MathDB

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Part of 2003 Vietnam National Olympiad

Problems(2)

$g(x) = f(x) f(1-x)$

Source: 41-st Vietnamese Mathematical Olympiad 2003

3/10/2007
Let f:R→Rf: \mathbb{R}\to\mathbb{R} is a function such that f(cot⁡x)=cos⁡2x+sin⁡2xf( \cot x ) = \cos 2x+\sin 2x for all 0<x<π0 < x < \pi. Define g(x)=f(x)f(1−x)g(x) = f(x) f(1-x) for −1≤x≤1-1 \leq x \leq 1. Find the maximum and minimum values of gg on the closed interval [−1,1].[-1, 1].
functiontrigonometrycalculusalgebra proposedalgebra
integer system (has root? )

Source: 41-st Vietnamese Mathematical Olympiad 2003

3/10/2007
Find the largest positive integer nn such that the following equations have integer solutions in x,y1,y2,...,ynx, y_{1}, y_{2}, ... , y_{n} : (x+1)2+y12=(x+2)2+y22=...=(x+n)2+yn2.(x+1)^{2}+y_{1}^{2}= (x+2)^{2}+y_{2}^{2}= ... = (x+n)^{2}+y_{n}^{2}.
number theory proposednumber theory