1962 IMO Problems
Find the smallest natural number which has the following properties:
(a) Its decimal representation has 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 .
Determine all real numbers which satisfy the inequality:
Consider the cube ( and are the upper and lower bases, respectively, and edges , , , are parallel). The point moves at constant speed along the perimeter of the square in the direction , and the point moves at the same rate along the perimeter of the square in the direction . Points and begin their motion at the same instant from the starting positions and , respectively. Determine and draw the locus of the midpoints of the segments .
Solve the equation .
On the circle there are given three distinct points . Construct (using only straightedge and compass) a fourth point on such that a circle can be inscribed in the quadrilateral thus obtained.
Consider an isosceles triangle. Let be the radius of its circumscribed circle and the radius of its inscribed circle. Prove that the distance between the centers of these two circles is
The tetrahedron has the following property: there exist five spheres, each tangent to the edges , or to their extensions.
(a) Prove that the tetrahedron is regular.
(b) Prove conversely that for every regular tetrahedron five such spheres exist.