Two brothers were left some money, amounting to an exact number of
pounds, to divide between them. DEE undertook the division. "But
your heap is larger than mine!" cried DUM...

The sum of the numbers 4 and 1 [1/3] is the same as the product of 4 and 1 [1/3]; that is to say 4 + 1 [1/3] = 4 × 1 [1/3]. What other numbers have the sum equal to the product and can this be so for any whole numbers?

Weekly Problem 36 - 2006

Stage: 3 Challenge Level:

$x^2+x$ is the largest.

Given that $x$ is in $(0,1)$, we may deduce that $x^2+x> x^2>
x^3> x^4$ and also that $x^2+x> x^2+x^3 = x(x+x^2)$.