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Shape and Territory

If for any triangle ABC tan(A - B) + tan(B - C) + tan(C - A) = 0 what can you say about the triangle?

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Napoleon's Hat

Three equilateral triangles ABC, AYX and XZB are drawn with the point X a moveable point on AB. The points P, Q and R are the centres of the three triangles. What can you say about triangle PQR?

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The Root Cause

Prove that if a is a natural number and the square root of a is rational, then it is a square number (an integer n^2 for some integer n.)

Mind Your Ps and Qs

Stage: 5 Short Challenge Level: Challenge Level:2 Challenge Level:2

To do this question you need to be really sure what the two symbols $\Rightarrow$ and $\Leftrightarrow$ mean.

$p\Rightarrow q$ essentially means that IF $p$ is True THEN $q$ is true.

$p\Leftrightarrow q$ means that $p$ is true if and only if $q$ is true. Alternatively, they are either both true or both false.

To get started, pick one of the simpler looking statements for $p$.

Go through the other statements in turn and see which of them work as $q$ in $p\Rightarrow q$. You will soon be able to pair-off possible partners.

Finally, don't forget to read the details of the question carefully!