IMO 1960

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 $N$ .

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

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: $$\tan{\alpha}=\dfrac{4nh}{(n^2-1)a}.$$

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 $ABCDA'B'C'D'$ (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 $ZY=2XZ$ .

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.