International Competitions 1968

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.

Find all natural numbers $x$ such that the product of their digits (in decimal notation) is equal to $x^2 - 10x - 22$.

Consider the system of equations $$\begin{aligned} ax_1^2 + bx_1 + c &= x_2 \\ ax_2^2 + bx_2 + c &= x_3 \\ &\vdots \\ ax_{n-1}^2 + bx_{n-1} + c &= x_n \\ ax_n^2 + bx_n + c &= x_1, \end{aligned}$$ with unknowns $x_1, x_2, \ldots, x_n$, 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.

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.

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.

For every natural number $n$, evaluate the sum $$\sum_{k=0}^{\infty} \left[ \frac{n + 2^k}{2^{k+1}} \right] = \left[ \frac{n + 1}{2} \right] + \left[ \frac{n + 2}{4} \right] + \cdots + \left[ \frac{n + 2^k}{2^{k+1}} \right] + \cdots$$

(The symbol $[x]$ denotes the greatest integer not exceeding $x$.)