# 1968 IMO Problems

Problems of the 10th IMO 1968 in USSR.

## Problem 1

Prove that there is one and only one triangle whose side lengths are consecutive integers, and one of whose angles is twice as large as another.

## Problem 2

Find all natural numbers such that the product of their digits (in decimal notation) is equal to .

## Problem 3

Consider the system of equations with unknowns where are real and . Let . Prove that for this system

(a) if , there is no solution,

(b) if , there is exactly one solution,

(c) if , there is more than one solution.

## Problem 4

Prove that in every tetrahedron there is a vertex such that the three edges meeting there have lengths which are the sides of a triangle.

## Problem 5

Let be a real-valued function defined for all real numbers such that, for some positive constant , the equation holds for all .

(a) Prove that the function is periodic (i.e., there exists a positive number such that for all ).

(b) For , give an example of a non-constant function with the required properties.

## Problem 6

For every natural number , evaluate the sum (The symbol denotes the greatest integer not exceeding .)