Let $x$ be an angle between $0^o$ and $90^o$ so that $$\frac{\sin^4 x}{9}+\frac{\cos^4 x}{16 }=\frac{1}{25} .$$Then what is $\tan x$?

Every officially published book used to have an ISBN code (International Standard Book Number) which consisted of $10$ symbols. Such code looked like this: $$a_1a_2 . . . a_9a_{10}$$with $a_1, . . . , a_9 \in \{0, 1, . . . , 9\}$ and $a_{10} \in \{0, 1, . . . , 9, X\}$. The symbol $X$ stood for the number $10$. With a valid ISBN code was $$a_1 + 2a2 + . . . + 9a_9 + 10a_{10}$$a multiple of $11$. Prove the following statements. (a) If one symbol is changed in a valid ISBN code, the result is no valid ISBN code. (b) When two different symbols swap places in a valid ISBN code then the result is not a valid ISBN.

The point $M$ is the center of a regular pentagon $ABCDE$. The point $P$ is an inner point of the line segment $[DM]$. The circumscribed circle of triangle $\vartriangle ABP$ intersects the side $[AE]$ at point $Q$ (different from $A$). The perpendicular from $P$ on $CD$ intersects the side $[AE] $ at point $S$. Prove that $PS$ is the bisector of $\angle APQ$.

There are $n$ hoops on a circle. Rik numbers all hoops with a natural number so that all numbers from $1$ to $n$ occur exactly once. Then he makes one walk from hoop to hoop. He starts in hoop $1$ and then follows the following rule: if he gets to hoop $k$, then he walks to the hoop that places $k$ clockwise without getting into the intermediate hoops. The walk ends when Rik has to walk to a hoop he has already been to. The length of the walk is the number of hoops he passed on the way. For example, for $n = 6$ Rik can take a walk of length $5$ as the hoops are numbered as shown in the figure. (a) Determine for every even $n$ how Rik can number the hoops so that he has one walk of length $n$. (b) Determine for every odd $n$ how Rik can number the hoops so that he has one walk of length $n - 1$. (c) Show that for an odd $n$ there is no such numbering of the hoops that Rik can make a walk of length $n$.