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Dividing the Field

A farmer has a field which is the shape of a trapezium as illustrated below. To increase his profits he wishes to grow two different crops. To do this he would like to divide the field into two trapeziums each of equal area. How could he do this?

Two Circles

Draw two circles, each of radius 1 unit, so that each circle goes through the centre of the other one. What is the area of the overlap?

Star Gazing

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

Sinking Feeling

Age 14 to 16 Short Challenge Level:

Two vases are cylindrical in shape.
The larger vase has diameter 20 cm. 
The smaller vase has diameter 10 cm and height 16 cm.

The larger vase is partially filled with water.
Then the empty smaller vase, with the open end at the top, is slowly pushed down into the water, which flows over its rim.

When the smaller vase is pushed right down, it is half full of water.

What was the original depth of the water in the larger vase?


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