NL-NWO 1963

In a plane are given a straight line $\ell$ and a point $P$ not located on $\ell$ . Is there a circle in this plane such that there exist more than three different points $S$ on $\ell$ with the property that the perpendicular bisector of $PS$ is tangent to the circle ? Explain the answer.

The straight lines $k$ and $\ell$ intersect at right angles. A line intersects $k$ in $A$ and $\ell$ in $B$ . Consider all straight line segments $PQ$ ( $P$ on $k$ and $Q$ on $\ell$ ), which makes an angle of $45^o$ with $AB$ . (a) Determine the locus of the midpoints of the line segments $PQ$ , (b) If the perpendicular bisector of such a line segment $PQ$ intersects the line $k$ at $K$ and the line $\ell$ at $L$ , then prove that $KL \ge PQ$ .

Twenty numbers $a_1,a_2,..,a_{20}$ satisfy: $$a_k \ge 7k \,\,\,\, for \,\,\,\, k = 1,2,..., 20$$ $$a_1+a_2+...+a_{20}=1518$$ Prove that among the numbers $k = 1,2,... ,20$ there are no more than seventeen, for which $a_k \ge 20k -2k^2$ .

One considers for $n > 2$ the polynomial: $$(x^2-x+1)^n - (x^2-x+2)^n+ (1+x)^n+(2-x)^n$$ Show that the degree of this polynomial is $2n - 2$ . The polynomial is written in the form $$a_0+a_1x+a_2x^2+...+a_{2n-2}x^{2n-2}$$ Prove that $a_2+a_3+...+a_{2n-2}=0$

You want to color the side faces of a cube in such a way that each face is colored evenly. Six colors are available: red, white, blue, yellow, purple, orange. Two cube colors are called the same if one arises from the other by a rotation of the cube. (a) How many different cube colorings are there, using six colors? (b) How many different cube colorings are there, using exactly five colors?