### Building Tetrahedra

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

### 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?

### Areas and Ratios

What is the area of the quadrilateral APOQ? Working on the building blocks will give you some insights that may help you to work it out.

# Sum = Product = Quotient

##### Stage: 4 Short Challenge Level:

$b$ 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.