MathDB

6

Part of 2001 IMC

Problems(2)

IMC 2001 Problem 6

Source: IMC 2001 Day 1 Problem 6

10/30/2020
Suppose that the differentiable functions a,b,f,g:R→Ra, b, f, g:\mathbb{R} \rightarrow \mathbb{R} satisfy f(x)≥0,f′(x)≥0,g(x)≥0,g′(x)≥0 for all x∈R, f(x)\geq 0, f'(x) \geq 0,g(x)\geq 0, g'(x) \geq 0 \text{ for all } x \in \mathbb{R}, lim⁡x→∞a(x)=A≥0,lim⁡x→∞b(x)=B≥0,lim⁡x→∞f(x)=lim⁡x→∞g(x)=∞,\lim_{x\rightarrow \infty} a(x)=A\geq 0,\lim_{x\rightarrow \infty} b(x)=B\geq 0, \lim_{x\rightarrow \infty} f(x)=\lim_{x\rightarrow \infty} g(x)=\infty, and f′(x)g′(x)+a(x)f(x)g(x)=b(x).\frac{f'(x)}{g'(x)}+a(x)\frac{f(x)}{g(x)}=b(x). Prove that lim⁡x→∞f(x)g(x)=BA+1\lim_{x\rightarrow\infty}\frac{f(x)}{g(x)}=\frac{B}{A+1}.
functionreal analysis
IMC 2001 Problem 12

Source: IMC 2001 Day 2 Problem 6

10/30/2020
For each positive integer nn, let fn(ϑ)=sin⁡(ϑ)⋅sin⁡(2ϑ)⋅sin⁡(4ϑ)⋯sin⁡(2nϑ)f_{n}(\vartheta)=\sin(\vartheta)\cdot \sin(2\vartheta) \cdot \sin(4\vartheta)\cdots \sin(2^{n}\vartheta). For each real ϑ\vartheta and all nn, prove that ∣fn(ϑ)∣≤23∣fn(π3)∣|f_{n}(\vartheta)| \leq \frac{2}{\sqrt{3}}|f_{n}(\frac{\pi}{3})|
inequalitiesderivative