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Prove that for any positive numbers $x_1$, $x_2$,..., $x_n$
$${(x_1 + x_2 + ... + x_n)\left(\frac{1}{ x_1} + {1\over x_2} + ... + {1\over x_n}\right) \geq n^2 }$$
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?
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.