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Tennis

A tennis ball is served from directly above the baseline (assume the ball travels in a straight line). What is the minimum height that the ball can be hit at to ensure it lands in the service area?

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Tied Up

In a right angled triangular field, three animals are tethered to posts at the midpoint of each side. Each rope is just long enough to allow the animal to reach two adjacent vertices. Only one animal can reach all points in the field. Which one is it and why?

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Lying and Cheating

Follow the instructions and you can take a rectangle, cut it into 4 pieces, discard two small triangles, put together the remaining two pieces and end up with a rectangle the same size. Try it!

How Big?

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

The following solution came from Alistair Jacklin, class 9S, King Edward Camp Hill School for Boys. Kang Hong Joo of The Chinese High School, Singapore and Neil Wright of Madras College, St Andrew's and Jessica Zhang, Woodbridge School, Essex used the same method.

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The first thing I did was to split the large triangle into 3 sections, a square and 2 smaller triangles. Next I then worked out the area of the large triangle using the formula:-

Height * Width * 0.5 = area of the triangle
3 * 4 * 0.5 = area of a triangle
6 = area of a large triangle

With this I used a formula to show the area of this triangle by adding the areas of the other parts:

x 2 + 0.5 x(3 - x) + 0.5 x(4 - x) = 6.

The formula is then simplified in several easy steps to give:-

x 2 + 1.5 x - 0.5 x 2 + 2 x - 0.5 x 2 = 6
3.5 x = 6
x = 12/7

The area of the square is then x 2 = 144/49 exactly. Rounded to three decimal places this is 2.938.