MathDB

Problem 4

Part of 2006 VJIMC

Problems(2)

sum_i sum_j a_ij=n where A is nxn

Source: VJIMC 2006 1.4

6/28/2021
Let A=[aij]n×nA=[a_{ij}]_{n\times n} be a matrix with nonnegative entries such that ∑i=1n∑j=1naij=n.\sum_{i=1}^n\sum_{j=1}^na_{ij}=n. (a) Prove that ∣det⁡A∣≤1|\det A|\le1. (b) If ∣det⁡A∣=1|\det A|=1 and λ∈C\lambda\in\mathbb C is an arbitrary eigenvalue of AA, show that ∣λ∣=1|\lambda|=1.
matrixlinear algebra
f(x)/x=∞ for x->∞, integral is convergent of sin(f)

Source: VJIMC 2006 2.4

6/29/2021
Let f:[0,∞)→Rf:[0,\infty)\to\mathbb R ba a strictly convex continuous function such that lim⁡x→+∞f(x)x=+∞.\lim_{x\to+\infty}\frac{f(x)}x=+\infty.Prove that the improper integral ∫0+∞sin⁡(f(x))dx\int^{+\infty}_0\sin(f(x))\text dx is convergent but not absolutely convergent.
calculusintegration