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Bang's Theorem

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

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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?

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I keep three circular medallions in a rectangular box in which they just fit with each one touching the other two. The smallest one has radius 4 cm and touches one side of the box, the middle sized one has radius 9 cm and touches two sides of the box and the largest one touches three sides of the box. What is the radius of the largest one?

Square LCM

Stage: 4 Short Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3
Since HCF$(m,n)=12$ we can write $m=12a$, $n=12b$ where $a$ and $b$ are coprime. The lowest common multiple of $m$ and $n$ must be $12ab$.

Since $12ab=4\times 3ab$, $3ab$ must be a square number, and since $a$, $b$ are coprime exactly one of them is a multiple of $3$. Without loss of generality suppose that $a$ is divisible by $3$ and write $a=3c$.

$3ab=9cb$ is a square number, so $cb$ is a square number, which implies that both $b$ and $c$ are square numbers, say $c=d^2$ and $b=e^2$. Therefore $m=12a=36d^2$ and $n=12e^2$.

Then $\frac{m}{4}=9d^2$ and $\frac{n}{3}=4e^2$ are square numbers, whereas $\frac{m}{3}=12d^2$, $\frac{n}{4}=3e^2$ and $mn=36\times 12d^2e^2$ are not.

This problem is taken from the UKMT Mathematical Challenges.
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