# Polish Mathematical Olympiad 1977

1. Let $ABCD$ be a tetrahedron with $\angle BAD = 60^{\circ}$, $\angle BAC = 40^{\circ}$, $\angle ABD = 80^{\circ}$, $\angle ABC = 70^{\circ}$. Prove that the lines $AB$ and $CD$ are perpendicular.
2. Let $s \geq 3$ be a given integer. A sequence $K_n$ of circles and a sequence $W_n$ of convex $s$-gons satisfy $K_n \supset W_n \supset K_{n+1}$ for all $n = 1, 2, ...$. Prove that the sequence of the radii of the circles $K_n$ converges to zero.
3. Consider the set $A = \{0, 1, 2, . . . , 2^{2n} - 1\}$. The function $f : A \rightarrow A$ is given by $$f(x_0 + 2x_1 + 2^2x_2 + ... + 2^{2n-1}x_{2n-1})=(1 - x_0) + 2x_1 + 2^2(1 - x_2) + 2^3x_3 + ... + 2^{2n-1}x_{2n-1}$$ for every sequence $(x_0, x_1, . . . , x_{2n-1})$. Show that if $a_1, a_2, . . . , a_9$ are consecutive terms of an arithmetic progression, then the sequence $f(a_1), f(a_2), . . . , f(a_9)$ is not increasing.
4. A function $h : \mathbb{R} \rightarrow \mathbb{R}$ is differentiable and satisfies $h(ax) = bh(x)$ for all $x$, where $a$ and $b$ are given positive numbers and $0 \not = |a| \not = 1$. Suppose that $h'(0) \not = 0$ and the function $h'$ is continuous at $x = 0$. Prove that $a = b$ and that there is a real number $c$ such that $h(x) = cx$ for all $x$.
5. Show that for every convex polygon there is a circle passing through three consecutive vertices of the polygon and containing the entire polygon
6. Consider the polynomial $W(x) = (x - a)^kQ(x)$, where $a \neq 0$, $Q$ is a nonzero polynomial, and $k$ a natural number. Prove that $W$ has at least $k + 1$ nonzero coefficients.
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