International Mathematical Olympiad 2001 — Official contest

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 $\angle CAB + \angle COP \lt 90^\circ$.

Let $a$, $b$, $c$ be positive reals. Prove that $$
\frac{a}{\sqrt{a^2+8bc}} + \frac{b}{\sqrt{b^2+8ca}} + \frac{c}{\sqrt{c^2+8ab}} \ge 1.
$$

Twenty-one girls and twenty-one boys took part in a mathematical competition. It turned out that each contestant solved at most six problems, and 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. Show that there is a problem that was solved by at least three girls and at least three boys.

Let $n \gt 1$ be an odd integer and let $c_1$, $c_2$, …, $c_n$ 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 two permutations $a \neq b$ of $\{1,2,\dots,n\}$ such that $n!$ is a divisor of $S(a)-S(b)$.

Let $ABC$ be a triangle. Let $\overline{AP}$ bisect $\angle BAC$ and let $\overline{BQ}$ bisect $\angle ABC$, with $P$ on $\overline{BC}$ and $Q$ on $\overline{AC}$. If $AB + BP = AQ + QB$ and $\angle BAC = 60^\circ$, what are the angles of the triangle?

Let $a \gt b \gt c \gt d \gt 0$ be integers satisfying $$
ac + bd = (b+d+a-c)(b+d-a+c).
$$ Prove that $ab + cd$ is not prime.