IMO 1962

Official IMO 1962 problems  ·  6/7 solved, 4 verified.

Problem 1   ✓ verified · 7m17s · Solution →

Find the smallest natural number $n$ with the following properties:

(a) Its decimal representation ends in 6.

(b) If the final digit 6 is moved to the front, the resulting number is four times as large.

Problem 2   ✓ verified · 25m30s · Solution →

Find all real numbers $x$ satisfying $$\sqrt{3-x} - \sqrt{x+1} > \frac{1}{2}.$$

Problem 3   ✓ verified · 2m18s · Solution →

A cube $ABCDA'B'C'D'$ is given. Point $X$ moves at constant speed along the square $ABCD$ in the direction $A \to B$; point $Y$ moves at the same speed along the square $BCC'B'$ in the direction $B' \to C'$. Initially $X = A$ and $Y = B'$. Find the locus of all midpoints of $XY$.

Problem 4   ✓ verified · 21m14s · Solution →

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

Problem 5   solved · 12m07s · Solution →

On a circle $k$ three points $A$, $B$, $C$ are given. Construct a fourth point $D$ on $k$ such that a circle can be inscribed in quadrilateral $ABCD$.

Problem 6

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

Problem 7   solved · 15m54s · Solution →

Prove that a tetrahedron $SABC$ has five spheres each tangent to all six lines determined by its edges if and only if it is regular.