International Mathematical Olympiad  – IMO 1968  Problems

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 $x$ such that the product of their digits (in decimal notation) is equal to $x^2 – 10x – 22$.

Problem 3

Consider the system of equations\[ax_1^2 + bx_1 + c = x_2\]\[ax_2^2 + bx_2 + c = x_3\]\[\cdots\]\[ax_{n-1}^2 + bx_{n-1} + c = x_n\]\[ax_n^2 + bx_n + c = x_1\]with unknowns $x_1, x_2, \cdots, x_n$ where $a, b, c$ are real and $a \neq 0$. Let $\Delta = (b – 1)^2 – 4ac$. Prove that for this system

(a) if $\Delta < 0$, there is no solution,

(b) if $\Delta = 0$, there is exactly one solution,

(c) if $\Delta > 0$, 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 $f$ be a real-valued function defined for all real numbers $x$ such that, for some positive constant $a$, the equation\[f(x + a) = \frac{1}{2} + \sqrt{f(x) – (f(x))^2}\]holds for all $x$.

(a) Prove that the function $f$ is periodic (i.e., there exists a positive number $b$ such that $f(x + b) = f(x)$ for all $x$).

(b) For $a = 1$, give an example of a non-constant function with the required properties.

Problem 6

For every natural number $n$, evaluate the sum\[\sum_{k = 0}^\infty\bigg[\frac{n + 2^k}{2^{k + 1}}\bigg] = \Big[\frac{n + 1}{2}\Big] + \Big[\frac{n + 2}{4}\Big] + \cdots + \bigg[\frac{n + 2^k}{2^{k + 1}}\bigg] + \cdots\](The symbol $[x]$ denotes the greatest integer not exceeding $x$.)