ n\in\{1,2,3,\ldots,2009\}$ such that

$$ 4n^6 + n^3 + 5$$

is divisible by $ 7$.
2009 Indonesia MO
For any real $ x$, let $ \lfloor x\rfloor$ be the largest integer that is not more than $ x$. Given a sequence of positive integers $ a_1,a_2,a_3,\ldots$ such that $ a_1>1$ and

$$ \left\lfloor\frac{a_1+1}{a_2}\right\rfloor=\left\lfloor\frac{a_2+1}{a_3}\right\rfloor=\left\lfloor\frac{a_3+1}{a_4}\right\rfloor=\cdots$$

Prove that

$$ \left\lfloor\frac{a_n+1}{a_{n+1}}\right\rfloor\leq1$$

holds for every positive integer $ n$.
2009 Indonesia MO
For every triangle $ ABC$, let $ D,E,F$ be a point located on segment $ BC,CA,AB$, respectively. Let $ P$ be the intersection of $ AD$ and $ EF$. Prove that:

$$ \frac{AB}{AF}\times DC+\frac{AC}{AE}\times DB=\frac{AD}{AP}\times BC$$
2009 Indonesia MO
In an island, there exist 7 towns and a railway system which connected some of the towns. Every railway segment connects 2 towns, and in every town there exists at least 3 railway segments that connects the town to another towns. Prove that there exists a route that visits 4 different towns once and go back to the original town. (Example: $ A-B-C-D-A$)
2009 Indonesia MO
In a drawer, there are at most $ 2009$ balls, some of them are white, the rest are blue, which are randomly distributed. If two balls were taken at the same time, then the probability that the balls are both blue or both white is $ \frac12$. Determine the maximum amount of white balls in the drawer, such that the probability statement is true?
2009 Indonesia MO
Find the lowest possible values from the function

$$ f(x) = x^{2008} - 2x^{2007} + 3x^{2006} - 4x^{2005} + 5x^{2004} - \cdots - 2006x^3 + 2007x^2 - 2008x + 2009$$

for any real numbers $ x$.
2009 Indonesia MO
A pair of integers $ (m,n)$ is called good if

$$ m\mid n^2 + n \ \text{and} \ n\mid m^2 + m$$

Given 2 positive integers $ a,b > 1$ which are relatively prime, prove that there exists a good pair $ (m,n)$ with $ a\mid m$ and $ b\mid n$, but $ a\nmid n$ and $ b\nmid m$.
2009 Indonesia MO
Given an acute triangle $ ABC$. The incircle of triangle $ ABC$ touches $ BC,CA,AB$ respectively at $ D,E,F$. The angle bisector of $ \angle A$ cuts $ DE$ and $ DF$ respectively at $ K$ and $ L$. Suppose $ AA_1$ is one of the altitudes of triangle $ ABC$, and $ M$ be the midpoint of $ BC$.

(a) Prove that $ BK$ and $ CL$ are perpendicular with the angle bisector of $ \angle BAC$.

(b) Show that $ A_1KML$ is a cyclic quadrilateral.
2008 Indonesia MO
Given triangle $ ABC$. Points $ D,E,F$ outside triangle $ ABC$ are chosen such that triangles $ ABD$, $ BCE$, and $ CAF$ are equilateral triangles. Prove that cicumcircles of these three triangles are concurrent.
2008 Indonesia MO
Prove that for $ x,y\in\mathbb{R^ + }$,

$ \frac {1}{(1 + \sqrt {x})^{2}} + \frac {1}{(1 + \sqrt {y})^{2}} \ge \frac {2}{x + y + 2}$
2008 Indonesia MO
Find all natural number which can be expressed in

$ \frac{a+b}{c}+\frac{b+c}{a}+\frac{c+a}{b}$

where $ a,b,c\in \mathbb{N}$ satisfy

$ \gcd(a,b)=\gcd(b,c)=\gcd(c,a)=1$
2008 Indonesia MO
Let $ A = \{1,2,\ldots,2008\}$

a) Find the number of subset of $ A$ which satisfy : the product of its elements is divisible by 7

b) Let $ N(i)$ denotes the number of subset of $ A$ which sum of its elements remains $ i$ when divided by 7. Prove that $ N(0) - N(1) + N(2) - N(3) + N(4) - N(5) + N(6)-N(7) = 0$



EDITED : thx for cosinator.. BTW, your statement and my correction give 80% hint of the solution :D
2008 Indonesia MO
Let $ m,n > 1$ are integers which satisfy $ n|4^m - 1$ and $ 2^m|n - 1$. Is it a must that $ n = 2^{m} + 1$?
2008 Indonesia MO
In a group of 21 persons, every two person communicate with different radio frequency. It's possible for two person to not communicate (means there's no frequency occupied to connect them). Only one frequency used by each couple, and it's unique for every couple. In every 3 persons, exactly two of them is not communicating to each other. Determine the maximum number of frequency required for this group. Explain your answer.
2008 Indonesia MO
Given triangle $ ABC$ with sidelengths $ a,b,c$. Tangents to incircle of $ ABC$ that parallel with triangle's sides form three small triangle (each small triangle has 1 vertex of $ ABC$). Prove that the sum of area of incircles of these three small triangles and the area of incircle of triangle $ ABC$ is equal to

$ \frac{\pi (a^{2}+b^{2}+c^{2})(b+c-a)(c+a-b)(a+b-c)}{(a+b+c)^{3}}$



(hmm,, looks familiar, isn't it? :wink: )
2008 Indonesia MO
Find all function $ f: \mathbb{N}\rightarrow\mathbb{N}$ satisfy $ f(mn)+f(m+n)=f(m)f(n)+1$ for all natural number $ n$
2007 Indonesia MO
Let $ ABC$ be a triangle with $ \angle ABC=\angle ACB=70^{\circ}$. Let point $ D$ on side $ BC$ such that $ AD$ is the altitude, point $ E$ on side $ AB$ such that $ \angle ACE=10^{\circ}$, and point $ F$ is the intersection of $ AD$ and $ CE$. Prove that $ CF=BC$.
2007 Indonesia MO
For every positive integer $ n$, $ b(n)$ denote the number of positive divisors of $ n$ and $ p(n)$ denote the sum of all positive divisors of $ n$. For example, $ b(14)=4$ and $ p(14)=24$. Let $ k$ be a positive integer greater than $ 1$.



(a) Prove that there are infinitely many positive integers $ n$ which satisfy $ b(n)=k^2-k+1$.



(b) Prove that there are finitely many positive integers $ n$ which satisfy $ p(n)=k^2-k+1$.
2007 Indonesia MO
Let $ a,b,c$ be positive real numbers which satisfy $ 5(a^2+b^2+c^2)<6(ab+bc+ca)$. Prove that these three inequalities hold: $ a+b>c$, $ b+c>a$, $ c+a>b$.
2007 Indonesia MO
A 10-digit arrangement $ 0,1,2,3,4,5,6,7,8,9$ is called beautiful if (i) when read left to right, $ 0,1,2,3,4$ form an increasing sequence, and $ 5,6,7,8,9$ form a decreasing sequence, and (ii) $ 0$ is not the leftmost digit. For example, $ 9807123654$ is a beautiful arrangement. Determine the number of beautiful arrangements.
2007 Indonesia MO
Let $ r$, $ s$ be two positive integers and $ P$ a 'chessboard' with $ r$ rows and $ s$ columns. Let $ M$ denote the maximum value of rooks placed on $ P$ such that no two of them attack each other.



(a) Determine $ M$.



(b) How many ways to place $ M$ rooks on $ P$ such that no two of them attack each other?



[Note: In chess, a rook moves and attacks in a straight line, horizontally or vertically.]
2007 Indonesia MO
Find all triples $ (x,y,z)$ of real numbers which satisfy the simultaneous equations



$$ x = y^3 + y - 8$$



$$y = z^3 + z - 8$$



$$ z = x^3 + x - 8.$$
2007 Indonesia MO
Points $ A,B,C,D$ are on circle $ S$, such that $ AB$ is the diameter of $ S$, but $ CD$ is not the diameter. Given also that $ C$ and $ D$ are on different sides of $ AB$. The tangents of $ S$ at $ C$ and $ D$ intersect at $ P$. Points $ Q$ and $ R$ are the intersections of line $ AC$ with line $ BD$ and line $ AD$ with line $ BC$, respectively.



(a) Prove that $ P$, $ Q$, and $ R$ are collinear.



(b) Prove that $ QR$ is perpendicular to line $ AB$.
2007 Indonesia MO
Let $ m$ and $ n$ be two positive integers. If there are infinitely many integers $ k$ such that $ k^2+2kn+m^2$ is a perfect square, prove that $ m=n$.
2006 Indonesia MO
Find all pairs $ (x,y)$ of real numbers which satisfy $ x^3-y^3=4(x-y)$ and $ x^3+y^3=2(x+y)$.
2006 Indonesia MO
Let $ a,b,c$ be positive integers. If $ 30|a+b+c$, prove that $ 30|a^5+b^5+c^5$.
2006 Indonesia MO
Let $ S$ be the set of all triangles $ ABC$ which have property: $ \tan A,\tan B,\tan C$ are positive integers. Prove that all triangles in $ S$ are similar.
2006 Indonesia MO
A black pawn and a white pawn are placed on the first square and the last square of a $ 1\times n$ chessboard, respectively. Wiwit and Siti move alternatingly. Wiwit has the white pawn, and Siti has the black pawn. The white pawn moves first. In every move, the player moves her pawn one or two squares to the right or to the left, without passing the opponent's pawn. The player who cannot move anymore loses the game. Which player has the winning strategy? Explain the strategy.
2006 Indonesia MO
In triangle $ ABC$, $ M$ is the midpoint of side $ BC$ and $ G$ is the centroid of triangle $ ABC$. A line $ l$ passes through $ G$, intersecting line $ AB$ at $ P$ and line $ AC$ at $ Q$, where $ P\ne B$ and $ Q\ne C$. If $ [XYZ]$ denotes the area of triangle $ XYZ$, show that $ \frac{[BGM]}{[PAG]}+\frac{[CMG]}{[QGA]}=\frac32$.
2006 Indonesia MO
Every phone number in an area consists of eight digits and starts with digit $ 8$. Mr Edy, who has just moved to the area, apply for a new phone number. What is the chance that Mr Edy gets a phone number which consists of at most five different digits?
2006 Indonesia MO
Let $ a,b,c$ be real numbers such that $ ab,bc,ca$ are rational numbers. Prove that there are integers $ x,y,z$, not all of them are $ 0$, such that $ ax+by+cz=0$.
2006 Indonesia MO
Find the largest $ 85$-digit integer which has property: the sum of its digits equals to the product of its digits.
2005 Indonesia MO
Let $ n$ be a positive integer. Determine the number of triangles (non congruent) with integral side lengths and the longest side length is $ n$.
2005 Indonesia MO
For an arbitrary positive integer $ n$, define $ p(n)$ as the product of the digits of $ n$ (in decimal). Find all positive integers $ n$ such that $ 11p(n)=n^2-2005$.
2005 Indonesia MO
Let $ k$ and $ m$ be positive integers such that $ \displaystyle\frac12\left(\sqrt{k+4\sqrt{m}}-\sqrt{k}\right)$ is an integer.



(a) Prove that $ \sqrt{k}$ is rational.



(b) Prove that $ \sqrt{k}$ is a positive integer.
2005 Indonesia MO
Let $ M$ be a point in triangle $ ABC$ such that $ \angle AMC=90^{\circ}$, $ \angle AMB=150^{\circ}$, $ \angle BMC=120^{\circ}$. The centers of circumcircles of triangles $ AMC,AMB,BMC$ are $ P,Q,R$, respectively. Prove that the area of $ \triangle PQR$ is greater than the area of $ \triangle ABC$.
2005 Indonesia MO
For an arbitrary real number $ x$, $ \lfloor x\rfloor$ denotes the greatest integer not exceeding $ x$. Prove that there is exactly one integer $ m$ which satisfy $ \displaystyle m-\left\lfloor \frac{m}{2005}\right\rfloor=2005$.
2005 Indonesia MO
Find all triples $ (x,y,z)$ of integers which satisfy



$ x(y + z) = y^2 + z^2 - 2$



$ y(z + x) = z^2 + x^2 - 2$



$ z(x + y) = x^2 + y^2 - 2$.
2005 Indonesia MO
Let $ ABCD$ be a convex quadrilateral. Square $ AB_1A_2B$ is constructed such that the two vertices $ A_2,B_1$ is located outside $ ABCD$. Similarly, we construct squares $ BC_1B_2C$, $ CD_1C_2D$, $ DA_1D_2A$. Let $ K$ be the intersection of $ AA_2$ and $ BB_1$, $ L$ be the intersection of $ BB_2$ and $ CC_1$, $ M$ be the intersection of $ CC_2$ and $ DD_1$, and $ N$ be the intersection of $ DD_2$ and $ AA_1$. Prove that $ KM$ is perpendicular to $ LN$.
2005 Indonesia MO
There are $ 90$ contestants in a mathematics competition. Each contestant gets acquainted with at least $ 60$ other contestants. One of the contestants, Amin, state that at least four contestants have the same number of new friends. Prove or disprove his statement.
2004 Indonesia MO
Determine the number of positive odd and even factor of $ 5^6-1$.
2004 Indonesia MO
When filled with an cold water using a particular cold water tap, a tank will be full in 14 minutes. In 21 minutes, the full tank could be emptied by opening a hole on the base of the tank. If the cold water tap and the hot water tap are opened simultaneously (allowing hot and cold water fill the tank), and the hole on the base of the tank is opened, the tank will be full in 12.6 minutes. Determine the number of minutes needed to fill the tank with hot water until the tank is full, assuming at first the tank is empty and the hole is closed.
2004 Indonesia MO
In how many ways can we change the sign $ *$ with $ +$ or $ -$, such that the following equation is true?

$$ 1 *2*3*4*5*6*7*8*9*10=29$$
2004 Indonesia MO
There exists 4 circles, $ a,b,c,d$, such that $ a$ is tangent to both $ b$ and $ d$, $ b$ is tangent to both $ a$ and $ c$, $ c$ is both tangent to $ b$ and $ d$, and $ d$ is both tangent to $ a$ and $ c$. Show that all these tangent points are located on a circle.
2004 Indonesia MO
$$ a+4b+9c+16d+25e+36f+49g=1$$

$$ 4a+9b+16c+25d+36e+49f+64g=12$$

$$ 9a+16b+25c+36d+49e+64f+81g=123$$

Determine the value of $ 16a+25b+36c+49d+64e+81f+100g$.
2004 Indonesia MO
Quadratic equation $ x^2+ax+b+1=0$ have 2 positive integer roots, for integers $ a,b$. Show that $ a^2+b^2$ is not a prime.
2004 Indonesia MO
Given triangle $ ABC$ with $ C$ a right angle, show that the diameter of the incenter is $ a+b-c$, where $ a=BC$, $ b=CA$, and $ c=AB$.
2004 Indonesia MO
8. Sebuah lantai luasnya 3 meter persegi ditutupi lima buah karpet dengan ukuran masing-masing 1 meter persegi. Buktikan bahwa ada dua karpet yang tumpang tindih dengan luas tumpang tindih minimal 0,2 meter persegi.



A floor of a certain room has a $ 3 \ m^2$  area. Then the floor is covered by 5 rugs, each has an area of $ 1 \ m^2$. Prove that there exists 2 overlapping rugs, with at least $ 0.2 \ m^2$ covered by both rugs.
2003 Indonesia MO
Prove that $a^9 - a$ is divisible by $6$ for all integers $a$.
2003 Indonesia MO
Let $ABCD$ be a quadrilateral, and $P,Q,R,S$ are the midpoints of $AB, BC, CD, DA$ respectively. Let $O$ be the intersection between $PR$ and $QS$. Prove that $PO = OR$ and $QO = OS$.
2003 Indonesia MO
Find all real numbers $x$ such that $\left\lfloor x^2 \right\rfloor + \left\lceil x^2 \right\rceil = 2003$.
2003 Indonesia MO
Given a $19 \times 19$ matrix where each component is either $1$ or $-1$. Let $b_i$ be the product of all components in the $i$-th row, and $k_i$ be the product of all components in the $i$-th column, for all $1 \le i \le 19$. Prove that for any such matrix, $b_1 + k_1 + b_2 + k_2 + \cdots + b_{19} + k_{19} \neq 0$.
2003 Indonesia MO
For any real numbers $a,b,c$, prove that

$$ 5a^2 + 5b^2 + 5c^2 \ge 4ab + 4ac + 4bc $$

and determine when equality occurs.
2003 Indonesia MO
The hall in a castle is a regular hexagon where its sides' length is 6 meters. The floor of the hall is to be tiled with equilateral triangular tiles where its sides' length is 50 centimeters. Each tile is divided into three congruent triangles by their altitudes up to its orthocenter (see below). Each of these small triangles are colored such that each tile has different colors and no two tiles have identical colorings. How many colors at least are required?



A tile's pattern is:

[asy]

draw((0,0.000)--(2,0.000));

draw((2,0.000)--(1,1.732));

draw((1,1.732)--(0,0.000));

draw((1,0.577)--(0,0.000));

draw((1,0.577)--(2,0.000));

draw((1,0.577)--(1,1.732));

[/asy]
2003 Indonesia MO
Let $k,m,n$ be positive integers such that $k > n > 1$ and $(k,n) = 1$. If $k-n | k^m - n^{m-1}$, prove that $k \le 2n - 1$.
2002 Indonesia MO
Prove that $n^4 - n^2$ is divisible by $12$ for all integers $n > 1$.
2002 Indonesia MO
Five dice are rolled. The product of the faces are then computed. Which result has a larger probability of occurring; $180$ or $144$?
2002 Indonesia MO
Find all solutions (real and complex) for $x,y,z$, given that:

$$ x+y+z = 6 \\
x^2+y^2+z^2 = 12 \\
x^3+y^3+z^3 = 24 $$
2002 Indonesia MO
Given a triangle $ABC$ where $AC > BC$, $D$ is located on the circumcircle of $ABC$ such that $D$ is the midpoint of the arc $AB$ that contains $C$. $E$ is a point on $AC$ such that $DE$ is perpendicular to $AC$. Prove that $AE = EC + CB$.
2002 Indonesia MO
Nine of the numbers $4, 5, 6, 7, 8, 12, 13, 16, 18, 19$ are going to be inputted to the empty cells in the following table:

$\begin{array} {|c|c|c|} \cline{1-3}

10 & & \\ \cline{1-3}

& & 9 \\ \cline{1-3}

& 3 & \\ \cline{1-3}

11 & & 17 \\ \cline{1-3}

& 20 & \\ \cline{1-3}

\end{array}$

such that each row sums to the same number, and each column sums to the same number. Determine all possible arrangements.
2002 Indonesia MO
Find all primes $p$ such that $4p^2+1$ and $6p^2+1$ are both primes.
2002 Indonesia MO
Let $ABCD$ be a rhombus where $\angle DAB = 60^\circ$, and $P$ be the intersection between $AC$ and $BD$. Let $Q,R,S$ be three points on the boundary of $ABCD$ such that $PQRS$ is a rhombus. Prove that exactly one of $Q,R,S$ lies on one of $A,B,C,D$.
2021 Iran MO (2nd Round)
There are two distinct Points $A$ and $B$ on a line. We color a point $P$ on segment $AB$, distinct from $A,B$ and midpoint of segment $AB$ to red. In each move , we can reflect one of the red point wrt $A$ or $B$ and color the midpoint of the resulting point and the point we reflected from ( which is one of $A$ or $B$ ) to red. For example , if we choose $P$ and the reflection of $P$ wrt to $A$ is $P'$ , then midpoint of $AP'$ would be red. Is it possible to make the midpoint of $AB$ red after a finite number of moves?
2021 Iran MO (2nd Round)
Call a positive integer $n$ "Fantastic" if none of its digits are zero and it is possible to remove one of its digits and reach to an integer which is a divisor of $n$ . ( for example , 25 is fantastic , as if we remove digit 2 , resulting number would be 5 which is divisor of 25 ) Prove that the number of Fantastic numbers is finite.
2021 Iran MO (2nd Round)
Circle $\omega$ is inscribed in quadrilateral $ABCD$ and is tangent to segments $BC, AD$ at $E,F$ , respectively.$DE$ intersects $\omega$ for the second time at $X$. if the circumcircle of triangle $DFX$ is tangent to lines $AB$ and $CD$ , prove that quadrilateral $AFXC$ is cyclic.
2021 Iran MO (2nd Round)
$n$ points are given on a circle $\omega$. There is a circle with radius smaller than $\omega$ such that all these points lie inside or on the boundary of this circle. Prove that we can draw a diameter of $\omega$ with endpoints not belonging to the given points such that all the $n$ given points remain in one side of the diameter.