CMO 1994

Let $ABCD$ be a trapezoid with $AB\parallel CD$ . Points $E,F$ lie on segments $AB,CD$ respectively. Segments $CE,BF$ meet at $H$ , and segments $ED,AF$ meet at $G$ . Show that $S_{EHFG}\le \dfrac{1}{4}S_{ABCD}$ . Determine, with proof, if the conclusion still holds when $ABCD$ is just any convex quadrilateral.

There are $m$ pieces of candy held in $n$ trays( $n,m\ge 4$ ). An operation is defined as follow: take out one piece of candy from any two trays respectively, then put them in a third tray. Determine, with proof, if we can move all candies to a single tray by finite operations.

Find all functions $f:[1,\infty )\rightarrow [1,\infty)$ satisfying the following conditions: (1) $f(x)\le 2(x+1)$ ; (2) $f(x+1)=\dfrac{1}{x}[(f(x))^2-1]$ .

Let $f(z)=c_0z^n+c_1z^{n-1}+ c_2z^{n-2}+\cdots +c_{n-1}z+c_n$ be a polynomial with complex coefficients. Prove that there exists a complex number $z_0$ such that $|f(z_0)|\ge |c_0|+|c_n|$ , where $|z_0|\le 1$ .

For arbitrary natural number $n$ , prove that $\sum^n_{k=0}C^k_n2^kC^{[(n-k)/2]}_{n-k}=C^n_{2n+1}$ , where $C^0_0=1$ and $[\dfrac{n-k}{2}]$ denotes the integer part of $\dfrac{n-k}{2}$ .

Let $M$ be a point which has coordinates $(p\times 1994,7p\times 1994)$ in the Cartesian plane ( $p$ is a prime). Find the number of right-triangles satisfying the following conditions: (1) all vertexes of the triangle are lattice points, moreover $M$ is on the right-angled corner of the triangle; (2) the origin ( $0,0$ ) is the incenter of the triangle.