DE-MO 2002

Find all real numbers $a,b$ satisfying the following system of equations \begin{align*} 2a^2 -2ab+b^2 &=a 4a^2 -5ab +2b^2 & =b. \end{align*}

Minimal distance of a finite set of different points in space is length of the shortest segment, whose both ends belong to this set and segment has length greater than $0$ . a) Prove there exist set of $8$ points on sphere with radius $R$ , whose minimal distance is greater than $1,15R$ . b) Does there exist set of $8$ points on sphere with radius $R$ , whose minimal distance is greater than $1,2R$ ?

Prove that for all primes $p$ true is equality $$\sum_{k=1}^{p-1}\left\lfloor\frac{k^3}{p}\right\rfloor=\frac{(p-2)(p-1)(p+1)}{4}$$

Given a positive real number $a_1$ , we recursively define $a_{n+1} = 1+a_1 a_2 \cdots \cdot a_n.$ Furthermore, let $$b_n = \frac{1}{a_1 } + \frac{1}{a_2 } +\cdots + \frac{1}{a_n }.$$ Prove that $b_n < \frac{2}{a_1}$ for all positive integers $n$ and that this is the smallest possible bound.

Theo Travel, who has $5$ children, has already visited $8$ countries of the eurozone. From every country, he brought $5$ not necessarily distinct coins home. Moreover, among these $40$ coins there are exactly $5$ of every value ( $1,2,5,10,20,$ and $50$ ct, $1$ and $2$ euro). He wants to give each child $8$ coins such that they are from different countries and that each child gets the same amount of money. Is this always possible?