MathDB

3

Part of 2014 IMC

Problems(2)

IMC 2014, Problem 3

Source: IMC 2014

7/27/2016
Let nn be a positive integer. Show that there are positive real numbers a0,a1,…,ana_0, a_1, \dots, a_n such that for each choice of signs the polynomial ±anxn±an−1xn−1±⋯±a1x±a0\pm a_nx^n\pm a_{n-1}x^{n-1} \pm \dots \pm a_1x \pm a_0 has nn distinct real roots.
(Proposed by Stephan Neupert, TUM, München)
IMCalgebrapolynomialcollege contestsreal analysis
IMC 2014, Problem 8

Source: IMC 2014

7/27/2016
Let f(x)=sin⁡xxf(x)=\frac{\sin x}{x}, for x>0x>0, and let nn be a positive integer. Prove that ∣f(n)(x)∣<1n+1|f^{(n)}(x)|<\frac{1}{n+1}, where f(n)f^{(n)} denotes the nthn^{\mathrm{th}} derivative of ff.
(Proposed by Alexander Bolbot, State University, Novosibirsk)
IMCcollege contestscalculusderivativefunction