1965 IMO Problems

Problem 1

Determine all values $x$ in the interval $0\leq x\leq 2\pi$ which satisfy the inequality $2\cos x \leq \left| \sqrt{1+\sin 2x} - \sqrt{1-\sin 2x } \right| \leq \sqrt{2}.$

Solution

Problem 2

Consider the system of equations $a_{11}x_1 + a_{12}x_2 + a_{13}x_3 = 0$ $a_{21}x_1 + a_{22}x_2 + a_{23}x_3 = 0$ $a_{31}x_1 + a_{32}x_2 + a_{33}x_3 = 0$ with unknowns $x_1$ , $x_2$ , $x_3$ . The coefficients satisfy the conditions:

(a) $a_{11}$ , $a_{22}$ , $a_{33}$ are positive numbers;

(b) the remaining coefficients are negative numbers;

(c) in each equation, the sum of the coefficients is positive.

Prove that the given system has only the solution $x_1 = x_2 = x_3 = 0$ .

Solution

Problem 3

Given the tetrahedron $ABCD$ whose edges $AB$ and $CD$ have lengths $a$ and $b$ respectively. The distance between the skew lines $AB$ and $CD$ is $d$ , and the angle between them is $\omega$ . Tetrahedron $ABCD$ is divided into two solids by plane $\varepsilon$ , parallel to lines $AB$ and $CD$ . The ratio of the distances of $\varepsilon$ from $AB$ and $CD$ is equal to $k$ . Compute the ratio of the volumes of the two solids obtained.

Solution

Problem 4

Find all sets of four real numbers $x_1$ , $x_2$ , $x_3$ , $x_4$ such that the sum of any one and the product of the other three is equal to $2$ .

Solution

Problem 5

Consider $\triangle OAB$ with acute angle $AOB$ . Through a point $M \neq O$ perpendiculars are drawn to $OA$ and $OB$ , the feet of which are $P$ and $Q$ respectively. The point of intersection of the altitudes of $\triangle OPQ$ is $H$ . What is the locus of $H$ if $M$ is permitted to range over (a) the side $AB$ , (b) the interior of $\triangle OAB$ ?

Solution

Problem 6

In a plane a set of $n$ points ( $n\geq 3$ ) is given. Each pair of points is connected by a segment. Let $d$ be the length of the longest of these segments. We define a diameter of the set to be any connecting segment of length $d$ . Prove that the number of diameters of the given set is at most $n$ .

Solution

1965 IMO (Problems) • Resources
Preceded by 1964 IMO	1 • 2 • 3 • 4 • 5 • 6	Followed by 1966 IMO
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1965 IMO Problems

Contents

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6