ZA-SAMO 2007

Determine whether $\frac{1}{\sqrt{2}} - \frac{1}{\sqrt{6}}$ is less than or greater than $\frac{3}{10}$ .

Consider the equation $x^4 = ax^3 + bx^2 + cx + 2007$ , where $a,b,c$ are real numbers. Determine the largest value of $b$ for which this equation has exactly three distinct solutions, all of which are integers.

In acute-angled triangle $ABC$ , the points $D,E,F$ are on sides $BC,CA,AB$ , respectively such that $\angle AFE = \angle BFD, \angle FDB = \angle EDC, \angle DEC = \angle FEA$ . Prove that $AD$ is perpendicular to $BC$ .

Let $ABC$ be a triangle and $PQRS$ a square with $P$ on $AB$ , $Q$ on $AC$ , and $R$ and $S$ on $BC$ . Let $H$ on $BC$ such that $AH$ is the altitude of the triangle from $A$ to base $BC$ . Prove that:

(a) $\frac{1}{AH} +\frac{1}{BC}=\frac{1}{PQ}$ (b) the area of $ABC$ is twice the area of $PQRS$ iff $AH=BC$

Let $Z$ and $R$ denote the sets of integers and real numbers, respectively. Let $f: Z \rightarrow R$ be a function satisfying: (i) $f(n) \ge 0$ for all $n \in Z$ (ii) $f(mn)=f(m)f(n)$ for all $m,n \in Z$ (iii) $f(m+n) \le max(f(m),f(n))$ for all $m,n \in Z$

(a) Prove that $f(n) \le 1$ for all $n \in Z$ (b) Find a function $f: Z \rightarrow R$ satisfying (i), (ii),(iii) and $0<f(2)<1$ and $f(2007) = 1$

Prove that it is not possible to write numbers $1,2,3,...,25$ on the squares of $5$ x $5$ chessboard such that any neighboring numbers differ by at most $4$ .