Solve, for $x,y\in\mathbb R$,
$$\begin{cases}\sqrt{\sin^2x+\dfrac1{\sin^2x}}+\sqrt{\cos^2y+\dfrac1{\cos^2y}}=\sqrt{\dfrac{20y}{x+y}},\\[4pt]\sqrt{\sin^2y+\dfrac1{\sin^2y}}+\sqrt{\cos^2x+\dfrac1{\cos^2x}}=\sqrt{\dfrac{20x}{x+y}}.\end{cases}$$
Let $ABC$ be a scalene triangle. Its incircle $(I)$ touches $BC,CA,AB$ at $D,E,F$, respectively. The line through $E$ perpendicular to $BI$ meets $(I)$ again at $K\ne E$; the line through $F$ perpendicular to $CI$ meets it again at $L\ne F$. Let $J$ be the midpoint of $KL$.
(a) Prove that $D,I,J$ are collinear.
(b) Fix $B,C$ and let $A$ vary with $AB/AC=k$ constant. Let $M\ne E,N\ne F$ be the other intersections of $IE,IF$ with $(I)$. Let $P=MN\cap IB$, $Q=MN\cap IC$. Prove that the perpendicular bisector of $PQ$ always passes through a fixed point.
Some natural numbers are initially written on a line. At each step, first identify every adjacent pair in the current left-to-right order, then insert their sum between each pair. After 2013 steps, how many occurrences of 2013 are present if initially:
(a) the numbers are $1,1000$;
(b) the numbers are $1,2,\ldots,1000$ in increasing order?
Find all functions $f:\mathbb R\to\mathbb R$ such that $f(0)=0$, $f(1)=2013$, and, for all real $x,y$,
$$(x-y)\bigl(f(f^2(x))-f(f^2(y))\bigr)$$
$$=(f(x)-f(y))\bigl(f^2(x)-f^2(y)\bigr),$$
where $f^2(x)=(f(x))^2$.
Let acute triangle $ABC$ be inscribed in $(O)$ and let $D$ be on the arc $BC$ not containing $A$. A variable line $\Delta$ through the orthocenter $H$ meets the circumcircles of $ABH$ and $ACH$ again at $M\ne H$ and $N\ne H$, respectively.
(a) Determine $\Delta$ for which the area of $AMN$ is greatest.
(b) Let $d_1$ pass through $M$ perpendicular to $DB$, and $d_2$ through $N$ perpendicular to $DC$. Prove that their intersection $P$ lies on a fixed circle.
Determine the number of ordered sextuples $(a,b,c,a',b',c')$ with all entries in $\{0,1,\ldots,14\}$ satisfying
$$\begin{cases}ab+a'b'\equiv1\pmod{15},\\ac+a'c'\equiv1\pmod{15},\\bc+b'c'\equiv1\pmod{15}.\end{cases}$$