International Mathematical Olympiad  – IMO 1960  Problems

Problem 1

Determine all three-digit numbers $N$ having the property that $N$ is divisible by 11, and $\dfrac{N}{11}$ is equal to the sum of the squares of the digits of $N$.

Problem 2

For what values of the variable $x$ does the following inequality hold:\[ \dfrac{4x^2}{(1-\sqrt{2x+1})^2}<2x+9 \ ? \]

Problem 3

In a given right triangle $ABC$, the hypotenuse $BC$, of length $a$, is divided into $n$ equal parts ($n$ and odd integer). Let $\alpha$ be the acute angel subtending, from $A$, that segment which contains the mdipoint of the hypotenuse. Let $h$ be the length of the altitude to the hypotenuse fo the triangle. Prove that:\[ \tan{\alpha}=\dfrac{4nh}{(n^2-1)a}. \]

Problem 4

Construct triangle $ABC$, given $h_a$, $h_b$ (the altitudes from $A$ and $B$), and $m_a$, the median from vertex $A$.

Problem 5

Consider the cube $ABCDA’B’C’D’$ (with face $ABCD$ directly above face $A’B’C’D’$).

a) Find the locus of the midpoints of the segments $XY$, where $X$ is any point of $AC$ and $Y$ is any piont of $B’D’$;

b) Find the locus of points $Z$ which lie on the segment $XY$ of part a) with $ZY=2XZ$.

Problem 6

Consider a cone of revolution with an inscribed sphere tangent to the base of the cone. A cylinder is circumscribed about this sphere so that one of its bases lies in the base of the cone. let $V_1$ be the volume of the cone and $V_2$ be the volume of the cylinder.

a) Prove that $V_1 \neq V_2$;

b) Find the smallest number $k$ for which $V_1=kV_2$; for this case, construct the angle subtended by a diamter of the base of the cone at the vertex of the cone.

Problem 7

An isosceles trapezoid with bases $a$ and $c$ and altitude $h$ is given.

a) On the axis of symmetry of this trapezoid, find all points $P$ such that both legs of the trapezoid subtend right angles at $P$;

b) Calculate the distance of $p$ from either base;

c) Determine under what conditions such points $P$ actually exist. Discuss various cases that might arise.