IMO 2001

Problem 1. Let $ABC$ be an acute-angled triangle with $O$ as its circumcenter. Let $P$ on line $BC$ be the foot of the altitude from $A$. Assume that $\angle BCA\ge \angle ABC+30^{\circ }$. Prove that $\angle CAB+\angle COP<90^{\circ }$.

Problem 2. Prove that

for all positive real numbers $a,b$, and $c$.

Problem 3. Twenty-one girls and twenty-one boys took part in a mathematical competition. It turned out that

(a) each contestant solved at most six problems, and

(b) for each pair of a girl and a boy, there was at least one problem that was solved by both the girl and the boy.

Prove that there is a problem that was solved by at least three girls and at least three boys.

Problem 4. Let $n$ be an odd integer greater than 1 and let $c_1,c_2,\dots ,c_n$ be integers. For each permutation $a=(a_1,a_2,\dots ,a_n)$ of $\{1,2,\dots ,n\}$, define $S(a)={\sum }_{i=1}^n{c}_i{a}_i$. Prove that there exist permutations $b$ and $c$, $b\ne c$, such that $n!$ divides $S(b)-S(c)$.

Problem 5. In a triangle $ABC$, let segment $AP$ bisect $\angle BAC$, with $P$ on side $BC$, and let segment $BQ$ bisect $\angle ABC$, with $Q$ on side $CA$. It is known that $\angle BAC=60^{\circ }$ and that $AB+BP=AQ+QB$. What are the possible angles of triangle $ABC$?

Problem 6. Let $a>b>c>d$ be positive integers and suppose

$ac+bd=(b+d+a-c)(b+d-a+c).$

Prove that $ab+cd$ is not prime.