KMO 1997

Let $f(n)$ be the number of ways to express positive integer $n$ as a sum of positive odd integers. Compute $f(n).$ (If the order of odd numbers are different, then it is considered as different expression.)

For positive integer $n,$ let $a_n=\sum_{k=0}^{[\frac{n}{2}]}\binom{n-2}{k}(-\frac{1}{4})^k.$ Find $a_{1997}.$ (For real $x,$ $[x]$ is defined as largest integer that does not exceeds $x.$ )

Let $ABCDEF$ be a convex hexagon such that $AB=BC,CD=DE, EF=FA.$ Prove that $\frac{BC}{BE}+\frac{DE}{DA}+\frac{FA}{FC}\ge\frac{3}{2}$ and find when equality holds.

For any prime number $p>2,$ and an integer $a$ and $b,$ if $1+\frac{1}{2^3}+\frac{1}{3^3}+\cdots+\frac{1}{(p-1)^3}=\frac{a}{b},$ prove that $a$ is divisible by $p.$

Let $a,b,c$ be the side lengths of any triangle $\triangle ABC$ opposite to $A,B$ and $C,$ respectively. Let $x,y,z$ be the length of medians from $A,B$ and $C,$ respectively. If $T$ is the area of $\triangle ABC$ , prove that $\frac{a^2}{x}+\frac{b^2}{y}+\frac{c^2}{z}\ge\sqrt{\sqrt{3}T}$

Find all polynomial $P(x,y)$ for any reals $x,y$ such that (i) $x^{100}+y^{100}\le P(x,y)\le 101(x^{100}+y^{100})$ (ii) $(x-y)P(x,y)=(x-1)P(x,1)+(1-y)P(1,y).$

Let $X,Y,Z$ be the points outside the $\triangle ABC$ such that $\angle BAZ=\angle CAY,\angle CBX=\angle ABZ,\angle ACY=\angle BCX.$ Prove that the lines $AX, BY, CZ$ are concurrent.

For any positive integers $x,y,z$ and $w,$ prove that $x^2,y^2,z^2$ and $w^2$ cannot be four consecutive terms of arithmetic sequence.