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...

Weekly Problem 30 - 2010

Stage: 3 and 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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