Let $n,k$ are integers and $p$ is a prime number. Find all $(n,k,p)$ such that $|6n^2-17n-39|=p^k$

Let $ABC$ is acute angled triangle and $K,L$ is points on $AC,BC$ respectively such that $\angle{AKB}=\angle{ALB}$. $P$ is intersection of $AL$ and $BK$ and $Q$ is the midpoint of segment $KL$. Let $T,S$ are the intersection $AL,BK$ with $(ABC)$ respectively. Prove that $TK,SL,PQ$ are concurrent.

Find all $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $f(x+f(x))=f(-x)$ and for all $x \leq y$ it satisfies $f(x) \leq f(y)$

Initially, Aslı distributes $1000$ balls to $30$ boxes as she wishes. After that, Aslı and Zehra make alternated moves which consists of taking a ball in any wanted box starting with Aslı. One who takes the last ball from any box takes that box to herself. What is the maximum number of boxes can Aslı guarantee to take herself regardless of Zehra's moves?

Prove that for all $a,b,c$ positive real numbers $\dfrac{a^4+1}{b^3+b^2+b}+\dfrac{b^4+1}{c^3+c^2+c}+\dfrac{c^4+1}{a^3+a^2+a} \ge 2$

A marble is placed on each $33$ unit square of a $10*10$ chessboard. After that, the number of marbles in the same row or column with that square is written on each of the remaining empty unit squares. What is the maximum sum of the numbers written on the board?

Let $ABC$ is triangle and $D \in AB$,$E \in AC$ such that $DE//BC$. Let $(ABC)$ meets with $(BDE)$ and $(CDE)$ at the second time $K,L$ respectively. $BK$ and $CL$ intersect at $T$. Prove that $TA$ is tangent to the $(ABC)$

For a prime number $p$. Can the number of n positive integers that make the expression \[\dfrac{n^3+np+1}{n+p+1}\]an integer be $777$?