MathDB

6

Part of 1999 IMC

Problems(2)

ugly but hard

Source: IMC 1999 day 1 problem 6

11/19/2005
(a) Let p>1p>1 a real number. Find a real constant cpc_p for which the following statement holds: If f:[−1,1]→Rf: [-1,1]\rightarrow\mathbb{R} is a continuously differentiable function with f(1)>f(−1)f(1)>f(-1) and ∣f′(y)∣≤1∀y∈[−1,1]|f'(y)|\le1 \forall y\in[-1,1], then ∃x∈[−1,1]:f′(x)>0\exists x\in[-1,1]: f'(x)>0 so that ∀y∈[−1,1]:∣f(y)−f(x)∣≤cpf′(x)p∣y−x∣\forall y\in[-1,1]: |f(y)-f(x)|\le c_p\sqrt[p]{f'(x)}|y-x|. (b) What if p=1p=1?
functionreal analysisreal analysis unsolved
IMC 1999 Problem 12

Source: IMC 1999 Day 2 Problem 6

10/27/2020
Let AA be a subset of Z/nZ\mathbb{Z}/n\mathbb{Z} with at most ln⁡(n)100\frac{\ln(n)}{100} elements. Define f(r)=∑s∈Ae2πirsnf(r)=\sum_{s\in A} e^{\dfrac{2 \pi i r s}{n}}. Show that for some r≠0r \ne 0 we have ∣f(r)∣≥∣A∣2|f(r)| \geq \frac{|A|}{2}.
fourier seriescomplex numberspigeonhole principleIMC