Write $\mathbb R$ for the set of real numbers. A function $f:\mathbb R\to\mathbb R$ satisfies
$$f(\cot x)=\cos2x+\sin2x$$
for every $0\lt x\lt \pi$. Determine $g(x)=f(x)f(1-x)$ for $-1\le x\le1$. Find the maximum and minimum values of $g(x)$ on $[-1,1]$.
Two circles $C_1,C_2$ touch externally at $M$, and the radius of $C_2$ is greater than that of $C_1$. A point $A$ moves on $C_2$ and is not on the line through the two centers. Points $B,C$ on $C_1$ are chosen so that $AB,AC$ are tangent to $C_1$. Lines $BM,CM$ meet $C_2$ again at $E,F$, respectively. Let $D$ be the intersection of the tangent at $A$ and the line $EF$. Prove that the locus of $D$ as $A$ varies is a straight line.
Let $S_n$ be the number of permutations $(a_1,a_2,\ldots,a_n)$ of $(1,2,\ldots,n)$ satisfying $1\le|a_k-k|\le2$ for every $k$. Prove that
$$\frac74S_{n-1}\lt S_n\lt 2S_{n-1}$$
for every integer $n\gt 6$.
Find the largest positive integer $n$ for which the following equations have integer solutions $x,y_1,y_2,\ldots,y_n$:
$$(x+1)^2+y_1^2=(x+2)^2+y_2^2=\cdots=(x+n)^2+y_n^2.$$
Define the polynomials
$$p(x)=4x^3-2x^2-15x+9,\qquad q(x)=12x^3+6x^2-7x+1.$$
Prove that each polynomial has at least three distinct real roots. Let $A$ be the largest root of $p(x)$ and $B$ the largest root of $q(x)$. Prove that $A^2+3B^2=4$.
Write $\mathbb R^+$ for the positive real numbers. Let $\mathcal F$ be the set of all functions $f:\mathbb R^+\to\mathbb R^+$ satisfying
$$f(3x)\ge f(f(2x))+x$$
for every $x\gt 0$. Find the largest real constant $\alpha$ such that $f(x)\ge\alpha x$ for every $f\in\mathcal F$ and every $x\in\mathbb R^+$.