CAPS 2004

Show that real numbers, $p, q, r$ satisfy the condition $p^4(q-r)^2 + 2p^2(q+r) + 1 = p^4$ if and only if the quadratic equations $x^2 + px + q = 0$ and $y^2 - py + r = 0$ have real roots (not necessarily distinct) which can be labeled by $x_1,x_2$ and $y_1,y_2$ , respectively, in such a way that $x_1y_1 - x_2y_2 = 1$ .

Show that for each natural number $k$ there exist only finitely many triples $(p, q, r)$ of distinct primes for which $p$ divides $qr-k$ , $q$ divides $pr-k$ , and $r$ divides $pq - k$ .

A point P in the interior of a cyclic quadrilateral ABCD satisfies ∠BPC = ∠BAP + ∠PDC. Denote by E, F and G the feet of the perpendiculars from P to the lines AB, AD and DC, respectively. Show that the triangles FEG and PBC are similar.

Solve in real numbers the system of equations: \begin{align*} \frac{1}{xy}&=\frac{x}{z}+1 \frac{1}{yz}&=\frac{y}{x}+1 \frac{1}{zx}&=\frac{z}{y}+1 \end{align*}

Points $K,L,M$ on the sides $AB,BC,CA$ respectively of a triangle $ABC$ satisfy $\frac{AK}{KB} = \frac{BL}{LC} = \frac{CM}{MA}$ . Show that the triangles $ABC$ and $KLM$ have a common orthocenter if and only if $\triangle ABC$ is equilateral.

On the table there are $k \ge 3$ heaps of $1, 2, \dots , k$ stones. In the first step, we choose any three of the heaps, merge them into a single new heap, and remove $1$ stone from this new heap. Thereafter, in the $i$ -th step ( $i \ge 2$ ) we merge some three heaps containing more than $i$ stones in total and remove $i$ stones from the new heap. Assume that after a number of steps a single heap of $p$ stones remains on the table. Show that the number $p$ is a perfect square if and only if so are both $2k + 2$ and $3k + 1$ . Find the least $k$ with this property.