1966 IMO Problems

Problem 1

In a mathematical contest, three problems, $A,B,C$ were posed. Among the participants ther were 25 students who solved at least one problem each. Of all the contestants who did not solve problem $A$ , the number who solved $B$ was twice the number who solved $C$ . The number of students who solved only problem $A$ was one more than the number of students who solved $A$ and at least one other problem. Of all students who solved just one problem, half did not solve problem $A$ . How many students solved only problem $B$ ?

Solution

Problem 2

Let $a,b,c$ be the lengths of the sides of a triangle, and $\alpha, \beta, \gamma$ respectively, the angles opposite these sides. Prove that if $a+b=\tan{\frac{\gamma}{2}}(a\tan{\alpha}+b\tan{\beta})$ the triangle is isosceles.

Solution

Problem 3

Prove that the sum of the distances of the vertices of a regular tetrahedron from the center of its circumscribed sphere is less than the sum of the distances of these vertices from any other point in space.

Solution

Problem 4

Prove that for every natural number $n$ , and for every real number $x \neq \frac{k\pi}{2^t}$ ( $t=0,1, \dots, n$ ; $k$ any integer) $\frac{1}{\sin{2x}}+\frac{1}{\sin{4x}}+\dots+\frac{1}{\sin{2^nx}}=\cot{x}-\cot{2^nx}$

Solution

Problem 5

Solve the system of equations $|a_1-a_2|x_2+|a_1-a_3|x_3+|a_1-a_4|x_4=1$ $|a_2-a_1|x_1+|a_2-a_3|x_3+|a_2-a_4|x_4=1$ $|a_3-a_1|x_1+|a_3-a_2|x_2+|a_3-a_4|x_4=1$ $|a_4-a_1|x_1+|a_4-a_2|x_2+|a_4-a_3|x_3=1$ where $a_1, a_2, a_3, a_4$ are four different real numbers.

Solution

Problem 6

Let $ABC$ be a triangle, and let $P$ , $Q$ , $R$ be three points in the interiors of the sides $BC$ , $CA$ , $AB$ of this triangle. Prove that the area of at least one of the three triangles $AQR$ , $BRP$ , $CPQ$ is less than or equal to one quarter of the area of triangle $ABC$ .

Alternative formulation: Let $ABC$ be a triangle, and let $P$ , $Q$ , $R$ be three points on the segments $BC$ , $CA$ , $AB$ , respectively. Prove that

$\min\left\{\left|AQR\right|,\left|BRP\right|,\left|CPQ\right|\right\}\leq\frac14\cdot\left|ABC\right|$ ,