1975 IMO Problems
Problems of the 17th IMO 1975 in Bulgaria.
Let be real numbers such that Prove that, if is any permutation of , then
Let be an infinite increasing sequence of positive integers. Prove that for every there are infinitely many which can be written in the form with positive integers and .
On the sides of an arbitrary triangle , triangles are constructed externally with . Prove that and .
When is written in decimal notation, the sum of its digits is . Let be the sum of the digits of . Find the sum of the digits of . ( and are written in decimal notation.)
Determine, with proof, whether or not one can find 1975 points on the circumference of a circle with unit radius such that the distance between any two of them is a rational number.
Find all polynomials , in two variables, with the following properties:
(i) for a positive integer and all real (that is, is homogeneous of degree ),
(ii) for all real