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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:

By considering the graph of $y=\sin x$ prove that, for $0\leq x \leq \pi/2$, $${2x\over \pi} \leq \sin x \leq x.$$ By considering the graph of $y=\tan x$ prove that, for $0 < a < b < \pi/2$, $${\tan a \over \tan b} < {a\over b}.$$

Can you find similar inequalities which hold for different ranges of $x$?