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a polynomial with all real roots

Source: Italy TST 2009 p4

March 10, 2012
algebrapolynomialalgebra unsolved

Problem Statement

Let nn be an even positive integer. An nn-degree monic polynomial P(x)P(x) has nn real roots (not necessarily distinct). Suppose yy is a positive real number such that for any real number t<yt<y, we have P(t)>0P(t)>0. Prove that P(0)1n−P(y)1n≥y.P(0)^{\frac{1}{n}}-P(y)^{\frac{1}{n}}\ge y.
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