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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.

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Dodgy Proofs

What is wrong with these dodgy proofs?

Exhaustion

Stage: 5 Challenge Level: Challenge Level:1

Find the positive integer solutions of the equation
 

\[ \left(1 + \frac{1}{a}\right)\left(1 + \frac{1}{b}\right) \left(1+ \frac{1}{c}\right) = 2. \]
 
 

This problem was taken from the Hungarian magazine KoMaL. There are many other challenging problems in English on the KoMal website.
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