Let $ABCD$ be a trapezoid with $AB \parallel CD$. A point $T$ which is inside the trapezoid satisfies $ \angle ATD = \angle CTB$. Let line $AT$ intersects circumcircle of $ACD$ at $K$ and line $BT$ intersects circumcircle of $BCD$ at $L$.($K \neq A$ , $L \neq B$) Prove that $KL \parallel AB$.

There is a school with $n$ students. Suppose that every student has exactly $2023$ friends and every couple of student that are not friends has exactly $2022$ friends in common. Then find all values of $n$

For all $n>1$, let $f(n)$ be the biggest divisor of $n$ except itself. Does there exists a positive integer $k$ such that the equality $n-f(n)=k$ has exactly $2023$ solutions?

Let $k$ be a positive integer and $S$ be a set of sets which have $k$ elements. For every $A,B \in S$ and $A\neq B$ we have $A \Delta B \in S$. Find all values of $k$ when $|S|=1023$ and $|S|=2023$. Note:$A \Delta B = (A \setminus B) \cup (B \setminus A)$

Let $ABC$ be a scalene triangle with circumcentre $O$, incentre $I$ and orthocentre $H$. Let the second intersection point of circle which passes through $O$ and tangent to $IH$ at point $I$, and the circle which passes through $H$ and tangent to $IO$ at point $I$ be $M$. Prove that $M$ lies on circumcircle of $ABC$.

Let $a,b,c,d$ be positive real numbers. What is the minimum value of $$ \frac{(a^2+b^2+2c^2+3d^2)(2a^2+3b^2+6c^2+6d^2)}{(a+b)^2(c+d)^2}$$

Let us call an integer sequence $\{ a_1,a_2, \dots \}$ nice if there exist a function $f: \mathbb{Z^+} \to \mathbb{Z^+} $ such that $$a_i \equiv a_j \pmod{n} \iff i\equiv j \pmod{f(n)}$$for all $i,j,n \in \mathbb{Z^+}$. Find all nice sequences.

Initially the equation $$\star \frac{1}{x-1} \star \frac{1}{x-2} \star \frac{1}{x-4} ... \star \frac{1}{x-2^{2023}}=0$$ is written on the board. In each turn Aslı and Zehra deletes one of the stars in the equation and writes $+$ or $-$ instead. The first move is performed by Aslı and continues in order. What is the maximum number of real solutions Aslı can guarantee after all the stars have been replaced by signs?

The points $ A,B,K,L,X$ lies of the circle $\Gamma$ in that order such that the arcs $\widehat{BK}$ and $\widehat{KL}$ are equal. The circle that passes through $A$ and tangent to $BK$ at $B$ intersects the line segment $KX$ at $P$ and $Q$. The circle that passes through $A$ and tangent to $BL$ at $B$ intersect the line segment $BX$ for the second time at $T$. Prove that $\angle{PTB} = \angle{XTQ}$