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Conference rusure::math

Title:	Mathematics at DEC

Moderator:	RUSURE::EDP

Created:	Mon Feb 03 1986
Last Modified:	Fri Jun 06 1997
Last Successful Update:	Fri Jun 06 1997
Number of topics:	2083
Total number of notes:	14613

1964.0. "Crux Mathematicorum 2020" by RUSURE::EDP (Always mount a scratch monkey.) Tue Apr 11 1995 13:57

    Proposed by Christopher J. Bradley, Clifton College, Bristol, U.K.

    Let a, b, c, d be DISTINCT real numbers such that

    	a/b + b/c + c/d + d/a = 4 and ac = bd.

    Find the maximum value of a/c + b/d + c/a + d/b.

T.R Title User Personal
Name Date Lines

1964.1 -12 HERON::BUCHANAN Et tout sera bien et Mon Apr 17 1995 11:50 17

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1964.1	-12	HERON::BUCHANAN	Et tout sera bien et	`Mon Apr 17 1995 11:50`	17
	> Let a, b, c, d be DISTINCT real numbers such that > > a/b + b/c + c/d + d/a = 4 and ac = bd. > > Find the maximum value of a/c + b/d + c/a + d/b. wlog, a = 1. get rid of c, by setting it = bd Then we want to maximize f(b)f(d) subject to f(b)+f(d)=4 where f(x) = x + 1/x. Set f(b) = y. So we are looking to maximize y(4-y). Examining f shows that y cannot lie in (-2,2) or (2,6). If y = 2, then a,b,c,d are not distinct. So y cannot lie in (-2,6). The maximum is achieved when y = -2 or 6.

>    Let a, b, c, d be DISTINCT real numbers such that
>
>    	a/b + b/c + c/d + d/a = 4 and ac = bd.
>
>    Find the maximum value of a/c + b/d + c/a + d/b.

	wlog, a = 1.
	get rid of c, by setting it = bd

	Then we want to maximize f(b)*f(d) subject to f(b)+f(d)=4
where f(x) = x + 1/x.

	Set f(b) = y. So we are looking to maximize y*(4-y). Examining f
shows that y cannot lie in (-2,2) or (2,6). If y = 2, then a,b,c,d are not
distinct. So y cannot lie in (-2,6).

	The maximum is achieved when y = -2 or 6.