Bang's Theorem

If all the faces of a tetrahedron have the same perimeter then show that they are all congruent.

Rudolff's Problem

A group of 20 people pay a total of £20 to see an exhibition. The admission price is £3 for men, £2 for women and 50p for children. How many men, women and children are there in the group?

Weekly Problem 33 - 2008

Short Challenge Level:

Quince, quance, quonce

(1) $7$ quince weigh the same as $4$ quonce, so a quince weighs less than a quonce.

(2) $5$ quonce weigh the same as $6$ quance, so a quance weighs less than a quonce.

So quonce is the heaviest. But $20$ quonce weigh the same as $35$ quince (from (1) multiplied by $5$), and $20$ quonce weigh the same as $24$ quance (from (2) multiplied by $4$), so $35$ quince weigh the same as $24$ quance, so a quance weighs more than a quince.

This problem is taken from the UKMT Mathematical Challenges.

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