CAPS 1999

The altitudes through the vertices $A,B,C$ of an acute-angled triangle $ABC$ meet the opposite sides at $D,E,F,$ respectively. The line through $D$ parallel to $EF$ meets the lines $AC$ and $AB$ at $Q$ and $R$ , respectively. The line $EF$ meets $BC$ at $P$ . Prove that the circumcircle of the triangle $PQR$ passes through the midpoint of $BC$ .

Find all natural numbers $k$ for which there exists a set $M$ of ten real numbers such that there are exactly $k$ pairwise non-congruent triangles whose side lengths are three (not necessarily distinct) elements of $M$ .

Find all positive integers $k$ for which the following assertion holds: If $F(x)$ is polynomial with integer coefficients ehich satisfies $F(c) \leq k$ for all $c \in \{0,1, \cdots,k+1 \}$ , then $$F(0)= F(1) = \cdots =F(k+1).$$

Find all functions $f: (1,\infty)\text{to R}$ satisfying $f(x)-f(y)=(y-x)f(xy)$ for all $x,y>1$ .

Prove that for any integer $n \ge 3$ , the least common multiple of the numbers $1,2, ... ,n$ is greater than $2^{n-1}$ .