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Be Reasonable

Prove that sqrt2, sqrt3 and sqrt5 cannot be terms of ANY arithmetic progression.

Good Approximations

Solve quadratic equations and use continued fractions to find rational approximations to irrational numbers.

Rational Roots

Given that a, b and c are natural numbers show that if sqrt a+sqrt b is rational then it is a natural number. Extend this to 3 variables.

Spirostars

Age 16 to 18 Challenge Level:

First prove that the line segments are chords of a circle. Then prove that the angle of turn between one chord and the next is equal to the angle subtended by the chord at the centre of the circle.

You do not need to have met the Logo programming language previously in order to do this question. You can download the free MSWlogo program from http://www.softronix.com/logo.html

If you have not used Logo before then, from the first part of this simple introduction to Logo for beginners , you can learn enough in five minutes to use Logo to experiment in drawing the spirostars.

Try repeat 1000 [fd 200 rt 360*sqrt 2]

What happens?