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        riddles >> putnam exam (pure math) >> A Year with Gauss
        
(Message started by: THUDandBLUNDER on Jan 18^th, 2005, 2:49am)

Title: A Year with Gauss
Post by THUDandBLUNDER on Jan 18^th, 2005, 2:49am

If Gauss proved at least one new theorem every day, but never more than 50 new theorems in any month, prove that there was a sequence of consecutive days in a year when Gauss proved exactly 125 new theorems.

Title: Re: A Year with Gauss
Post by TenaliRaman on Jan 18^th, 2005, 10:32am

::[hide]
Let d_i denote the number of theorems proved by Gauss on the ith day. Let D_i denote the number of theorems proved in the first i days.

First of all,
[sum] d_i <= 600 (50 theorem per month)

Now, gauss proves atleast one theorem per day, therefore
1<=D₁<D₂<...<D₃₆₅<=600

Add 125 to each term of the above,
126<=D₁+125<D₂+125<...<D₃₆₅+125<=725

Now consider the set,
G = {D₁, D₂,..., D₃₆₅, D₁+125, D₂+125,..., D₃₆₅+125}

#G = 730
Each of the terms takes values from 1 to 725

By Pigeonhole Principle,
There are atleast two terms which take the same value and hence proved.
[/hide]::

-- AI