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.

At a Glance

The area of a regular pentagon looks about twice as a big as the pentangle star drawn within it. Is it?

Contact

A circular plate rolls in contact with the sides of a rectangular tray. How much of its circumference comes into contact with the sides of the tray when it rolls around one circuit?

Punky Fish

Age 14 to 16 ShortChallenge Level

Starting with females
Every fish has at least 8 stripes
10 fish with 8 stripes have 80 stripes
There are 6 more stripes
As a male fish has one extra stripe, these 6 stripes must belong to 6 male fish.

So there are 6 males and 4 females.

Therefore the ratio of male to female fish is 3:2.

Starting with males
Now $86=9\times9 + 5$
$=9\times 8+9+5 = 9\times8+14$
$=9\times 7+18+5 = 9\times7+23$
$=9\times 6+27+5 = 9\times6+32=9\times6+8\times 4$

Therefore the ratio of male to female fish is $3:2$.

This problem is taken from the UKMT Mathematical Challenges.
You can find more short problems, arranged by curriculum topic, in our short problems collection.