International Mathematical Olympiad  – IMO 1967  Problems

Problem 1

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\[a\leq \cos \alpha + \sqrt{3} \sin \alpha .\]

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\[(c_{m+1} – c_k)(c_{m+2}- c_k)\cdots (c_{m+n}- c_k)\]is divisible by the product $c_1 c_2\cdots c_n$.

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 $ABC$ that are similar to $\triangle A_1 B_1 C_1$ (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 $BC$, $B_0$ on $CA$, and $C_0$ on $AB$). Of all such possible triangles, determine the one with maximum area, and construct it.

Problem 5

Consider the sequence $\{ c_n \}$, where\[c_1 = a_1 + a_2 + \cdots + a_8\]\[c_2 = a_1^2 + a_2^2 + \cdots + a_8^2\]\[\cdots\]\[c_n = a_1^n + a_2^n + \cdots + a_8^n\]\[\cdots\]in which $a_1$, $a_2$, $\cdots$, $a_8$ 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?