Olympiad problems

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 $max\{ |S|\}$.

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 , 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 $max\{ |x|,|y|,|z|\} \le 1$.

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 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 $\angle BXC > 90^{\circ}$