FKMO 1997

A word is a sequence of 0 and 1 of length 8. Let $x$ and $y$ be two words differing in exactly three places. Prove that the number of words differing from each of $x$ and $y$ in at least five places is 188.

The incircle of a triangle $A_1A_2A_3$ is centered at $O$ and meets the segment $OA_j$ at $B_j$ , $j = 1, 2, 3$ . A circle with center $B_j$ is tangent to the two sides of the triangle having $A_j$ as an endpoint and intersects the segment $OB_j$ at $C_j$ . Prove that $$\frac{OC_1+OC_2+OC_3}{A_1A_2+A_2A_3+A_3A_1} \leq \frac{1}{4\sqrt{3}}$$ and find the conditions for equality.

Find all pairs of functions $f, g: \mathbb R \to \mathbb R$ such that [list] (i) if $x < y$ , then $f(x) < f(y)$ ;

(ii) $f(xy) = g(y)f(x) + f(y)$ for all $x, y \in \mathbb R$ . [/list]

Given a positive integer $n$ , find the number of $n$ -digit natural numbers consisting of digits 1, 2, 3 in which any two adjacent digits are either distinct or both equal to 3.

For positive numbers $a_1,a_2,\dots,a_n$ , we define $$A=\frac{a_1+a_2+\cdots+a_n}{n}, G=\sqrt[n]{a_1\cdots a_n}, H=\frac{n}{a_1^{-1}+\cdots+a_n^{-1}}$$ Prove that

(i) $\frac{A}{H}\leq -1+2\left(\frac{A}{G}\right)^n$ , for n even

(ii) $\frac{A}{H}\leq -\frac{n-2}{n}+\frac{2(n-1)}{n}\left(\frac{A}{G}\right)^n$ , for $n$ odd

Let $p_1,p_2,\dots,p_r$ be distinct primes, and let $n_1,n_2,\dots,n_r$ be arbitrary natural numbers. Prove that the number of pairs of integers $(x, y)$ such that $$x^3 + y^3 = p_1^{n_1}p_2^{n_2}\cdots p_r^{n_r}$$ does not exceed $2^{r+1}$ .