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        riddles >> putnam exam (pure math) >> Chords of length 1/n
        
(Message started by: Aryabhatta on Nov 22^nd, 2004, 12:24pm)

Title: Chords of length 1/n
Post by Aryabhatta on Nov 22^nd, 2004, 12:24pm

f:[0,1] -> [bbr] is a continuous function such that f(0) = f(1) -- ( 1 )

([bbr] = set of Reals)

Show that for all natural numbers n, the graph of f has a 'horizontal' chord of length 1/n. ( horizontal = parallel to the x-axis )

This leads to the open ended question:

Given any real [alpha] [in] [0,1] - {1,1/2,1/3,....} is there some f ( satisfying ( 1 ) ) for which there is no horizontal chord of length [alpha]?

Title: Re: Chords of length 1/n
Post by Obob on Nov 22^nd, 2004, 1:03pm

Define g:[0,(n-1)/n] [smiley=to.gif] [smiley=bbr.gif] by g(x)=f(x+1/n)-f(x), and suppose that g has no root. Then either g(x)>0 for every x or g(x)<0 for every x, as g is a continuous function. Then g(0)>0, so f(1/n)>f(0). Since g(1/n)>0, we have f(2/n)>f(1/n). Continuing in this manner, we get f(0)<f(1/n)<f(2/n)<[smiley=cdots.gif]<f(1)=f(0), a contradiction. Therefore g has a root, and f has a horizontal chord of length 1/n.

For every other number [smiley=alpha.gif] there is a continuous function f with f(0)=f(1) for which there is no horizontal chord of length [smiley=alpha.gif]. Let n be a positive integer for which 1/(n+1)<[smiley=alpha.gif]<1/n. Then n[smiley=alpha.gif]<1, so 0<1-n[smiley=alpha.gif]<[smiley=alpha.gif]. Now define f(k[smiley=alpha.gif])=k and f(1-k[smiley=alpha.gif])=-k (including when k=0 in both cases), and extend f to all of [0,1] by making it piecewise linear (we can do this since all the numbers 1-k[smiley=alpha.gif] and j[smiley=alpha.gif] are all distinct). Now f(0)=f(1)=0, f is continuous, and it is easily verified that f(x+[smiley=alpha.gif])-f(x)=1[smiley=ne.gif]0 for every x. Hence f has no chord of length [smiley=alpha.gif].

Title: Re: Chords of length 1/n
Post by Aryabhatta on Nov 22^nd, 2004, 3:55pm

Nice job Obob.
Your counterexample for [alpha] is really nice.

Doesn't it seem like a surprising result though?

This property is true only for 1,1/2,1/3,... !
not even for a single other rational like 2/5!

Seems like a really strange result to me, but I guess I will have to get used to it ;D