1969 IMO Problems
Problems of the 11th IMO 1969 in Romania.
Problem 1
Prove that there are infinitely many natural numbers with the following property: the number
is not prime for any natural number
.
Problem 2
Let be real constants,
a real variable, and
Given that
, prove that
for some integer
.
Problem 3
For each value of , find necessary and sufficient conditions on the number
so that there exists a tetrahedron with k edges of length
, and the remaining
edges of length 1.
Problem 4
A semicircular arc is drawn on
as diameter.
is a point on
other than
and
, and
is the foot of the perpendicular from
to
. We consider three circles,
, all tangent to the line
. Of these,
is inscribed in
, while
and
are both tangent to
and to
, one on each side of
. Prove that
,
and
have a second tangent in common.
Problem 5
Given points in the plane such that no three are collinear. Prove that there are at least
convex quadrilaterals whose vertices are four of the given points.
Problem 6
Prove that for all real numbers , with
, the inequality
is satisfied. Give necessary and sufficient conditions for equality.