FI-MO 1999

Show that the equation $x^3 + 2y^2 + 4z = n$ has an integral solution $(x, y, z)$ for all integers $n.$

Suppose that the positive numbers $a_1, a_2,.. , a_n$ form an arithmetic progression; hence $a_{k+1}- a_k = d,$ for $k = 1, 2,... , n - 1.$ Prove that $$\frac{1}{a_1a_2}+\frac{1}{a_2a_3}+...+\frac{1}{a_{n-1}a_n}=\frac{n-1}{a_1a_n}.$$

Determine how many primes are there in the sequence $$101, 10101, 1010101 ....$$

Three unit circles have a common point $O.$ The other points of (pairwise) intersection are $A, B$ and $C$ . Show that the points $A, B$ and $C$ are located on some unit circle.

An ordinary domino tile can be identified as a pair $(k,m)$ where numbers $k$ and $m$ can get values $0, 1, 2, 3, 4, 5$ and $6.$ Pairs $(k,m)$ and $(m, k)$ determine the same tile. In particular, the pair $(k, k)$ determines one tile. We say that two domino tiles match, if they have a common component. Generalized n-domino tiles $m$ and $k$ can get values $0, 1,... , n.$ What is the probability that two randomly chosen $n$ -domino tiles match?