International Mathematical Olympiad · 1977

Problems

Problem 1Inside square $ABCD$ , construct equilateral triangles $ABK$ , $BCL$ , $CDM$ , and $DAN$ . Prove that the midpoints of $KL,LM,MN,NK$ and of $AK,BK,BL,CL,CM,DM,DN,AN$ are the twelve vertices of a regular dodecagon.Solutions: 1
Problem 2In a finite sequence of real numbers, every sum of seven successive terms is negative and every sum of eleven successive terms is positive. Determine the maximum length.Solutions: 1
Problem 3Let $n>2$ and let $V_n=\{1+kn:k=1,2,\ldots\}$ . An element of $V_n$ is indecomposable if it is not a product of two elements of $V_n$ . Prove that some element of $V_n$ has two distinct factorizations into indecomposable elements, where order is ignored.Solutions: 1
Problem 4Let $f(\theta)=1-a\cos\theta-b\sin\theta-A\cos2\theta-B\sin2\theta$ , where $a,b,A,B$ are real. Prove that if $f(\theta)\ge0$ for all real $\theta$ , then $a^2+b^2\le2$ and $A^2+B^2\le1$ .Solutions: 1
Problem 5Let $a,b$ be positive integers. Dividing $a^2+b^2$ by $a+b$ gives quotient $q$ and remainder $r$ . Find all pairs $(a,b)$ such that $q^2+r=1977$ .Solutions: 1
Problem 6Let $f:\mathbb{Z}_{>0}\to\mathbb{Z}_{>0}$ . Prove that if $f(n+1)>f(f(n))$ for every positive integer $n$ , then $f(n)=n$ for every positive integer $n$ .Solutions: 1