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Symmetrical Minima. (TOMATO Subjective 33)

Problem: Let \(k\) be a fixed odd positive integer. Find the minimum value of \(x^2 + y^2\), where \(x,y\) are non-negative integers and \(x+y=k\).

 

Solution: We have \(y=k-x\). Therefore we get an equation in \(x\) where \(k\) is a constant, precisely \(f(x) = x^2 + (k-x)^2\).

To minimise, we differentiate \(f(x)\) w.r.t \(x\).

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1 comment

  1. This one is also pretty simple through Cauchy Schwartz

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