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Age 16 to 18

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Well done to Dylan from Brooke Weston and Joshua from Bohunt Sixth Form in the UK who both sent full solutions.

Joshua sketched the graph of $y=\sin{(\cos x)}$ by considering the minimum and maximum values of $\cos x,$ where the function should be increasing and decreasing, and where the function should cross the co-ordinate axes:

Dylan sketched the graph of $y=\sin{(\cos x)}$ by considering the periodicity and symmetry of $\cos x$ as well as the minima, maxima and roots of the function:

Dylan and Joshua both used the same method they'd used before for the graph of $y=\cos{(\sin x)}.$ This is Joshua's work:

This is Dylan's work:

Dylan and Joshua both completed the rest of the problem - building up to showing that $\cos{(\sin x)}\gt\sin{(\cos x)}$ (or, in Dylan's words, $g(x)\gt f(x)$) for all $x.$ This is Dylan's work:

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.