MathDB

1

Part of 2010 IMC

Problems(2)

IMC 2010 Problem 1, Day 2

Source:

7/27/2010
(a)(a) A sequence x1,x2,…x_1,x_2,\dots of real numbers satisfies xn+1=xncos⁡xn for all n≥1.x_{n+1}=x_n \cos x_n \textrm{ for all } n\geq 1. Does it follows that this sequence converges for all initial values x1?x_1? (5 points)
(b)(b) A sequence y1,y2,…y_1,y_2,\dots of real numbers satisfies yn+1=ynsin⁡yn for all n≥1.y_{n+1}=y_n \sin y_n \textrm{ for all } n\geq 1. Does it follows that this sequence converges for all initial values y1?y_1? (5 points)
trigonometryreal analysisreal analysis unsolved
IMC 2010 - Problem 1

Source:

7/26/2010
Let 0<a<b0 < a < b. Prove that ∫ab(x2+1)e−x2dx≥e−a2−e−b2\int_a^b (x^2+1)e^{-x^2} dx \geq e^{-a^2} - e^{-b^2}.
integrationreal analysisreal analysis unsolved