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Find the image of the function M:X->R

Source: GMB-IMAR 2005, Seniors, Problem 2

October 10, 2005
functioninequalities unsolvedinequalities

Problem Statement

Let n≥3n \geq 3 be an integer and let a,b∈Ra,b\in\mathbb{R} such that nb≥a2nb\geq a^2. We consider the set X={(x1,x2,…,xn)∈Rn∣∑k=1nxk=a, ∑k=1nxk2=b}. X = \left\{ (x_1,x_2,\ldots,x_n)\in\mathbb{R}^n \mid \sum_{k=1}^n x_k = a, \ \sum_{k=1}^n x_k^2 = b \right\} . Find the image of the function M:X→RM: X\to \mathbb{R} given by M(x1,x2,…,xn)=max⁡1≤k≤nxk. M(x_1,x_2,\ldots,x_n) = \max_{1\leq k\leq n} x_k . Dan Schwarz
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