ZA-SAMO 2008

Determine the number of positive divisors of $2008^8$ that are less than $2008^4$ .

Let $ABCD$ be a convex quadrilateral with the property that $AB$ extended and $CD$ extended intersect at a right angle. Prove that $AC\cdot BD>AD\cdot BC$ .

Let $a,b,c$ be positive real numbers. Prove that $$(a+b)(b+c)(c+a)\ge 8(a+b-c)(b+c-a)(c+a-b)$$ and determine when equality occurs.

A pack of $2008$ cards, numbered from $1$ to $2008$ , is shuffled in order to play a game in which each move has two steps:

(i) the top card is placed at the bottom;

(ii) the new top card is removed.

It turns out that the cards are removed in the order $1,2,\dots,2008$ . Which card was at the top before the game started?

Triangle $ABC$ has orthocentre $H$ . The feet of the perpendiculars from $H$ to the internal and external bisectors of $\hat{A}$ are $P$ and $Q$ respectively. Prove that $P$ is on the line that passes through $Q$ and the midpoint of $BC$ . (Note: The ortohcentre of a triangle is the point where the three altitudes intersect.)

Find all function pairs $(f,g)$ where each $f$ and $g$ is a function defined on the integers and with values, such that, for all integers $a$ and $b$ , $$f(a+b)=f(a)g(b)+g(a)f(b) g(a+b)=g(a)g(b)-f(a)f(b).$$