International Mathematical Olympiad  – IMO 1962  Problems

Problem 1

Find the smallest natural number $n$ which has the following properties:

a) Its decimal representation has a 6 as the last digit.

b) If the last digit 6 is erased and placed in front of the remaining digits, the resulting number is four times as large as the original number $n$.

Problem 2

Determine all real numbers $x$ which satisfy the inequality:\[ \sqrt{3-x}-\sqrt{x+1}>\dfrac{1}{2} \]

Problem 3

Consider the cube $ABCDA’B’C’D’$ ($ABCD$ and $A’B’C’D’$ are the upper and lower bases, repsectively, and edges $AA’, BB’, CC’, DD’$ are parallel). The point $X$ moves at a constant speed along the perimeter of the square $ABCD$ in the direction $ABCDA$, and the point $Y$ moves at the same rate along the perimiter of the square $B’C’CB$ in the direction $B’C’CBB’$. Points $X$ and $Y$ begin their motion at the same instant from the starting positions $A$ and $B’$, respectively. Determine and draw the locus of the midpionts of the segments $XY$.

Problem 4

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

Problem 5

On the circle $K$ there are given three distinct points $A,B,C$. Construct (using only a straightedge and a compass) a fourth point $D$ on $K$ such that a circle can be inscribed in the quadrilateral thus obtained.

Problem 6

Consider an isosceles triangle. let $R$ be the radius of its circumscribed circle and $r$ be the radius of its inscribed circle. Prove that the distance $d$ between the centers of these two circle is\[ d=\sqrt{R(R-2r)} \]

Problem 7

The tetrahedron $SABC$ has the following property: there exist five spheres, each tangent to the edges $SA, SB, SC, BC, CA, AB,$ or to their extensions.

a) Prove that the tetrahedron $SABC$ is regular.

b) Prove conversely that for every regular tetrahedron five such spheres exist.