### 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.

A 1 metre cube has one face on the ground and one face against a wall. A 4 metre ladder leans against the wall and just touches the cube. How high is the top of the ladder above the ground?

### Days and Dates

Investigate how you can work out what day of the week your birthday will be on next year, and the year after...

# HCF Expression

##### Stage: 4 Short Challenge Level:

For all positive integer values of $p$ and $q$, $2p^2 q$ and $3pq^2$ have a common factor of $pq$.
They will also have an additional common factor of $2$ if $q =2$ and an additional common factor of $3$ if $p=3$.
As the values of $p$ and $q$ are to be chosen from $2, 3$ and $5$, the largest possible value of the highest common factor will occur when $p=3$ and $q=5$.
For these values of $p$ and $q$, $2p^2 q$ and $3pq^2$ have values $90$ and $225$ respectively, giving a highest common factor of $45$.

This problem is taken from the UKMT Mathematical Challenges.
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