1962 IMO Problems
Contents
Day I
Problem 1
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 .
Problem 2
Determine all real numbers which satisfy the inequality:
Problem 3
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
.
Day II
Problem 4
Solve the equation .
Problem 5
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.
Problem 6
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
![$d=\sqrt{r(r-2p)}$](http://latex.artofproblemsolving.com/0/5/1/051a64f293508aad41b5262146f6db9cb1553d06.png)
Problem 7
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.