NL-NWO 1969

Determine the smallest $n$ such that $n \equiv (a - 1)$ mod $a$ for all $a \in \{2,3,..., 10\}$ .

Prove that for all $n \in N$ , $x^2 + y^2 = z^n$ has solutions with $x,y,z \in N$ .

Given a quadrilateral $ABCD$ with $AB = BD = DC$ and $AC = BC$ . On $BC$ lies point $E$ such that $AE = AB$ . Prove that $ED = EB$ .

An angle $< 45^o$ is given in the plane of the drawing. Furthermore, the projection $P_1$ of a point $P$ lying above the plane of the drawing and the distance from $P$ to $P_1$ are given. $P_1$ lies within the given angle. On the legs of the given angle, construct points $A$ and $B$ , respectively, such that the triangle $PAB$ has a minimal perimeter.

a) Prove that for $n = 2,3,4,...$ holds: $$\sin a + \sin 2a + ...+ \sin (n-1)a=\frac{\cos a \left(\frac{a}{2}\right) - \cos \left(n-\frac{1}{2}\right) a}{2 \sin \left(\frac{a}{2}\right)}$$ b) A point on the circumference of a wheel, which, remaining in a vertical plane, rolls along a horizontal path, describes, at one revolution of the wheel, a curve having a length equal to four times the diameter of the wheel. Prove this by first considering tilting a regular $n$ -gon.