MathDB

Problem 3

Part of 2010 VJIMC

Problems(2)

integral is bounded

Source: VJIMC 2010 1.3

6/3/2021
Prove that there exist positive constants c1c_1 and c2c_2 with the following properties: a) For all real k>1k>1, ∣∫011−x2cos⁡(kx)dx∣<c1k3/2.\left|\int^1_0\sqrt{1-x^2}\cos(kx)\text dx\right|<\frac{c_1}{k^{3/2}}.b) For all real k>1k>1, ∣∫011−x2sin⁡(kx)dx∣<c2k.\left|\int^1_0\sqrt{1-x^2}\sin(kx)\text dx\right|<\frac{c_2}k.
calculusintegration
invertible: A,A+B,A+2B,...,A+(2n)B implies A+(2n+1)B is invertible

Source: VJIMC 2010 2.3

6/4/2021
Let AA and BB be two n×nn\times n matrices with integer entries such that all of the matrices A,A+B,A+2B,A+3B,…,A+(2n)BA,\enspace A+B,\enspace A+2B,\enspace A+3B,\enspace\ldots,\enspace A+(2n)Bare invertible and their inverses have integer entries, too. Show that A+(2n+1)BA+(2n+1)B is also invertible and that its inverse has integer entries.
matrixlinear algebra