International Mathematical Olympiad · 1968

Problems

Problem 1Prove 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.Solutions: 1
Problem 2Find all natural numbers $x$ such that the product of their digits (in decimal notation) is equal to $x^2 - 10x - 22$ .Solutions: 1
Problem 3Consider the following system of equations in the unknowns $x_1, x_2, \ldots, x_n$ , where $a, b, c$ are real numbers with $a \neq 0$ : $ax_1^2+bx_1+c=x_2,\quad ax_2^2+bx_2+c=x_3,\quad \ldots,\quad ax_{n-1}^2+bx_{n-1}+c=x_n,\quad ax_n^2+bx_n+c=x_1.$ Let $\Delta=(b-1)^2-4ac$ . Prove that: (a) if $\Delta<0$ , the system has no solution; (b) if $\Delta=0$ , the system has exactly one solution; (c) if $\Delta>0$ , the system has more than one solution.Solutions: 1
Problem 4Prove that in every tetrahedron there is a vertex such that the three edges meeting there have lengths which are the sides of a triangle.Solutions: 1
Problem 5Let $f$ be a real-valued function defined for all real numbers $x$ such that, for some positive constant $a$ , the equation $f(x+a)=\frac12+\sqrt{f(x)-f(x)^2}$ holds for all $x$ . (a) Prove that $f$ is periodic. (b) For $a=1$ , give an example of a non-constant function with the required property.Solutions: 1
Problem 6Let $[x]$ denote the greatest integer not exceeding $x$ . If $n$ is a positive integer, express $[\frac{n+1}{2}]+[\frac{n+2}{4}]+[\frac{n+4}{8}]+\cdots$ as a function of $n$ .Solutions: 1