Olympiad problems

2023 Indonesia Regional, 3

Tags: combinatorics , Indonesia , RMO , maximization , algebra

Find the maximum value of an integer $B$ such that for every 9 distinct natural number with the sum of $2023$, there must exist a sum of 4 of the number that is greater than or equal to $B$

2023 Indonesia MO, 3

Tags: combinatorics , Game Theory , Perfect Square , Indonesia , Indonesia MO

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 TST, C

Tags: combinatorics , maximization , Indonesia , Indonesia TST

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 $max\{ |S|\}$.

2024 Indonesia TST, N

Tags: function , number theory , Indonesia , Indonesia TST , complete residue system , residue , FE NT

Find all functions $f:\mathbb{N} \rightarrow \mathbb{N}$ such that for every prime number $p$ and natural number $x$, $$\{ x,f(x),\cdots f^{p-1}(x) \} $$ is a complete residue system modulo $p$. With $f^{k+1}(x)=f(f^k(x))$ for every natural number $k$ and $f^1(x)=f(x)$. [i]Proposed by IndoMathXdZ[/i]

2024 Indonesia Regional, 1

Tags: Inequality , inequalities , algebra , Indonesia , RMO

Given a real number $C\leqslant 2$. Prove that for every positive real number $x,y$ with $xy=1$, the following inequality holds: \[ \sqrt{\frac{x^2+y^2}{2}} + \frac{C}{x+y} \geqslant 1 + \frac{C}{2}.\] [i]Proposed by Fajar Yuliawan, Indonesia[/i]

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 $A'B'$.

2023 Indonesia MO, 5

Tags: number theory , greatest common divisor , least common multiple , Indonesia , Indonesia MO

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

Tags: inequalities , Hi , Indonesia , Indonesia MO

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]

2024 Indonesia TST, A

Tags: algebra , polynomial , Indonesia , Indonesia TST

Find all second degree polynomials $P(x)$ such that for all $a \in\mathbb{R} , a \geq 1$, then $P(a^2+a) \geq a.P(a+1)$

2024 Indonesia TST, N

Tags: function , number theory , Indonesia , Indonesia TST , complete residue system , residue , FE NT

Find all functions $f:\mathbb{N} \rightarrow \mathbb{N}$ such that for every prime number $p$ and natural number $x$, $$\{ x,f(x),\cdots f^{p-1}(x) \} $$ is a complete residue system modulo $p$. With $f^{k+1}(x)=f(f^k(x))$ for every natural number $k$ and $f^1(x)=f(x)$. [i]Proposed by IndoMathXdZ[/i]

2024 Indonesia TST, A

Tags: algebra , inequalities , Indonesia , Indonesia TST , absolute value

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, 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.

2023 Indonesia TST, G

Tags: projective geometry , geometry , Indonesia , Indonesia TST

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 $\angle ABC = \angle AQP$

2024 Indonesia Regional, 2

Tags: domino , combinatorics , Chessboard , board , Boards , Indonesia , RMO

Given an $n \times n$ board which is divided into $n^2$ squares of size $1 \times 1$, all of which are white. Then, Aqua selects several squares from this board and colors them black. Ruby then places exactly one $1\times 2$ domino on the board, so that the domino covers exactly two squares on the board. Ruby can rotate the domino into a $2\times 1$ domino. After Aqua colors, it turns out there are exactly $2024$ ways for Ruby to place a domino on the board so that it covers exactly $1$ black square and $1$ white square. Determine the smallest possible value of $n$ so that Aqua and Ruby can do this. [i]Proposed by Muhammad Afifurrahman, Indonesia [/i]

2024 Indonesia Regional, 4

Tags: number theory , Quadratic Residues , quadratic reciprocity , Indonesia , RMO

Find the number of positive integer pairs $1\leqslant a,b \leqslant 2027$ that satisfy \[ 2027 \mid a^6+b^5+b^2.\] (Note: For integers $a$ and $b$, the notation $a \mid b$ means that there is an integer $c$ such that $ac=b$.) [i]Proposed by Valentio Iverson, Indonesia[/i]

2023 Indonesia MO, 4

Tags: number theory , Digits , NT construction , Indonesia , Indonesia MO

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.

2022 Indonesia Regional, 4

Tags: geometry , Indonesia , RMO , 2022 , circumcircle , geometric transformation , reflection

Suppose $ABC$ is a triangle with circumcenter $O$. Point $D$ is the reflection of $A$ with respect to $BC$. Suppose $\ell$ is the line which is parallel to $BC$ and passes through $O$. The line through $B$ and parallel to $CD$ meets $\ell$ at $B_1$. Lines $CB_1$ and $BD$ intersect at point $B_2$. The line through $C$ parallel to $BD$ and $\ell$ meet at $C_1$. Finally, $BC_1$ and $CD$ intersects at point $C_2$. Prove that points $A, B_2, C_2, D$ lie on a circle.

2024 Indonesia Regional, 3

Tags: geometry , Indonesia , RMO

Given a triangle $ABC$, points $X,Y,$ and $Z$ are the midpoints of $BC,CA,$ and $AB$ respectively. The perpendicular bisector of $AB$ intersects line $XY$ and line $AC$ at $Z_1$ and $Z_2$ respectively. The perpendicular bisector of $AC$ intersects line $XZ$ and line $AB$ at $Y_1$ and $Y_2$ respectively. Let $K$ be a point such that $KZ_1 = KZ_2$ and $KY_1 = KY_2$. Prove that $KB=KC$.

2024 Indonesia TST, G

Tags: Tstst , geometry , incenter , Indonesia , Indonesia TST

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 TST, A

Tags: algebra , inequalities proposed , Indonesia , Indonesia TST

Let $a,b,c$ positive real numbers and $a+b+c = 1$. Prove that \[a^2 + b^2 + c^2 + \frac{3}{\frac{1}{a} + \frac{1}{b} + \frac{1}{c}} \ge 2(ab + bc + ac)\]

2023 Indonesia TST, G

Tags: geometry , Indonesia , Indonesia TST

Incircle of triangle $ABC$ tangent to $AB$ and $AC$ on $E$ and $F$ respectively. If $X$ is the midpoint of $EF$, prove $\angle BXC > 90^{\circ}$

2023 Indonesia Regional, 5

Tags: geometry , Indonesia , regional olympiad , RMO

Given $\triangle ABC$ and points $D$ and $E$ at the line $BC$, furthermore there are points $X$ and $Y$ inside $\triangle ABC$. Let $P$ be the intersection of line $AD$ and $XE$, and $Q$ be the intersection of line $AE$ and $YD$. If there exist a circle that passes through $X, Y, D, E$, and $$\angle BXE + \angle BCA = \angle CYD + \angle CBA = 180^{\circ}$$ Prove that the line $BP$, $CQ$, and the perpendicular bisector of $BC$ intersect at one point.

2022 Indonesia MO, 2

Tags: algebra , polynomial , number theory , Indonesia , Indonesia MO

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

Tags: geometry , polygon , Inequality , convex , inequalities , Indonesia , Indonesia MO

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.

2023 Indonesia TST, A

Tags: algebra , polynomial , Polynomials , Integer Polynomial , integer polynomials , Indonesia , Indonesia TST

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 $a>b$