International Competitions 1960

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

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

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

Construct triangle $ABC$, given $h_{a}, h_{b}$ (the altitudes from $A$ and $B$) and $m_{a}$, the median from vertex $A$.

Consider the cube $ABCDA^{\prime}B^{\prime}C^{\prime}D^{\prime}$ (with face $ABCD$ directly above face $A^{\prime}B^{\prime}C^{\prime}D^{\prime}$).

(a) Find the locus of the midpoints of segments $XY$, where $X$ is any point of $AC$ and $Y$ is any point of $B^{\prime}D^{\prime}$.

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

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 $V_{2}$ 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 diameter of the base of the cone at the vertex of the cone.

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.)