Tri-split

A point P is selected anywhere inside an equilateral triangle. What can you say about the sum of the perpendicular distances from P to the sides of the triangle? Can you prove your conjecture?

Areas and Ratios

What is the area of the quadrilateral APOQ? Working on the building blocks will give you some insights that may help you to work it out.

Six Discs

Six circular discs are packed in different-shaped boxes so that the discs touch their neighbours and the sides of the box. Can you put the boxes in order according to the areas of their bases?

Weekly Problem 9 - 2006

Stage: 4 Challenge Level:

$\frac{5}{12}$

Let $AB$ and $AD$ be of length $b$ and $h$ respectively. Then the area of $ABCD = bh$ and the area of $$ABQP = \frac{1}{2}b\left(\frac{h}{2} + \frac{h}{3}\right)= \frac{5bh}{12}$$

Thus the required ratio is $\frac{5}{12}$.

This problem is taken from the UKMT Mathematical Challenges.

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