1969 IMO Problems
Problems of the 11th IMO 1969 in Romania.
Prove that there are infinitely many natural numbers with the following property: the number is not prime for any natural number .
Let be real constants, a real variable, and Given that , prove that for some integer .
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.
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.
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.
Prove that for all real numbers , with , the inequality is satisfied. Give necessary and sufficient conditions for equality.