International Mathematical Olympiad · 1973

Problems

Problem 1Let $O$ lie on a line $l$ . The vectors $\overrightarrow{OP_1},\ldots,\overrightarrow{OP_n}$ are unit vectors, and the points $P_i$ and the line $l$ lie in one plane, with every $P_i$ in the same half-plane bounded by $l$ . Prove that, when $n$ is odd, $\left|\sum_{i=1}^n\overrightarrow{OP_i}\right|\ge 1$ .Solutions: 1
Problem 2Does there exist a finite set $M$ of points in space, not all in one plane, such that for every two points $A,B\in M$ there are two other points $C,D\in M$ for which the lines $AB$ and $CD$ are parallel but distinct?Solutions: 1
Problem 3Determine the minimum of $a^2+b^2$ over real $a,b$ for which $x^4+ax^3+bx^2+ax+1=0$ has at least one real solution.Solutions: 1
Problem 4A soldier must investigate mines in an equilateral triangular region. His detector has radius equal to one-half the triangle’s altitude, and he starts at one vertex. Determine the shortest path that checks the whole region.Solutions: 1
Problem 5Let $G$ be a set of non-constant functions $f:\mathbb R\to\mathbb R$ of the form $f(x)=ax+b$ , with real $a,b$ and $a\ne0$ . Suppose: if $f,g\in G$ then $g\circ f\in G$ ; each inverse $f^{-1}$ belongs to $G$ ; and every $f\in G$ has a fixed point. Prove that there is $k\in\mathbb R$ fixed by every $f\in G$ .Solutions: 1
Problem 6Let $a_1,\ldots,a_n$ be positive numbers and let $0<q<1$ . Find real numbers $b_1,\ldots,b_n$ such that (1) $a_k<b_k$ for every $k$ ; (2) $q<b_{k+1}/b_k<1/q$ for $k=1,\ldots,n-1$ ; and (3) $\sum_{k=1}^n b_k<\frac{1+q}{1-q}\sum_{k=1}^n a_k$ .Solutions: 1