International Mathematical Olympiad  – IMO 1966  Problems

Problem 1

In a mathematical contest, three problems, $A,B,C$ were posed. Among the participants there 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 $B$?

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.

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.

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)\[\frac{1}{\sin{2x}}+\frac{1}{\sin{4x}}+\dots+\frac{1}{\sin{2^nx}}=\cot{x}-\cot{2^nx}\]

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.

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

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

where the abbreviation $\left|P_1P_2P_3\right|$ denotes the (non-directed) area of an arbitrary triangle $P_1P_2P_3$.