Code to Zero

Find all 3 digit numbers such that by adding the first digit, the square of the second and the cube of the third you get the original number, for example 1 + 3^2 + 5^3 = 135.

Always Two

Find all the triples of numbers a, b, c such that each one of them plus the product of the other two is always 2.

Parabella

This is a beautiful result involving a parabola and parallels.

Powerful Factors

Age 16 to 18 Challenge Level:

Use the following identities:

$x^2-y^2 \equiv (x-y)(x+y)$

and

$x^3+y^3 \equiv (x+y)(x^2-xy+y^2)$

to find the highest power of $2$ and the highest power of $3$ which divide $5^{36}-1$.