CAPS 1996

Show that an integer $p > 3$ is a prime if and only if for every two nonzero integers $a,b$ exactly one of the numbers $N_1 = a+b-6ab+\frac{p-1}{6}$ , $N_2 = a+b+6ab+\frac{p-1}{6}$ is a nonzero integer.

Let ⋆ be a binary operation on a nonempty set $M$ . That is, every pair $(a,b) \in M$ is assigned an element $a$ ⋆ $b$ in $M$ . Suppose that ⋆ has the additional property that $(a$ ⋆ $b)$ ⋆ $b= a$ and $a$ ⋆ $(a$ ⋆ $b)= b$ for all $a,b \in M$ . (a) Show that $a$ ⋆ $b = b$ ⋆ $a$ for all $a,b \in M$ . (b) On which finite sets $M$ does such a binary operation exist?

The base of a regular quadrilateral pyramid $\pi$ is a square with side length $2a$ and its lateral edge has length a $\sqrt{17}$ . Let $M$ be a point inside the pyramid. Consider the five pyramids which are similar to $\pi$ , whose top vertex is at $M$ and whose bases lie in the planes of the faces of $\pi$ . Show that the sum of the surface areas of these five pyramids is greater or equal to one fifth the surface of $\pi$ , and find for which $M$ equality holds.

Decide whether there exists a function $f : Z \rightarrow Z$ such that for each $k =0,1, ...,1996$ and for any integer $m$ the equation $f (x)+kx = m$ has at least one integral solution $x$ .

Two sets of intervals $A ,B$ on the line are given. The set $A$ contains $2m-1$ intervals, every two of which have an interior point in common. Moreover, every interval from $A$ contains at least two disjoint intervals from $B$ . Show that there exists an interval in $B$ which belongs to at least $m$ intervals from $A$ .

The points $E$ and $D$ lie in the interior of sides $AC$ and $BC$ , respectively, of a triangle $ABC$ . Let $F$ be the intersection of the lines $AD$ and $BE$ .Show that the area of the traingles $ABC$ and $ABF$ satisfies:

$\frac{S_{ABC}}{S_{ABF}} = \frac{\mid{AC}\mid}{\mid{AE} \mid} + \frac{\mid{BC}\mid}{\mid{BD}\mid} - 1$ .