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Sum Equals Product

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: Challenge Level:3 Challenge Level:3 Challenge Level:3

$x^2+x$ 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 $x^2+x> x^2+x^3 = x(x+x^2)$.

This problem is taken from the UKMT Mathematical Challenges.

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