JMC 2016 Challenge cont.

20.** **A cyclic quadrilateral has all four vertices on the circumference of a circle.

Brahmagupta (598–670AD) gave the following formula for the area, *A*, of a cyclic quadrilateral whose edges have lengths *a*, *b*, *c*, *d*:

where *s *is half of the perimeter of the quadrilateral.

What is the area of the cyclic quadrilateral with sides of length 4cm, 5cm, 7cm and 10cm?

21**. **The diagram shows a pentagon drawn on a square grid. All vertices of the pentagon and triangle are grid points.

What fraction of the area of the pentagon is shaded?

23**. **The diagram shows the first few squares of a ‘spiral’ sequence of squares. All but the first three squares have been numbered.

After the first six squares, the sequence is continued by placing the next square alongside three existing squares – the largest existing square and two others.

The three smallest squares have sides of length 1.

What is the side length of the 12th square?

25**. **Beatrix places dominoes on a 5 × 5 board, either horizontally or vertically, so that each domino covers two small squares.

She stops when she cannot place another domino, as in the example shown in the diagram.

When Beatrix stops, what is the largest possible number of squares that may be still uncovered?

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