International Mathematical Olympiad 1960 — Contest Problems

(BUL) Find all the three-digit numbers for which one obtains, when dividing the number by $11$, the sum of the squares of the digits of the initial number.

(HUN) For which real numbers $x$ does the following inequality hold:
$$\frac{4x^2}{(1-\sqrt{1+2x})^2}<2x+9?$$

(ROM) A right-angled triangle $ABC$ is given for which the hypotenuse $BC$ has length $a$ and is divided into $n$ equal segments, where $n$ is odd. Let $\alpha$ be the angle with which the point $A$ sees the segment containing the middle of the hypotenuse. Prove that
$$\tan\alpha=\frac{4nh}{(n^2-1)a},$$
where $h$ is the height of the triangle.

(HUN) Construct a triangle $ABC$ whose lengths of heights $h_a$ and $h_b$ (from $A$ and $B$, respectively) and length of median $m_a$ (from $A$) are given.

(CZS) A cube $ABCDA'B'C'D'$ is given.

(a) Find the locus of all midpoints of segments $XY$, where $X$ is any point on segment $AC$ and $Y$ any point on segment $B'D'$.

(b) Find the locus of all points $Z$ on segments $XY$ such that $\overrightarrow{ZY}=2\overrightarrow{XZ}$.

(BUL) An isosceles trapezoid with bases $a$ and $b$ and height $h$ is given.

(a) On the line of symmetry construct the point $P$ such that both (non-base) sides are seen from $P$ with an angle of $90^\circ$.

(b) Find the distance of $P$ from one of the bases of the trapezoid.

(c) Under what conditions for $a,b$, and $h$ can the point $P$ be constructed (analyze all possible cases)?

(GDR) A sphere is inscribed in a regular cone. Around the sphere a cylinder is circumscribed so that its base is in the same plane as 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 $V_1=V_2$ is impossible.

(b) Find the smallest $k$ for which $V_1=kV_2$, and in this case construct the angle at the vertex of the cone.