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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?

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Exhaustion

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

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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.

Climbing

Age 16 to 18 Challenge Level:
The graph of $y=\sin x$ lies above the line $y=2x/\pi$ joining the origin to the point $(\pi/2, 1)$. The graph of $\sin x$ lies below the line $y=x$. Why? What does this tell you about $\sin x$?