1967 IMO Problems

Problems of the 9th IMO 1967 in Yugoslavia.

Day I

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

Solution

Problem 2

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

Solution

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 $c_1 c_2\cdots c_n$ .

Solution

Day II

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.

Solution

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 $c_n = 0$ .

Solution

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?

Solution

1967 IMO (Problems) • Resources
Preceded by 1966 IMO	1 • 2 • 3 • 4 • 5 • 6	Followed by 1968 IMO
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1967 IMO Problems

Contents

Day I

Problem 1

Problem 2

Problem 3

Day II

Problem 4

Problem 5

Problem 6