International Mathematical Olympiad 1967

Let $ABCD$ be a parallelogram with side lengths $AB = a$, $AD = 1$, and with $\angle BAD = \alpha$. If $\triangle ABD$ is acute, prove that the four circles of radius 1 with centers $A, B, C, D$ cover the parallelogram if and only if $$a \leq \cos \alpha + \sqrt{3} \sin \alpha.$$

Prove that if one and only one edge of a tetrahedron is greater than 1, then its volume is $\leq 1/8$.

Let $k, m, n$ be natural numbers such that $m + k + 1$ is a prime greater than $n + 1$. Let $c_s = s(s + 1)$. Prove that the product $$(c_{m+1} - c_k)(c_{m+2} - c_k) \cdots (c_{m+n} - c_k)$$ is divisible by the product $c_1c_2\cdots c_n$.

Let $A_0B_0C_0$ and $A_1B_1C_1$ be any two acute-angled triangles. Consider all triangles $ABC$ that are similar to $\triangle A_1B_1C_1$ (so that vertices $A_1, B_1, C_1$ correspond to vertices $A, B, C$, respectively) and circumscribed about triangle $A_0B_0C_0$ (where $A_0$ lies on $BC$, $B_0$ on $CA$, and $AC_0$ on $AB$). Of all such possible triangles, determine the one with maximum area, and construct it.

Consider the sequence $\{c_n\}$, where $$\begin{aligned} c_1 &= a_1 + a_2 + \cdots + a_8 \\ c_2 &= a_1^2 + a_2^2 + \cdots + a_8^2 \\ &\cdots \\ c_n &= a_1^n + a_2^n + \cdots + a_8^n \\ &\cdots \end{aligned}$$ in which $a_1, a_2, \ldots, a_8$ are real numbers not all equal to zero. Suppose that an infinite number of terms of the sequence $\{c_n\}$ are equal to zero. Find all natural numbers $n$ for which $c_n = 0$.

In a sports contest, there were $m$ medals awarded on $n$ successive days ($n > 1$). On the first day, one medal and $1/7$ of the remaining $m - 1$ medals were awarded. On the second day, two medals and $1/7$ of the now remaining medals were awarded; and so on. On the $n$-th and last day, the remaining $n$ medals were awarded. How many days did the contest last, and how many medals were awarded altogether?