International Mathematical Olympiad · 2001

Problems

Problem 1Let $ABC$ be an acute-angled triangle with circumcentre $O$ . Let $P$ on $BC$ be the foot of the altitude from $A$ . Suppose that $\angle BCA\ge\angle ABC+30^\circ$ . Prove that $\angle CAB+\angle COP<90^\circ$ .Solutions: 1
Problem 2For all positive real numbers $a,b,c$ , prove $\frac{a}{\sqrt{a^2+8bc}}+\frac{b}{\sqrt{b^2+8ca}}+\frac{c}{\sqrt{c^2+8ab}}\ge1$ .Solutions: 1
Problem 3Twenty-one girls and twenty-one boys took part in a mathematical contest. Each contestant solved at most six problems. For each girl and each boy, at least one problem was solved by both of them. Prove that there was a problem solved by at least three girls and at least three boys.Solutions: 1
Problem 4Let $n>1$ be an odd integer and let $k_1,k_2,\dots,k_n$ be integers. For each permutation $\mathbf a=(a_1,\dots,a_n)$ of $1,\dots,n$ , define $S(\mathbf a)=\sum_{i=1}^n k_i a_i$ . Prove that there are distinct permutations $\mathbf b,\mathbf c$ such that $n!$ divides $S(\mathbf b)S(\mathbf c)$ .Solutions: 1
Problem 5In triangle $ABC$ , $AP$ bisects $\angle BAC$ with $P$ on $BC$ , and $BQ$ bisects $\angle ABC$ with $Q$ on $CA$ . Given $\angle BAC=60^\circ$ and $AB+BP=AQ+QB$ , determine the possible angles of $ABC$ .Solutions: 1
Problem 6Let $a>b>c>d>0$ be integers satisfying $ac+bd=(b+d+a-c)(b+d-a+c)$ . Prove that $ab+cd$ is not prime.Solutions: 1