Olympiad problems

2021 Thailand Online MO, P1

There is a fence that consists of $n$ planks arranged in a line. Each plank is painted with one of the available $100$ colors. Suppose that for any two distinct colors $i$ and $j$, there is a plank with color $i$ located to the left of a (not necessarily adjacent) plank with color $j$. Determine the minimum possible value of $n$.

2021 Thailand Online MO, P10

Tags: combinatorics , Thailand online MO , desi chudayi , o physicsknight muth mrnejrha

Each cell of the board with $2021$ rows and $2022$ columns contains exactly one of the three letters $T$, $M$, and $O$ in a way that satisfies each of the following conditions: [list] [*] In total, each letter appears exactly $2021\times 674$ of times on the board. [*] There are no two squares that share a common side and contain the same letter. [*] Any $2\times 2$ square contains all three letters $T$, $M$, and $O$. [/list] Prove that each letter $T$, $M$, and $O$ appears exactly $674$ times on every row.

2022 Thailand Online MO, 5

Tags: geometry , similar triangles , geometry solved , Thailand , Thailand online MO , Angle Chasing , reflection

Let $ABC$ be an acute triangle with circumcenter $O$ and orthocenter $H$. Let $M_B$ and $M_C$ be the midpoints of $AC$ and $AB$, respectively. Place points $X$ and $Y$ on line $BC$ such that $\angle HM_BX = \angle HM_CY = 90^{\circ}$. Prove that triangles $OXY$ and $HBC$ are similar.

2021 Thailand Online MO, P5

Tags: algebra , polynomial , Thailand online MO

Prove that there exists a polynomial $P(x)$ with real coefficients and degree greater than 1 such that both of the following conditions are true $\cdot$ $P(a)+P(b)+P(c)\ge 2021$ holds for all nonnegative real numbers $a,b,c$ such that $a+b+c=3$ $\cdot$ There are infinitely many triples $(a,b,c)$ of nonnegative real numbers such that $a+b+c=3$ and $P(a)+P(b)+P(c)= 2021$