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Is There a Theorem?

Draw a square. A second square of the same size slides around the first always maintaining contact and keeping the same orientation. How far does the dot travel?

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Center Path

Four rods of equal length are hinged at their endpoints to form a rhombus. The diagonals meet at X. One edge is fixed, the opposite edge is allowed to move in the plane. Describe the locus of the point X and prove your assertion.

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The Old Goats

A rectangular field has two posts with a ring on top of each post. There are two quarrelsome goats and plenty of ropes which you can tie to their collars. How can you secure them so they can't fight each other but can reach every corner of the field?

Defined Distances

Stage: 3 Short Challenge Level: Challenge Level:1

We plot $J$ and $K$ $13$ apart. There are a number of options for where to add $L$ and $M$, but working systematically we find that only one option is consistent with $MJ$ being $12$. The final layout looks like

The distance between the points furthest apart is $11+2+12=25$.

This problem is taken from the UKMT Mathematical Challenges.
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