Inequalities

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

Show without recourse to any calculating aid that 7^{1/2} + 7^{1/3} + 7^{1/4} < 7 and 4^{1/2} + 4^{1/3} + 4^{1/4} > 4 . Sketch the graph of f(x) = x^{1/2} + x^{1/3} + x^{1/4} -x

Which of these continued fractions is bigger and why?

Three fences of different lengths form three sides of an enclosure. What arrangement maximises the area?

Generalise this inequality involving integrals.

10 must remain within easy reach...

Powers of numbers might look large, but which of these is the largest...

Find the maximum value of 1/p + 1/q + 1/r where this sum is less than 1 and p, q, and r are positive integers.

An inequality involving integrals of squares of functions.