1964 IMO Problems
Problems of the 6th IMO 1964 in USSR.
Contents
Day I
Problem 1
(a) Find all positive integers for which
is divisible by
.
(b) Prove that there is no positive integer for which
is divisible by
.
Problem 2
Suppose are the sides of a triangle. Prove that
Problem 3
A circle is inscribed in a triangle with sides
. Tangents to the circle parallel to the sides of the triangle are contructed. Each of these tangents cuts off a triangle from
. In each of these triangles, a circle is inscribed. Find the sum of the areas of all four inscribed circles (in terms of
).
Day II
Problem 4
Seventeen people correspond by mail with one another - each one with all the rest. In their letters only three different topics are discussed. Each pair of correspondents deals with only one of these topics. Prove that there are at least three people who write to each other about the same topic.
Problem 5
Suppose five points in a plane are situated so that no two of the straight lines joining them are parallel, perpendicular, or coincident. From each point perpendiculars are drawn to all the lines joining the other four points. Determine the maximum number of intersections that these perpendiculars can have.
Problem 6
In tetrahedron , vertex
is connected with
, the centroid of
. Lines parallel to
are drawn through
and
. These lines intersect the planes
and
in points
and
, respectively. Prove that the volume of
is one third the volume of
. Is the result true if point
is selected anywhere within
?