MathDB

5

Part of 1994 IMC

Problems(2)

IMC 1994 D1 P5

Source:

3/6/2017
a) Let f∈C[0,b]f\in C[0,b], g∈C(R)g\in C(\mathbb R) and let gg be periodic with period bb. Prove that ∫0bf(x)g(nx) dx\int_0^b f(x) g(nx)\,\mathrm dx has a limit as n→∞n\to\infty and lim⁡n→∞∫0bf(x)g(nx) dx=1b∫0bf(x) dx⋅∫0bg(x) dx\lim_{n\to\infty}\int_0^b f(x)g(nx)\,\mathrm dx=\frac 1b \int_0^b f(x)\,\mathrm dx\cdot\int_0^b g(x)\,\mathrm dx
b) Find lim⁡n→∞∫0πsin⁡x1+3cos⁡2nx dx\lim_{n\to\infty}\int_0^\pi \frac{\sin x}{1+3\cos^2nx}\,\mathrm dx
IMCcalculusreal analysis
IMC problem 5

Source: IMC 1994

9/24/2014
problem 5. Let x1,x2,…,xkx_1, x_2,\ldots, x_k be vectors of mm-dimensional Euclidean space, such that x1+x2+…+xk=0x_1+x_2+\ldots + x_k=0. Show that there exists a permutation π\pi of the integers {1,2,…,k}\{ 1, 2, \ldots, k \} such that: ∥∑i=1nxπ(i)∥≤(∑i=1k∥xi∥2)1/2\left\lVert \sum_{i=1}^n x_{\pi (i)}\right\rVert \leq \left( \sum_{i=1}^k \lVert x_i \rVert ^2\right)^{1/2}for each n=1,2,…,kn=1, 2, \ldots, k. Note that ∥⋅∥\lVert \cdot \rVert denotes the Euclidean norm. (18 points).
vectorinductionIMCcollege contests