Question bank IMO 1962

(POL) Find the smallest natural number $n$ with the following properties:
(a) In decimal representation it ends with $6$.
(b) If we move this digit to the front of the number, we get a number $4$ times larger.

(HUN) Find all real numbers $x$ for which
$$\sqrt{3-x}-\sqrt{x+1}>\frac12.$$

(CZS) A cube $ABCDA'B'C'D'$ is given. The point $X$ is moving at a constant speed along the square $ABCD$ in the direction from $A$ to $B$. The point $Y$ is moving with the same constant speed along the square $BCC'B'$ in the direction from $B'$ to $C'$. Initially, $X$ and $Y$ start out from $A$ and $B'$ respectively. Find the locus of all the midpoints of $XY$.

(ROM) Solve the equation
$$\cos^2x+\cos^2 2x+\cos^2 3x=1.$$

(BUL) On the circle $k$ three points $A$, $B$, and $C$ are given. Construct the fourth point on the circle $D$ such that one can inscribe a circle in $ABCD$.

(GDR) Let $ABC$ be an isosceles triangle with circumradius $r$ and inradius $\rho$. Prove that the distance $d$ between the circumcenter and incenter is given by
$$d=\sqrt{r(r-2\rho)}.$$

(USS) Prove that a tetrahedron $SABC$ has five different spheres that touch all six lines determined by its edges if and only if it is regular.