Found problems: 53
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 $P(2024)$.
2024 Indonesia TST, C
Given a sequence of integers $A_1,A_2,\cdots A_{99}$ such that for every sub-sequence that contains $m$ consecutive elements, there exist not more than $max\{ \frac{m}{3} ,1\}$ odd integers. Let $S=\{ (i,j) \ | i<j \}$ such that $A_i$ is even and $A_j$ is odd. Find $max\{ |S|\}$.
2022 Indonesia Regional, 1
Let $A$ and $B$ be sets such that there are exactly $144$ sets which are subsets of either $A$ or $B$. Determine the number of elements $A \cup B$ has.
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 $\omega$.
2023 Indonesia Regional, 2
Let $K$ be a positive integer such that there exist a triple of positive integers $(x,y,z)$ such that
\[x^3+Ky , y^3 + Kz, \text{and } z^3 + Kx\]
are all perfect cubes.
(a) Prove that $K \ne 2$ and $K \ne 4$
(b) Find the minimum value of $K$ that satisfies.
[i]Proposed by Muhammad Afifurrahman[/i]
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, 2
Determine all functions $f : \mathbb{R} \to \mathbb{R}$ such that the following equation holds for every real $x,y$:
\[ f(f(x) + y) = \lfloor x + f(f(y)) \rfloor. \]
[b]Note:[/b] $\lfloor x \rfloor$ denotes the greatest integer not greater than $x$.
2024 Indonesia TST, G
Given an acute triangle $ABC$. The incircle with center $I$ touches $BC,CA,AB$ at $D,E,F$ respectively. Let $M,N$ be the midpoint of the minor arc of $AB$ and $AC$ respectively. Prove that $M,F,E,N$ are collinear if and only if $\angle BAC =90$$^{\circ}$
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 Regional, 3
It is known that $x$ and $y$ are reals satisfying
\[ 5x^2 + 4xy + 11y^2 = 3. \]
Without using calculus (differentials/integrals), determine the maximum value of $xy - 2x + 5y$.
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.
2023 Indonesia TST, A
Find all Polynomial $P(x)$ and $Q(x)$ with Integer Coefficients satisfied the equation:
\[Q(a+b) = \frac{P(a) - P(b)}{a - b}\]
$\forall a, b \in \mathbb{Z}^+$ and $a>b$
2015 Indonesia MO Shortlist, N8
The natural number $n$ is said to be good if there are natural numbers $a$ and $b$ that satisfy $a + b = n$ and $ab | n^2 + n + 1$.
(a) Show that there are infinitely many good numbers.
(b) Show that if $n$ is a good number, then $7 \nmid n$.
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 Regional, 2
(a) Determine a natural number $n$ such that $n(n+2022)+2$ is a perfect square.
[hide=Spoiler]In case you didn't realize, $n=1$ works lol[/hide]
(b) Determine all natural numbers $a$ such that for every natural number $n$, the number $n(n+a)+2$ is never a perfect square.
2024 Indonesia TST, A
Find all second degree polynomials $P(x)$ such that for all $a \in\mathbb{R} , a \geq 1$, then
$P(a^2+a) \geq a.P(a+1)$
2023 Indonesia TST, G
Given an acute triangle $ABC$ with circumcenter $O$. The circumcircle of $BCH$ and a circle with diameter of $AC$ intersect at $P (P \neq C)$. A point $Q$ on segment of $PC$ such that $PB = PQ$. Prove that $\angle ABC = \angle AQP$
2023 Indonesia Regional, 1
Let $ABCD$ be a square with side length $43$ and points $X$ and $Y$ lies on sides $AD$ and $BC$ respectively such that the ratio of the area of $ABYX$ to the area of $CDXY$ is $20 : 23$ . Find the maximum possible length of $XY$.
2022 Indonesia MO, 1
Determine all functions $f: \mathbb{R} \to \mathbb{R}$ such that for any $x,y \in \mathbb{R}$ we have
\[ f(f(f(x)) + f(y)) = f(y) - f(x) \]
2023 Indonesia TST, G
Incircle of triangle $ABC$ tangent to $AB$ and $AC$ on $E$ and $F$ respectively. If $X$ is the midpoint of $EF$, prove $\angle BXC > 90^{\circ}$
2022 Indonesia Regional, 5
Numbers $1$ to $22$ are written on a board. A "move" is a procedure of picking two numbers $a,b$ on the board such that $b \geq a+2$, then erasing $a$ and $b$ to be replaced with $a+1$ and $b-1$. Determine the maximum possible number of moves that can be done on the board.
2023 Indonesia MO, 8
Let $a, b, c$ be three distinct positive integers. Define $S(a, b, c)$ as the set of all rational roots of $px^2 + qx + r = 0$ for every permutation $(p, q, r)$ of $(a, b, c)$. For example, $S(1, 2, 3) = \{ -1, -2, -1/2 \}$ because the equation $x^2+3x+2$ has roots $-1$ and $-2$, the equation $2x^2+3x+1=0$ has roots $-1$ and $-1/2$, and for all the other permutations of $(1, 2, 3)$, the quadratic equations formed don't have any rational roots.
Determine the maximum number of elements in $S(a, b, c)$.
2023 Indonesia TST, A
Let $a,b,c$ positive real numbers and $a+b+c = 1$. Prove that
\[a^2 + b^2 + c^2 + \frac{3}{\frac{1}{a} + \frac{1}{b} + \frac{1}{c}} \ge 2(ab + bc + ac)\]
2024 Indonesia TST, A
Given real numbers $x,y,z$ which satisfies
$$|x+y+z|+|xy+yz+zx|+|xyz| \le 1$$
Show that $max\{ |x|,|y|,|z|\} \le 1$.
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 $a_1, a_2, \dots, a_n$.