International Mathematical Olympiad · 1966

Problems

Problem 1Three problems $A$ , $B$ , and $C$ were given on a mathematics olympiad. All 25 students solved at least one of these problems. The number of students who solved $B$ and not $A$ is twice the number of students who solved $C$ and not $A$ . The number of students who solved only $A$ is greater by 1 than the number of students who, along with $A$ , solved at least one other problem. Among the students who solved only one problem, half solved $A$ . How many students solved only $B$ ?Solutions: 1
Problem 2If $a$ , $b$ , and $c$ are the sides and $\alpha$ , $\beta$ , and $\gamma$ are the respective angles of a triangle for which $a+b=\tan\frac{\gamma}{2}\left(a\tan\alpha+b\tan\beta\right)$ , prove that the triangle is isosceles.Solutions: 1
Problem 3Prove 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.Solutions: 1
Problem 4Prove the following equality for any natural number $n$ and any real number $x$ for which no denominator vanishes (that is, $x\ne \frac{k\pi}{2^t}$ for $t=0,1,\dots,n$ and integer $k$ ): $\frac{1}{\sin 2x}+\frac{1}{\sin 4x}+\frac{1}{\sin 8x}+\cdots+\frac{1}{\sin 2^nx} = \cot x - \cot 2^nx.$ Solutions: 1
Problem 5Solve the following system of equations for real numbers $x_1$ , $x_2$ , $x_3$ , $x_4$ : $\begin{aligned} |a_1-a_2|x_2 + |a_1-a_3|x_3 + |a_1-a_4|x_4 &= 1 \\ |a_2-a_1|x_1 + |a_2-a_3|x_3 + |a_2-a_4|x_4 &= 1 \\ |a_3-a_1|x_1 + |a_3-a_2|x_2 + |a_3-a_4|x_4 &= 1 \\ |a_4-a_1|x_1 + |a_4-a_2|x_2 + |a_4-a_3|x_3 &= 1 \end{aligned}$ where $a_1$ , $a_2$ , $a_3$ , $a_4$ are mutually distinct real numbers.Solutions: 1
Problem 6Let $M$ , $K$ , and $L$ be points chosen in the interiors of sides $AB$ , $BC$ , and $CA$ , respectively, of triangle $ABC$ . Prove that the area of at least one of the three triangles $\triangle AML$ , $\triangle KBM$ , and $\triangle LCK$ is less than or equal to one-fourth of the area of triangle $ABC$ .Solutions: 1