CMO 1995

Let $a_1,a_2,\cdots ,a_n; b_1,b_2,\cdots ,b_n (n\ge 3)$ be real numbers satisfying the following conditions: (1) $a_1+a_2+\cdots +a_n= b_1+b_2+\cdots +b_n$ ; (2) $0<a_1=a_2, a_i+a_{i+1}=a_{i+2}$ ( $i=1,2,\cdots ,n-2$ ); (3) $0<b_1\le b_2, b_i+b_{i+1}\le b_{i+2}$ ( $i=1,2,\cdots ,n-2$ ). Prove that $a_{n-1}+a_n\le b_{n-1}+b_n$ .

Let $f: \mathbb{N} \rightarrow \mathbb{N}$ be a function satisfying the following conditions: (1) $f(1)=1$ ; (2) $\forall n\in \mathbb{N}$ , $3f(n) f(2n+1) =f(2n) ( 1+3f(n) )$ ; (3) $\forall n\in \mathbb{N}$ , $f(2n) < 6 f(n)$ . Find all solutions of equation $f(k) +f(l)=293$ , where $k<l$ . ( $\mathbb{N}$ denotes the set of all natural numbers).

Find the minimun value of $\sum_{i=1}^{10} \sum_{j=1}^{10} \sum_{k=1}^{10}|k(x+y-10i)(3x-6y-36j)(19x+95y-95k)|$ , where $x,y$ are integers.

Given four spheres with their radii equal to $2,2,3,3$ respectively, each sphere externally touches the other spheres. Suppose that there is another sphere that is externally tangent to all those four spheres, determine the radius of this sphere.

Let $a_1,a_2,\cdots ,a_{10}$ be pairwise distinct natural numbers with their sum equal to 1995. Find the minimal value of $a_1a_2+a_2a_3+\cdots +a_9a_{10}+a_{10}a_1$ .

Let $n(n>1)$ be an odd. We define $x_k=(x^{(k)}_1,x^{(k)}_2,\cdots ,x^{(k)}_n)$ as follow: $x_0=(x^{(0)}_1,x^{(0)}_2,\cdots ,x^{(0)}_n)=(1,0,\cdots ,0,1)$ ; $x^{(k)}_i =\begin{cases}0, x^{(k-1)}_i=x^{(k-1)}_{i+1}, 1, x^{(k-1)}_i\not= x^{(k-1)}_{i+1},\end{cases}$ $i=1,2,\cdots ,n$ , where $x^{(k-1)}_{n+1}= x^{(k-1)}_1$ . Let $m$ be a positive integer satisfying $x_0=x_m$ . Prove that $m$ is divisible by $n$ .