NMC 1991

Determine the last two digits of the number

$$2^{5}+2^{5^{2}}+2^{5^{3}}+\cdots+2^{5^{1991}}$$

written in decimal notation.

In the trapezium $A B C D$ the sides $A B$ and $C D$ are parallel, and $E$ is a fixed point on the side $A B$. Determine the point $F$ on the side $C D$ so that the area of the intersection of the triangles $A B F$ and $C D E$ is as large as possible.

Figure 3.

Show that

$$\frac{1}{2^{2}}+\frac{1}{3^{2}}+\ldots+\frac{1}{n^{2}}<\frac{2}{3}$$

for all $n \geq 2$.

Let $f(x)$ be a polynomial with integer coefficients. We assume that there exists a positive integer $k$ and $k$ consecutive integers $n, n+1, \ldots, n+k-1$ so that none of the numbers $f(n), f(n+1), \ldots, f(n+k-1)$ is divisible by $k$. Show that the zeroes of $f(x)$ are not integers.