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 $n$.

Determine all integers $n>1$ that satisfy the following condition: for any positive integer $x$, if gcd$(x,n)=1$, then gcd$(x+101,n)=1$.

Let $a_1,a_2,\cdots$ be an infinity sequence of positive integers such that $a_1=2021$ and $$a_{n+1}=(a_1+a_2+\cdots+a_n)^2-1$$for all positive integers $n$. Prove that for any integer $n\ge 2$, $a_n$ is the product of at least $2n$ (not necessarily distinct) primes.

Let $ABC$ be an acute triangle such that $\angle B > \angle C$. Let $D$ and $E$ be the points on the segments $BC$ and $CA$, respectively, such that $AD$ bisects $\angle A$ and $BE\perp AC$. Finally, let $M$ be the midpoint of the side $BC$. Suppose that the circumcircle of $\triangle CDE$ intersects $AD$ again at a point $X$ different from $D$. Prove that $\angle XME = 90^{\circ} - \angle BAC$.

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 $P(a)+P(b)+P(c)= 2021$

Let $m<n$ be two positive integers and $x_m<x_{m+1}<\cdots<x_n$ be a sequence of rational numbers. Suppose that $kx_k$ is an integer for all integers $k$ which $m\leq k\leq n$. Prove that $$x_n-x_m\geq \frac{1}{m}-\frac{1}{n}.$$

Let $ABC$ be an acute triangle. Construct a point $X$ on the different side of $C$ with respect to the line $AB$ and construct a point $Y$ on the different side of $B$ with respect to the line $AC$ such that $BX=AC$, $CY=AB$, and $AX=AY$. Let $A'$ be the reflection of $A$ across the perpendicular bisector of $BC$. Suppose that $X$ and $Y$ lie on different sides of the line $AA'$, prove that points $A$, $A'$, $X$, and $Y$ lie on a circle.

Let $\mathbb N$ be the set of positive integers. Determine all functions $f:\mathbb N\times\mathbb N\to\mathbb N$ that satisfy both of the following conditions: $f(\gcd (a,b),c) = \gcd (a,f(c,b))$ for all $a,b,c \in \mathbb{N}$. $f(a,a) \geq a$ for all $a \in \mathbb{N}$.

For each positive integer $k$, denote by $\tau(k)$ the number of all positive divisors of $k$, including $1$ and $k$. Let $a$ and $b$ be positive integers such that $\tau(\tau(an)) = \tau(\tau(bn))$ for all positive integers $n$. Prove that $a=b$.

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: 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 $O$. Prove that each letter $T$, $M$, and $O$ appears exactly $674$ times on every row.