Question bank IMO 1968

Prove that there exists a unique triangle whose side lengths are consecutive natural numbers and one of whose angles is twice the measure of one of the others.

Find all positive integers $x$ for which $p(x)=x^2-10x-22$, where $p(x)$ denotes the product of the digits of $x$.

Let $a,b,c$ be real numbers. Prove that the system of equations
$$\begin{cases}ax_1^2+bx_1+c=x_2,\\ax_2^2+bx_2+c=x_3,\\\cdots\\ax_{n-1}^2+bx_{n-1}+c=x_n,\\ax_n^2+bx_n+c=x_1.\end{cases}$$
(a) has no real solutions if $(b-1)^2-4ac\lt 0$;

(b) has a unique real solution if $(b-1)^2-4ac=0$;

(c) has more than one real solution if $(b-1)^2-4ac>0$.

Prove that in any tetrahedron there is a vertex such that the lengths of its sides through that vertex are sides of a triangle.

Let $a>0$ be a real number and $f(x)$ a real function defined on all of $\mathbb R$, satisfying for all $x\in\mathbb R$,
$$f(x+a)=\frac12+\sqrt{f(x)-f(x)^2}.$$
(a) Prove that the function $f$ is periodic; i.e., there exists $b>0$ such that for all $x$, $f(x+b)=f(x)$.

(b) Give an example of such a nonconstant function for $a=1$.

Let $[x]$ denote the integer part of $x$, i.e., the greatest integer not exceeding $x$. If $n$ is a positive integer, express as a simple function of $n$ the sum
$$\left\lfloor\frac{n+1}{2}\right\rfloor+\left\lfloor\frac{n+2}{4}\right\rfloor+\cdots+\left\lfloor\frac{n+2^i}{2^{i+1}}\right\rfloor+\cdots.$$