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Golden Thoughts

Rectangle PQRS has X and Y on the edges. Triangles PQY, YRX and XSP have equal areas. Prove X and Y divide the sides of PQRS in the golden ratio.

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Building Tetrahedra

Can you make a tetrahedron whose faces all have the same perimeter?

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Ladder and Cube

A 1 metre cube has one face on the ground and one face against a wall. A 4 metre ladder leans against the wall and just touches the cube. How high is the top of the ladder above the ground?

Sum = Product = Quotient

Stage: 4 Short Challenge Level: Challenge Level:1

$b$ cannot be zero because we can't divide by zero. $ab=\frac{a}{b}\Rightarrow b^2=1$ so $b=\pm 1$.

If $b=1$ then the equation $a+b=ab$ becomes $a+1=a$ which is impossible. Hence we must have $b=-1$. Then $a+b=ab$ becomes $a-1=-a\Rightarrow a=\frac{1}{2}$.

Finally we check that $\frac{a}{b}=a+b$ holds for $(a,b)=(\frac{1}{2},-1)$ which it does, so there is exactly one pair which satisfies the conditions.

This problem is taken from the UKMT Mathematical Challenges.
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