IMO 1967

Problem 1. Let $A B C D$ be a parallelogram with side lengths $A B = a, A D = 1$ , and with $∠ B A D = α$ . If $Δ A B D$ is acute, prove that the four circles of radius $1$ with centers $A, B, C, D$ cover the parallelogram if and only if

$a \leq cos α + 3 sin α .$

Problem 2. Prove that if one and only one edge of a tetrahedron is greater than $1$ , then its volume is $\leq 1/8$ .

Problem 3. 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

$(c_{m + 1} - c_{k}) (c_{m + 2} - c_{k}) \dots (c_{m + n} - c_{k})$

is divisible by the product $c_{1} c_{2} \dots c_{n}$ .

Problem 4. Let $A_{0} B_{0} C_{0}$ and $A_{1} B_{1} C_{1}$ be any two acute-angled triangles. Consider all triangles $A B C$ that are similar to $Δ A_{1} B_{1} C_{1}$ (so that vertices $A_{1}, B_{1}, C_{1}$ correspond to vertices $A, B, C$ , respectively) and circumscribed about triangle $A_{0} B_{0} C_{0}$ (where $A_{0}$ lies on $B C, B_{0}$ on $C A$ , and $A C_{0}$ on $A B$ ). Of all such possible triangles, determine the one with maximum area, and construct it.

Problem 5. Consider the sequence ${c_{n}}$ , where

c_{1} = a_{1} + a_{2} + \dots + a_{8} c_{2} = a_{1}^{2} + a_{2}^{2} + \dots + a_{8}^{2} \dots c_{n} = a_{1}^{n} + a_{2}^{n} + \dots + a_{8}^{n} \dots

in which $a_{1}, a_{2}, \dots, a_{8}$ 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 $c_{n} = 0$ .

Problem 6. 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?

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