International Competitions 2001

Let $ABC$ be an acute-angled triangle with circumcentre $O$. Let $P$ on $BC$ be the foot of the altitude from $A$.

Suppose that $\angle BCA \geq \angle ABC + 30^{\circ}$.

Prove that $\angle CAB + \angle COP < 90^{\circ}$.

Prove that

$$\frac{a}{\sqrt{a^2 + 8bc}} + \frac{b}{\sqrt{b^2 + 8ca}} + \frac{c}{\sqrt{c^2 + 8ab}} \geq 1$$

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

Twenty-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 that was solved by at least three girls and at least three boys.

Let $n$ be an odd integer greater than 1, and let $k_1, k_2, \ldots, k_n$ be given integers. For each of the $n!$ permutations $a = (a_1, a_2, \ldots, a_n)$ of $1, 2, \ldots, n$, let

$$S(a) = \sum_{i=1}^{n} k_i a_i.$$

Prove that there are two permutations $b$ and $c, b \neq c$, such that $n!$ is a divisor of $S(b) - S(c)$.

In a triangle $ABC$, let $AP$ bisect $\angle BAC$, with $P$ on $BC$, and let $BQ$ bisect $\angle ABC$, with $Q$ on $CA$.

It is known that $\angle BAC = 60^{\circ}$ and that $AB + BP = AQ + QB$.

What are the possible angles of triangle $ABC$?

Let $a, b, c, d$ be integers with $a > b > c > d > 0$. Suppose that

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

Prove that $ab + cd$ is not prime.