Question bank APMO 2002

Let $a_1,a_2,\ldots,a_n$ be nonnegative integers, where $n$ is a positive integer, and let
$$A_n=\frac{a_1+\cdots+a_n}{n}.$$
Prove that
$$a_1!a_2!\cdots a_n!\ge(\lfloor A_n\rfloor!)^n,$$
where $\lfloor A_n\rfloor$ is the greatest integer not exceeding $A_n$, and $a!=1\cdot2\cdots a$ for $a\ge1$, with $0!=1$. When does equality hold?

Find all positive integers $a,b$ such that
$$\frac{a^2+b}{b^2-a}\quad\text{and}\quad\frac{b^2+a}{a^2-b}$$
are both integers.

Let $ABC$ be an equilateral triangle. Let $P\in AC$ and $Q\in AB$ be such that both triangles $ABP,ACQ$ are acute. Let $R,S$ be their respective orthocentres, and $T=BP\cap CQ$. Find all possible values of $\angle CBP$ and $\angle BCQ$ such that $TRS$ is equilateral.



Editorial convention: the intersection $T$ is unique and triangle $TRS$ is nondegenerate. The simultaneous choice $P=C,Q=B$ makes $BP=CQ=BC$ and is therefore excluded.

Let $x,y,z$ be positive numbers such that $1/x+1/y+1/z=1$. Show that
$$\sqrt{x+yz}+\sqrt{y+zx}+\sqrt{z+xy}\ge\sqrt{xyz}+\sqrt x+\sqrt y+\sqrt z.$$

Let $\mathbb R$ denote the real numbers. Find all functions $f:\mathbb R\to\mathbb R$ satisfying:
(i) only finitely many real $s$ satisfy $f(s)=0$;
(ii) $f(x^4+y)=x^3f(x)+f(f(y))$ for all real $x,y$.