MathDB

9

Part of 2012 Thailand Mathematical Olympiad

Problems(1)

P(x) >= (x + 1)^n for all x > 0, all roots are real

Source: Thailand Mathematical Olympiad 2012 p9

8/17/2020
Let nn be a positive integer and let P(x)=xn+an−1xn−1+...+a1x+1P(x) = x^n + a_{n-1}x^{n-1} +... + a_1x + 1 be a polynomial with positive real coefficients. Under the assumption that the roots of PP are all real, show that P(x)≥(x+1)nP(x) \ge (x + 1)^n for all x>0x > 0.
algebrapolynomialReal Rootsinequalities