### Golden Thoughts

Rectangle PQRS has X and Y on the edges. Triangles PQY, YRX and XSP have equal areas. Prove X and Y divide the sides of PQRS in the golden ratio.

### From All Corners

Straight lines are drawn from each corner of a square to the mid points of the opposite sides. Express the area of the octagon that is formed at the centre as a fraction of the area of the square.

### Star Gazing

Find the ratio of the outer shaded area to the inner area for a six pointed star and an eight pointed star.

# Punky Fish

##### Stage: 3 and 4 Short Challenge Level:

If there are f females and m males then $86=8f+9m=8(f+m)+m$. Now $86=8\times 10+6=8 \times 9+14...$ Only the first of these is feasible so m=6 and f=4. Therefore the ratio of male to female fish is $3:2$.

Alternatively, dividing the stripes into groups of $8$, there are $6$ left over. One left over is produced by each male, so there are $6$ or $14$ or... males. However, as $14 \times 9 = 126 > 86$, there must be six males. Then, this leaves $32$ stripes remaining, so there are $4$ females. Therefore the ratio of male to female fish is $3:2$.

This problem is taken from the UKMT Mathematical Challenges.
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