Hello All,

Two weeks ago we featured a problem submitted by Sandeep S. In our opinion it was such a superlative problem, that everyone of all levels should get to see it.

Here is the problem:

Sandeep's Harmonic SumsFor each positive integer \(n\), let \[H_n = \frac{1}{1} + \frac{1}{2} + \cdots + \frac{1}{n}.\] If \[ \sum_{n=4}^{\infty} \frac{1}{nH_nH_{n-1}} = \frac{a}{b} \] for relatively prime positive integers \(a\) and \(b\), find \(a+b\).

This problem is posed by Sandeep S.

The SolutionNote that \[\begin{align} \sum_{n=4}^{\infty}\frac{1}{nH_nH_{n-1}} &=\sum_{n=4}^{\infty}\frac{1/n}{H_nH_{n-1}} \\ &= \sum_{n=4}^{\infty}\frac{H_n - H_{n-1} }{H_nH_{n-1}} \\ &= \sum_{n=4}^{\infty} \left( \frac{1}{H_{n-1}}-\frac{1}{H_n} \right)\\ &= \frac{1}{H_3} \\ &= \frac{6}{11}. \\ \end{align} \]

Therefore the answer is \(6+11=17\).

The elegance of the solution blew us all away. Thanks Sandeep for such a cool problem!

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## Comments

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TopNewestWell, I am not going to derogate his method but I don't think I am blown away by this, coz I have done it in the exactly same manner. This was quite an easy problem for 180 points.

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The mind blowing aspect is not the extreme creativity of the solution(Like other commenters pointed out, it is not so creative). Is the fact that intimidating problems everywhere in life can have simple solutions.

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A note, this was from the February NIMO contest, featuring only student-written problems. Check it out: internetolympiad.com

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Yeah, the problem was part of the February 2013 NIMO (http://www.artofproblemsolving.com/Forum/resources.php?c=182&cid=194&year=2013). It's a cool problem. The original was posed by a user named ssilwa. Is that Sandeep?

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Sandeep is an indian name and we call our maths olympiad as INMO.So he is clean...

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Thanks for that link, can you provide a link with the answers too?

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Really , it was quite a simple problem and required nothing so creative . This telescopic series are standard way to solve these problems , i think there are many good problems by Zi song or others. :)

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i like this problem so much, it feels like blowing my mind away. i do beg for more creative problem such as this. thank you brilliant.org

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this question is from one of maths olympiad exam

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I think there should be at least one comment about limit process, convergence...

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I don't see how we go from line 3 to line 4 in the solution, can someone explain?

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I struggled with that too but if you start to write out the series in full it is (1/H3 - 1/H4) + (1/H4 - 1/H5) + (1/H5 - 1/H6) ....etc so you can see that after !/H3 all the subsequent terms cancel out....leaving 1/H3

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To elabroate, it leaves out all except 1/H3 and 1/H(infinity), but since the Harmonic series diverges then 1/H(infinity) is zero, so it's 1/H3 - 0 = 1/H3 = 6/11

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It's cool. Amazing

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a nice one

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yeah very cool

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Firstly, this is a standard partial fraction trick. Secondly I sincerely question "is there really a reason to be blown away by an algebraic manipulation even though it turns out to be genuine?" (I don't think so) And again to students submitting problems, as I've also earlier mentioned, please please mention the source if you are 'picking up' problems. Brilliant is now a large community and some one or the other will report it. Recently I reported such a case, and Calvin had to change the wording from "posed by

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This solution I think the best

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there is 9 dots in 3*3 form how many ways can we draw lines that passes at least 4 dots?

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It's one of those problems that if you know to rearrange parts and do a method like the one in the solution then it's easy to get, if not then it is basically impossible to do and when you look at the solution the only thing you learn is "well, I'll do something like that next time I encounter a problem like this." Not too amazing, really. Amazing problems are the ones which you can do in multiple ways depending on how you look at it, ones where you find the solution from small pieces of information you gather over examining pieces of the problem.

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I felt it quite easy

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but ., we go it wrong!

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yep

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Really awesome !

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Great technique bro!

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but very easy prob ...i solved in 5 mins

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Uh,to be honest, i tried it the same way.This is a stanadard approach taught in sequence and series problems.So i am not blown away.

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Telescopic sums to solve series really are a standard approach,the problem is to see it as one of them.

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Yes, I myself did it the same way so I cannot see what is "mind blowing" here.

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http://www.youtube.com/watch?v=eqjl-qRy71w.... this is mind blowing(not related to this problem)....

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Duh,I get the same answer using python 2.7 interpreter, i just combine summation,oo and symbols class in sympy module.:)

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