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Shades of Fermat's Last Theorem

The familiar Pythagorean 3-4-5 triple gives one solution to (x-1)^n + x^n = (x+1)^n so what about other solutions for x an integer and n= 2, 3, 4 or 5?

Exhaustion

Find the positive integer solutions of the equation (1+1/a)(1+1/b)(1+1/c) = 2

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.

Discrete Trends

Age 16 to 18 Challenge Level:

Show that if $n$ is a positive integer then

$$n^{1/n} < 1 + \sqrt {{2\over {n-1}}}.$$

Show that $n^{1/n}\rightarrow 1$ as $n\rightarrow \infty$.

Find the maximum value of $n^{1/n}$ and prove that it is indeed the maximum.