Let $n$ be a positive integer larger than $1$, and let $a_0,a_1,\dots,a_{n-1}$ be integers. It is known that the equation $$x^n+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+\cdots+a_1x+a_0=0$$has $n$ pairwise relatively prime integer roots. Prove that $a_0$ and $a_1$ are relatively prime.

We say that a real number is 'almost an integer' if it differs from an integer by at most $0.1$. For example, $2023$,$-2023.9$ and $2023.0822$ are almost integers. Show that among any $10$ real numbers, there exists two different real numbers whose difference is almost an integer.

Given $\Omega ABC$ with $AB<AC$, let $AD$ be the bisector of $\angle BAC$ with $D$ on the side $BC$. Let $\Gamma$ be a circle passing through $A$ and $D$ which is tangent to $BC$ at $D$. Suppose $\Gamma$ cuts the side $AB$ again at $E\ne A$. The tangent to the circumcircle of $\Delta BDE$ at $D$ intersects $\Gamma$ again at $F\ne D$. Let $P$ be the intersection point of the segments $EF$ and $AC$. Prove that $PD$ is perpendicular to $BC$.

After expanding the polynomial $(1+x+y)^{2023}$ and collecting the like terms, we obtain the expression $$a_0+a_1x+a_2y+a_3x^2+a_4xy+a_5y^2+\cdots+a_ky^{2023}$$Find the total number of $a_i$'s which are divisible by $5$.

Find all nonnegative real numbers $a,b,c$ such that $$\frac{4a+9b+25c}{2a+3b+5c}+\frac{4b+9c+25a}{2b+3c+5a}+\frac{4c+9a+25b}{2c+3a+5b}=10$$

Let $\Omega$ be the incircle of $\Delta ABC$. There are two smaller circles $\omega_1$ amd $\omega_2$ inside $\Delta ABC$. The circle $\omega_1$ is tangent to $\Omega$ at $P$, tangent to $BC$ at $D$, and also tangent to $AB$. The circle $\omega_2$ is tangent to $\Omega$ at $Q$, tangent to $BC$ at $E$, and also tangent to $AC$. Prove that $D,E,Q,P$ are concyclic.

Let $M$ be the midpoint of the side $BC$ of an acute $\Delta ABC$, and let $D$ be the foot of perpendicular from $C$ to $AM$. The circumcircle of $\Delta ABD$ intersects the side $BC$ again at $E\ne B$. Suppose $F$ is a point on the segment $AE$ such that $FB = FC$. Prove that $F$ is the midpoint of $AE$.

Given is an $n\times n$ board, with an integer written in each grid. For each move, I can choose any grid, and add $1$ to all $2n-1$ numbers in its row and column. Find the largest $N(n)$, such that for any initial choice of integers, I can make a finite number of moves so that there are at least $N(n)$ even numbers on the board.

Let $n$ be a positive integer. Prove that there exists a poisitve integer $m$ such that $$7^n \mid 3^m+5^m-1$$

Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ satisfying the following condition: for any real numbers $x$ and $y$, the number $f(x + f(y))$ is equal to $x + f(y)$ or $f(f(x)) + y$