ZA-SAMO 2002

Given a quadrilateral $ABCD$ such that $AB^2 + CD^2 = AD^2 + BC^2$ , prove that $AC \perp BD$ .

Find all triples of natural numbers $(a,b,c)$ such that $a$ , $b$ and $c$ are in geometric progression and $a + b + c = 111$ .

A small square $PQRS$ is contained in a big square. Produce $PQ$ to $A$ , $QR$ to $B$ , $RS$ to $C$ and $SP$ to $D$ so that $A$ , $B$ , $C$ and $D$ lie on the four sides of the large square in order, produced if necessary. Prove that $AC = BD$ and $AC \perp BD$ .

How many ways are there to express 1000000 as a product of exactly three integers greater than 1? (For the purpose of this problem, $abc$ is not considered different from $bac$ , etc.)

In acute-angled triangle $ABC$ , a semicircle with radius $r_a$ is constructed with its base on $BC$ and tangent to the other two sides. $r_b$ and $r_c$ are defined similarly. $r$ is the radius of the incircle of $ABC$ . Show that $$\frac{2}{r} = \frac{1}{r_a} + \frac{1}{r_b} + \frac{1}{r_c}.$$

Find all rational numbers $a$ , $b$ , $c$ and $d$ such that $$8a^2 - 3b^2 + 5c^2 + 16d^2 - 10ab + 42cd + 18a + 22b - 2c - 54d = 42,$$ $$15a^2 - 3b^2 + 21c^2 - 5d^2 + 4ab +32cd - 28a + 14b - 54c - 52d = -22.$$