Find all real numbers $x,y$ such that
$$\begin{cases}\sqrt{3x}\left(1+\dfrac1{x+y}\right)=2,\\[3pt]\sqrt{7y}\left(1-\dfrac1{x+y}\right)=4\sqrt2.\end{cases}$$
Let $SABC$ be a tetrahedron. Triangles $DAB,EBC,FCA$ in the plane of $ABC$ are similar respectively to $SAB,SBC,SCA$. Let $O$ be the circumcenter of $DEF$. Suppose $K$ is the “circumsphere of $SABC$ opposite $O$” (tangent to the planes $SAB,SBC,SCA,ABC$, on the opposite side of $ABC$ from $S$, but on the same side of $SAB$ as $C$, the same side of $SBC$ as $A$, and the same side of $SCA$ as $B$). Prove that $K$ touches the plane $ABC$ at $O$.
Let $n$ be a positive integer and let $k$ be a positive integer not exceeding $n$. Determine the number of ordered $k$-element tuples $(a_1,a_2,\ldots,a_n)$ satisfying:
(a) All $a_i$ are distinct and belong to $\{1,2,\ldots,n\}$.
(b) $a_r\gt a_s$ for some $r\lt s$.
(c) $a_r$ has the same parity as $r$ for some $r$.
Determine all functions $f$ on the positive integers, with positive integer values, satisfying
$$f(n)+f(n+1)=f(n+2)f(n+3)-1996$$
for every positive integer $n$.
Triangle $ABC$ has $BC=1$ and angle $A=x$. Let $f(x)$ be the least possible distance between its incenter and centroid. Determine $f(x)$. What is the largest value of $f(x)$ for $60^\circ\lt x\lt 180^\circ$?