Integers $a, b, c, d$ satisfy $a+b+c+d=0$. Show that $$n=(ab-cd)\cdot(bc-ad)\cdot(ca-bd)$$is a perfect square.

Show that for every integer $n\geq2$ there are two distinct powers of $n$ such that their sum is greater than $10^{2023}$ and their positive difference is divisible with $2023$.

Let there be a quadrilateral $ABCD$ such that $\angle CAD=45, \angle ACD=30, \angle BAC=\angle BCA=15$. Find $\angle DBC$.

Find all triplets of prime numbers $(m, n, p)$, that satisfy the system of equations: $$\left\{\begin{matrix} 2m-n+13p=2072,\\3m+11n+13p=2961.\end{matrix}\right.$$

Find all pairs of real numbers $(x, y)$, that satisfy the system of equations: $$\left\{\begin{matrix} 6(1-x)^2=\dfrac{1}{y} \\ \\6(1-y)^2=\dfrac{1}{x}.\end{matrix}\right.$$

Let there be a square $ABCD$. Points $E$ and $F$ are on sides $(BC)$ and $(AB)$ such that $BF=CE$. LInes $AE$ and $CF$ intersect in point $G$. Prove that $EF$ and $DG$ are perpendicular.

Find all triplets of integers $(a, b, c)$, that verify the equation $$|a+3|+b^2+4\cdot c^2-14\cdot b-12\cdot c+55=0.$$

Prove that the number $1$ can be written as a sum of $2023$ fractions of the form $\frac{1}{k_i}$, where all nonnegative integers $k_i (1\leq i\leq 2023)$ are distinct.

Solve the equation $$\left[\frac{x^2+1}{x}\right]-\left[\frac{x}{x^2+1}\right]=3.$$

Cirlce $\Omega$ is inscribed in triangle $ABC$ with $\angle BAC=40$. Point $D$ is inside the angle $BAC$ and is the intersection of exterior bisectors of angles $B$ and $C$ with the common side $BC$. Tangent form $D$ touches $\Omega$ in $E$. FInd $\angle BEC$.

Find all three digit positive integers that have distinct digits and after their greatest digit is switched to $1$ become multiples of $30$.

Let there be an integer $n\geq2$. In a chess tournament $n$ players play between each other one game. No game ended in a draw. Show that after the end of the tournament the players can be arranged in a list: $P_1, P_2, P_3,\ldots,P_n$ such that for every $i (1\leq i\leq n-1)$ the player $P_i$ won against player $P_{i+1}$.