Olympiad problems

2024 Indonesia MO, 4

Kobar and Borah are playing on a whiteboard with the following rules: They start with two distinct positive integers on the board. On each step, beginning with Kobar, each player takes turns changing the numbers on the board, either from $P$ and $Q$ to $2P-Q$ and $2Q-P$, or from $P$ and $Q$ to $5P-4Q$ and $5Q-4P$. The game ends if a player writes an integer that is not positive. That player is declared to lose, and the opponent is declared the winner. At the beginning of the game, the two numbers on the board are $2024$ and $A$. If it is known that Kobar does not lose on his first move, determine the largest possible value of $A$ so that Borah can win this game.

2024 Indonesia MO, 3

Tags: geometry , circumcircle , Indonesia , orthocenter , Indonesia MO , Inamo

The triangle $ABC$ has $O$ as its circumcenter, and $H$ as its orthocenter. The line $AH$ and $BH$ intersect the circumcircle of $ABC$ for the second time at points $D$ and $E$, respectively. Let $A'$ and $B'$ be the circumcenters of triangle $AHE$ and $BHD$ respectively. If $A', B', O, H$ are [b]not[/b] collinear, prove that $OH$ intersects the midpoint of segment $A'B'$.

2019 Indonesia MO, 5

Tags: Inamo , algebra

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 $a = b$.

2024 Indonesia MO, 5

Tags: combinatorics , Integers , number theory , Indonesia , Indonesia MO , Inamo

Each integer is colored with exactly one of the following colors: red, blue, or orange, and all three colors are used in the coloring. The coloring also satisfies the following properties: 1. The sum of a red number and an orange number results in a blue-colored number, 2. The sum of an orange and blue number results in an orange-colored number; 3. The sum of a blue number and a red number results in a red-colored number. (a) Prove that $0$ and $1$ must have distinct colors. (b) Determine all possible colorings of the integers which also satisfy the properties stated above.

2024 Indonesia MO, 1

Tags: algebra , Manipulation , system of equations , Indonesia , Inamo , Indonesia MO

Determine all positive real solutions $(a,b)$ to the following system of equations. \begin{align*} \sqrt{a} + \sqrt{b} &= 6 \\ \sqrt{a-5} + \sqrt{b-5} &= 4 \end{align*}

2023 Indonesia MO, 1

Tags: geometry , Indonesia , Indonesia MO , Inamo

An acute triangle $ABC$ has $BC$ as its longest side. Points $D,E$ respectively lie on $AC,AB$ such that $BA = BD$ and $CA = CE$. The point $A'$ is the reflection of $A$ against line $BC$. Prove that the circumcircles of $ABC$ and $A'DE$ have the same radii.

2019 Indonesia MO, 6

Tags: Inamo , geometry , power of a point

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.

2024 Indonesia MO, 2

Tags: number theory , Inamo , primes , factorizations , Indonesia , NT construction , Indonesia MO

The triplet of positive integers $(a,b,c)$ with $a<b<c$ is called a [i]fatal[/i] triplet if there exist three nonzero integers $p,q,r$ which satisfy the equation $a^p b^q c^r = 1$. As an example, $(2,3,12)$ is a fatal triplet since $2^2 \cdot 3^1 \cdot (12)^{-1} = 1$. The positive integer $N$ is called [i]fatal[/i] if there exists a fatal triplet $(a,b,c)$ satisfying $N=a+b+c$. (a) Prove that 16 is not [i]fatal[/i]. (b) Prove that all integers bigger than 16 which are [b]not[/b] an integer multiple of 6 are fatal.

2019 Indonesia MO, 3

Tags: Inamo , geometry

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 $PM = PN$.