In an acute-angled triangle $ABC$, $M$ is a point on the side $BC$, the line $AM$ meets the circumcircle $\omega$ of $ABC$ at the point $Q$ distinct from $A$. The tangent to $\omega$ at $Q$ intersects the line through $M$ perpendicular to the diameter $AK$ of $\omega$ at the point $P$. Let $L$ be the point on $\omega$ distinct from $Q$ such that $PL$ is tangent to $\omega$ at $L$. Prove that $L,M$ and $K$ are collinear.
There are $n=1681$ children, $a_1,a_2,\ldots,a_n$, seated clockwise in a circle on the floor. The teacher walks behind the children in the clockwise direction. She drops a candy behind the first child $a_1$. She then skips one child and drops a candy behind the third child, $a_3$. Now she skips two children and drops a candy behind the next child, $a_6$. She continues this way, at each stage skipping one child more than at the preceding stage before dropping a candy behind the next child. How many children will never receive a candy? Justify your answer.
Let $n\ge3$ be an integer. Prove that there exist positive integers $a_1,a_2,\ldots,a_n$, each at least $2$, such that $$a_1a_2\cdots\widehat{a_i}\cdots a_n\equiv1\pmod{a_i},\qquad i=1,\ldots,n.$$ Here $\widehat{a_i}$ means the term $a_i$ is omitted.
Is it possible to color each square on a $9\times9$ board so that each $2\times3$ or $3\times2$ block contains exactly $2$ black squares? If so, what is/are the possible total number(s) of black squares?
Let $A$ be a point on the circle $\omega$ centred at $B$ and $\Gamma$ a circle centred at $A$. For $i=1,2,3$, a chord $P_iQ_i$ of $\omega$ is tangent to $\Gamma$ at $S_i$ and another chord $P_iR_i$ of $\omega$ is perpendicular to $AB$ at $M_i$. Let $Q_iT_i$ be the other tangent from $Q_i$ to $\Gamma$ at $T_i$, and $N_i$ the intersection of $AQ_i$ with $M_iT_i$. Prove that $N_1,N_2,N_3$ are collinear.