Consider the following function $ f(x)=\frac{1}{1+2^{1-2x}}$. Compute the value of $$f\left(\frac{1}{10}\right)+f\left(\frac{2}{10}\right)+...+f\left(\frac{9}{10}\right).$$

Prove that if positive reals $x,y$ satisfy $x+y= 3$, $x,y \ge 1$ then $$9(x- 1)(y- 1) + (y^2 + y+ 1)(x + 1) + (x^2-x+ 1)(y- 1) \ge 9$$

Find all functions $f:R \to R$, such that $f(x)+f(y)=f(x+y)$, and there exists non-constant polynomials $P(x)$, $Q(x)$ such that $P(x)f(Q(x))=f(P(x)Q(x))$

Find all injective function $f: N \to N$ satisfying that for all positive integers $m,n$, we have: $f(n(f(m)) \le nm$

There is a regular $2021$-gon. We put a coin with heads up on every vertex of it. Every time, you can choose one vertex, and flip the coin on the vertices adjacent to it. Can you make all the coin tails up?

There are $3n$ $A$s and $2n$ $B$s in a string, where $n$ is a positive integer, prove that you can find a substring in this string that contains $3$ $A$s and $2$ $B$s.

Find the largest positive integer $n$ such that no two adjacent digits are the same, and for any two distinct digits $0 \leq a,b \leq 9 $, you can't get the string $abab$ just by removing digits from $n$.

There are n cards on a table numbered from $1$ to $n$, where $n$ is an even number. Two people take turns taking away the cards. The first player will always take the card with the largest number on it, but the second player will take a random card. Prove: the probability that the first player takes the card with the number $i$ is $ \frac{i-1}{n-1} $

Let $D,E,F$ be the midpoints of $BC$ ,$CA$, $AB$ in $\vartriangle ABC$ such that $AD= 9$, $BE= 12$, $CF= 15$. Calculate the area of $\vartriangle ABC$

Let $O$ be the circumcenter and $I$ be the incenter of $\vartriangle$, $P$ is the reflection from $I$ through $O$, the foot of perpendicular from $P$ to $BC,CA,AB$ is $X,Y,Z$, respectively. Prove that $AP^2+PX^2=BP^2+PY^2=CP^2+PZ^2$.

Given a triangle $ABC$, $D$ is the reflection from the perpendicular foot from $A$ to $BC$ through the midpoint of $BC$. $E$ is the reflection from the perpendicular foot from $B$ to $CA$ through the midpoint of $CA$. $F$ is the reflection from the perpendicular foot from $C$ to $AB$ through the midpoint of $AB$. Prove: $DE \perp AC$ if and only if $DF \perp AB$

Given a $\triangle ABC$, $\angle A=45^\circ$, $O$ is the circumcenter and $H$ is the orthocenter of $\triangle ABC$. $M$ is the midpoint of $\overline{BC}$, and $N$ is the midpoint of $\overline{OH}$. Prove that $\angle BAM=\angle CAN$.

Compute the remainder of $3^{2021}$ mod $15$

Prove: if $2^{2^n-1}-1$ is a prime, then $n$ is a prime.

Prove: for all positive integers $m, n$ $\frac 1m + \frac 1{m+1} + \dotsb + \frac 1 {m+n} $ is not an integer.

Prove: There exists a positive integer $n$ with $2021$ prime divisors, satisfying $n|2^n+1$.