Found problems: 22
2023 Indonesia MO, 3
A natural number $n$ is written on a board. On every step, Neneng and Asep changes the number on the board with the following rule: Suppose the number on the board is $X$. Initially, Neneng chooses the sign up or down. Then, Asep will pick a positive divisor $d$ of $X$, and replace $X$ with $X+d$ if Neneng chose the sign "up" or $X-d$ if Neneng chose "down". This procedure is then repeated. Asep wins if the number on the board is a nonzero perfect square, and loses if at any point he writes zero.
Prove that if $n \geq 14$, Asep can win in at most $(n-5)/4$ steps.
2024 Indonesia MO, 3
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 $A'B'$.
2023 Indonesia MO, 5
Let $a$ and $b$ be positive integers such that $\text{gcd}(a, b) + \text{lcm}(a, b)$ is a multiple of $a+1$. If $b \le a$, show that $b$ is a perfect square.
2022 Indonesia MO, 8
Determine the smallest positive real $K$ such that the inequality
\[ K + \frac{a + b + c}{3} \ge (K + 1) \sqrt{\frac{a^2 + b^2 + c^2}{3}} \]holds for any real numbers $0 \le a,b,c \le 1$.
[i]Proposed by Fajar Yuliawan, Indonesia[/i]
2020 Indonesia MO, 4
Problem 4. A chessboard with $2n \times 2n$ tiles is coloured such that every tile is coloured with one out of $n$ colours. Prove that there exists 2 tiles in either the same column or row such that if the colours of both tiles are swapped, then there exists a rectangle where all its four corner tiles have the same colour.
2024 Indonesia MO, 5
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.
2023 Indonesia MO, 4
Determine whether or not there exists a natural number $N$ which satisfies the following three criteria:
1. $N$ is divisible by $2^{2023}$, but not by $2^{2024}$,
2. $N$ only has three different digits, and none of them are zero,
3. Exactly 99.9% of the digits of $N$ are odd.
2020 Indonesia MO, 2
Problem 2. Let $P(x) = ax^2 + bx + c$ where $a, b, c$ are real numbers. If $$P(a) = bc, \hspace{0.5cm} P(b) = ac, \hspace{0.5cm} P(c) = ab$$ then prove that $$(a - b)(b - c)(c - a)(a + b + c) = 0.$$
2022 Indonesia MO, 2
Let $P(x)$ be a polynomial with integer coefficient such that $P(1) = 10$ and $P(-1) = 22$.
(a) Give an example of $P(x)$ such that $P(x) = 0$ has an integer root.
(b) Suppose that $P(0) = 4$, prove that $P(x) = 0$ does not have an integer root.
2024 Indonesia MO, 6
Suppose $A_1 A_2 \ldots A_n$ is an $n$-sided polygon with $n \geq 3$ and $\angle A_j \leq 180^{\circ}$ for each $j$ (in other words, the polygon is convex or has fewer than $n$ distinct sides).
For each $i \leq n$, suppose $\alpha_i$ is the smallest possible value of $\angle{A_i A_j A_{i+1}}$ where $j$ is neither $i$ nor $i+1$. (Here, we define $A_{n+1} = A_1$.) Prove that
\[ \alpha_1 + \alpha_2 + \cdots + \alpha_n \leq 180^{\circ} \] and determine all equality cases.
2020 Indonesia MO, 3
The wording is just ever so slightly different, however the problem is identical.
Problem 3. Determine all functions $f: \mathbb{N} \to \mathbb{N}$ such that $n^2 + f(n)f(m)$ is a multiple of $f(n) + m$ for all natural numbers $m, n$.
2024 Indonesia MO, 7
Suppose $P(x) = x^n + a_{n-1} x^{n-1} + \cdots + a_1x + a_0$ where $a_0, a_1, \ldots, a_{n-1}$ are reals for $n\geq 1$ (monic $n$th-degree polynomial with real coefficients). If the inequality
\[ 3(P(x)+P(y)) \geq P(x+y) \] holds for all reals $x,y$, determine the minimum possible value of $P(2024)$.
2023 Indonesia MO, 7
Given a triangle $ABC$ with $\angle ACB = 90^{\circ}$. Let $\omega$ be the circumcircle of triangle $ABC$. The tangents of $\omega$ at $B$ and $C$ intersect at $P$. Let $M$ be the midpoint of $PB$. Line $CM$ intersects $\omega$ at $N$ and line $PN$ intersects $AB$ at $E$. Point $D$ is on $CM$ such that $ED \parallel BM$. Show that the circumcircle of $CDE$ is tangent to $\omega$.
2024 Indonesia MO, 1
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, 6
Determine the number of permutations $a_1, a_2, \dots, a_n$ of $1, 2, \dots, n$ such that for every positive integer $k$ with $1 \le k \le n$, there exists an integer $r$ with $0 \le r \le n - k$ which satisfies
\[ 1 + 2 + \dots + k = a_{r+1} + a_{r+2} + \dots + a_{r+k}. \]
2022 Indonesia MO, 3
Let $ABCD$ be a rectangle. Points $E$ and $F$ are on diagonal $AC$ such that $F$ lies between $A$ and $E$; and $E$ lies between $C$ and $F$. The circumcircle of triangle $BEF$ intersects $AB$ and $BC$ at $G$ and $H$ respectively, and the circumcircle of triangle $DEF$ intersects $AD$ and $CD$ at $I$ and $J$ respectively. Prove that the lines $GJ, IH$ and $AC$ concur at a point.
2020 Indonesia MO, 1
Since this is already 3 PM (GMT +7) in Jakarta, might as well post the problem here.
Problem 1. Given an acute triangle $ABC$ and the point $D$ on segment $BC$. Circle $c_1$ passes through $A, D$ and its centre lies on $AC$. Whereas circle $c_2$ passes through $A, D$ and its centre lies on $AB$. Let $P \neq A$ be the intersection of $c_1$ with $AB$ and $Q \neq A$ be the intersection of $c_2$ with $AC$. Prove that $AD$ bisects $\angle{PDQ}$.
2023 Indonesia MO, 1
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.
2022 Indonesia MO, 1
Determine all functions $f: \mathbb{R} \to \mathbb{R}$ such that for any $x,y \in \mathbb{R}$ we have
\[ f(f(f(x)) + f(y)) = f(y) - f(x) \]
2024 Indonesia MO, 8
Let $n \ge 2$ be a positive integer. Suppose $a_1, a_2, \dots, a_n$ are distinct integers. For $k = 1, 2, \dots, n$, let
\[ s_k := \prod_{\substack{i \not= k, \\ 1 \le i \le n}} |a_k - a_i|, \]
i.e. $s_k$ is the product of all terms of the form $|a_k - a_i|$, where $i \in \{ 1, 2, \dots, n \}$ and $i \not= k$.
Find the largest positive integer $M$ such that $M$ divides the least common multiple of $s_1, s_2, \dots, s_n$ for any choices of $a_1, a_2, \dots, a_n$.
2024 Indonesia MO, 2
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.
2022 Indonesia MO, 4
Given a regular $26$-gon. Prove that for any $9$ vertices of that regular $26$-gon, then there exists three vertices that forms an isosceles triangle.