Let $ABC$ be an acute triangle. Let $D$ be a point on side $AB$ and $E$ a point on side $AC$ such that $BC\parallel DE$. Let $X$ be an interior point of $BCED$. Suppose rays $DX$ and $EX$ meet side $BC$ at points $P$ and $Q$, respectively, such that both $P,Q$ lie between $B,C$. Suppose the circumcircles of triangles $BQX$ and $CPX$ intersect at $Y\ne X$. Prove that $A,X,Y$ are collinear.
Consider a $100\times100$ table, and identify its cell in row $a$ and column $b$, $1\le a,b\le100$, with $(a,b)$. Let $51\le k\le99$ be an integer. A $k$-knight moves one cell vertically or horizontally and $k$ cells in the other direction: from $(a,b)$ to $(c,d)$ with $(|a-c|,|b-d|)$ equal to $(1,k)$ or $(k,1)$. It starts at $(1,1)$ and makes several moves. A sequence of moves is a sequence $(x_0,y_0)=(1,1),(x_1,y_1),\ldots,(x_n,y_n)$ such that all cells lie in the table and each successive pair is a legal move. Each visited cell is called reachable. For each $k$, find $L(k)$, the number of reachable cells.
Let $n$ be a positive integer and $a_1,a_2,\ldots,a_n$ positive real numbers. Prove that
$$\sum_{i=1}^n\frac1{2^i}\left(\frac2{1+a_i}\right)^{2^i}\ge\frac2{1+a_1a_2\cdots a_n}-\frac1{2^n}.$$
Prove that for every positive integer $t$ there is a unique permutation $a_0,a_1,\ldots,a_{t-1}$ of $0,1,\ldots,t-1$ such that, for every $0\le i\le t-1$, the binomial coefficient
$$\binom{t+i}{2a_i}$$
is odd and $2a_i\ne t+i$.
Line $\ell$ intersects sides $BC,AD$ of cyclic quadrilateral $ABCD$ at interior points $R,S$, respectively, and intersects ray $DC$ beyond $C$ at $Q$, and ray $BA$ beyond $A$ at $P$. Circumcircles of $QCR,QDS$ intersect at $N\ne Q$, while circumcircles of $PAS,PBR$ intersect at $M\ne P$. Let $MP$ and $NQ$ meet at $X$, $AB$ and $CD$ meet at $K$, and $BC$ and $AD$ meet at $L$. Prove that $X$ lies on $KL$.
AI-generated configuration (the line $KL$ is not added):
