AUSTRIAN-POLISH 1990

Find all solutions in positive integers to $a^A = b^B = c^C = 1990^{1990}abc$ , where $A = b^c, B = c^a, C = a^b$ .

Show that there are two real solutions to: $$\begin{cases} x + y^2 + z^4 = 0 y + z^2 + x^4 = 0 z + x^2 + y^5 = 0\end {cases}$$

Find all solutions in positive integers to: $$\begin{cases} x_1^4 + 14x_1x_2 + 1 = y_1^4 x_2^4 + 14x_2x_3 + 1 = y_2^4 ... x_n^4 + 14x_nx_1 + 1 = y_n^4 \end{cases}$$

Let $n>1$ be an integer and let $f_1$ , $f_2$ , ..., $f_{n!}$ be the $n!$ permutations of $1$ , $2$ , ..., $n$ . (Each $f_i$ is a bijective function from $\{1,2,...,n\}$ to itself.) For each permutation $f_i$ , let us define $S(f_i)=\sum^n_{k=1} |f_i(k)-k|$ . Find $\frac{1}{n!} \sum^{n!}_{i=1} S(f_i)$ .

$p(x)$ is a polynomial with integer coefficients. The sequence of integers $a_1, a_2, ... , a_n$ (where $n > 2$ ) satisfies $a_2 = p(a_1), a_3 = p(a_2), ... , a_n = p(a_{n-1}), a_1 = p(a_n)$ . Show that $a_1 = a_3$ .

$D_n$ is a set of domino pieces. For each pair of non-negative integers $(a, b)$ with $a \le b \le n$ , there is one domino, denoted $[a, b]$ or $[b, a]$ in $D_n$ . A ring is a sequence of dominoes $[a_1, b_1], [a_2, b_2], ... , [a_k, b_k]$ such that $b_1 = a_2, b_2 = a_3, ... , b_{k-1} = a_k$ and $b_k = a_1$ . Show that if $n$ is even there is a ring which uses all the pieces. Show that for n odd, at least $(n+1)/2$ pieces are not used in any ring. For $n$ odd, how many different sets of $(n+1)/2$ are there, such that the pieces not in the set can form a ring?

We are given a supply of $a \times b$ tiles with $a$ and $b$ distinct positive integers. The tiles are to be used to tile a $28 \times 48$ rectangle. Find $a, b$ such that the tile has the smallest possible area and there is only one possible tiling. (If there are two distinct tilings, one of which is a reflection of the other, then we treat that as more than one possible tiling. Similarly for other symmetries.) Find $a, b$ such that the tile has the largest possible area and there is more than one possible tiling.

$a_1, a_2, ... , a_n$ is a sequence of integers such that every non-empty subsequence has non-zero sum. Show that we can partition the positive integers into a finite number of sets such that if $x_i$ all belong to the same set, then $a_1x_1 + a_2x_2 + ... + a_nx_n$ is non-zero.