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[Staff Post] A really wonderful user submitted problem

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 Sums

For 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 Solution

Note 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!

Note by Peter Taylor
3 years, 9 months ago

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Well, 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. Christopher Johnboy · 3 years, 9 months ago

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@Christopher Johnboy 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. Sebastian Garrido · 3 years, 9 months ago

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A note, this was from the February NIMO contest, featuring only student-written problems. Check it out: internetolympiad.com Michael Tang · 3 years, 9 months ago

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@Michael Tang 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? Sotiri Komissopoulos · 3 years, 9 months ago

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@Sotiri Komissopoulos Sandeep is an indian name and we call our maths olympiad as INMO.So he is clean... Arshdeep Duggal · 3 years, 9 months ago

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@Sotiri Komissopoulos I know ssilwa from AoPS Fiyi Adebekun · 3 years, 9 months ago

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@Sotiri Komissopoulos Thanks for that link, can you provide a link with the answers too? Pranav Arora · 3 years, 9 months ago

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@Pranav Arora http://www.artofproblemsolving.com/Forum/viewtopic.php?p=2944235#p2944235 Nur Muhammad Shafiullah · 3 years, 9 months ago

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@Nur Muhammad Shafiullah I meant the solution to the other problems given in the link posted by Sotiri K. Pranav Arora · 3 years, 9 months ago

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@Pranav Arora You can click on the problem number of any problem on the link given by him to see the full solution. Nur Muhammad Shafiullah · 3 years, 9 months ago

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@Nur Muhammad Shafiullah Thanks! Pranav Arora · 3 years, 9 months ago

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@Nur Muhammad Shafiullah thanks Oikj P · 3 years, 9 months ago

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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. :) Shivang Jindal · 3 years, 9 months ago

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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 Joshua Richard Theodoroes · 3 years, 9 months ago

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this question is from one of maths olympiad exam Anubhav Singh · 3 years, 9 months ago

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I think there should be at least one comment about limit process, convergence... Victor Chaves · 3 years, 9 months ago

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I don't see how we go from line 3 to line 4 in the solution, can someone explain? Ovi N. · 3 years, 9 months ago

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@Ovi N. 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 Liam Ashby · 3 years, 9 months ago

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@Liam Ashby 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 Michael Tong · 3 years, 9 months ago

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It's cool. Amazing Pebrudal Zanu · 3 years, 9 months ago

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a nice one Namrata Sak · 3 years, 9 months ago

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yeah very cool Christian Baldo · 3 years, 9 months ago

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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 " to "shared by ". Abhishek De · 3 years, 9 months ago

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This solution I think the best Harry Setiawan · 3 years, 9 months ago

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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? Mohammad Davoodi · 3 years, 9 months ago

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I think everyone who was not mind blown is jealous of the ingenuity of Sandeep Fiyi Adebekun · 3 years, 9 months ago

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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. Michael Tong · 3 years, 9 months ago

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I felt it quite easy Shourya Pandey · 3 years, 9 months ago

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@Shourya Pandey but ., we go it wrong! Vamsi Krishna Appili · 3 years, 9 months ago

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@Shourya Pandey yep Superman Son · 3 years, 9 months ago

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Really awesome ! Ritvik Choudhary · 3 years, 9 months ago

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Great technique bro! Piyal De · 3 years, 9 months ago

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but very easy prob ...i solved in 5 mins R Kumar · 3 years, 9 months ago

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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. Arshdeep Duggal · 3 years, 9 months ago

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@Arshdeep Duggal Telescopic sums to solve series really are a standard approach,the problem is to see it as one of them. Rafael Saboya · 3 years, 9 months ago

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@Arshdeep Duggal Yes, I myself did it the same way so I cannot see what is "mind blowing" here. Pranav Arora · 3 years, 9 months ago

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@Pranav Arora http://www.youtube.com/watch?v=eqjl-qRy71w.... this is mind blowing(not related to this problem).... Krishna Jha · 3 years, 9 months ago

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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.:) Mharfe Micaroz · 3 years, 9 months ago

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