Maths problem

Thursday 9th July 2020

\(a\), \(b\), \(c\), \(d\), \(e\) and \(f\) are positive real numbers with the property that \(a+b+c+d+e+f=20\).

What is the smallest possible value of \(\sqrt{a^2+1^2}+\sqrt{b^2+2^2}+\sqrt{c^2+3^2}+\sqrt{d^2+4^2}+\sqrt{e^2+5^2}+\sqrt{f^2+6^2}\;\)?

Does the diagram below help you?

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Geometrically, the sum \(\sqrt{a^2+1^2}+\sqrt{b^2+2^2}+\sqrt{c^2+3^2}+\sqrt{d^2+4^2}+\sqrt{e^2+5^2}+\sqrt{f^2+6^2}\;\) is the total length of the six line segments from (0,0) to (20,21). This total length is minimised when the line segments line up (see diagram below), so that the total length is \(\sqrt{20^2+21^2}=29\).

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By Chris Luke

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