# [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
5 years 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.

- 5 years ago

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.

- 4 years, 12 months ago

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

- 5 years ago

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?

- 5 years ago

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

- 5 years ago

- 5 years ago

http://www.artofproblemsolving.com/Forum/viewtopic.php?p=2944235#p2944235

I meant the solution to the other problems given in the link posted by Sotiri K.

- 5 years ago

You can click on the problem number of any problem on the link given by him to see the full solution.

Thanks!

- 5 years ago

thanks

- 5 years ago

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. :)

- 5 years ago

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

this question is from one of maths olympiad exam

- 5 years ago

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

- 4 years, 12 months ago

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

- 5 years ago

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

- 5 years ago

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

- 5 years ago

It's cool. Amazing

- 5 years ago

a nice one

- 5 years ago

yeah very cool

- 5 years ago

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 ".

- 5 years ago

This solution I think the best

- 5 years ago

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

- 5 years ago

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.

- 5 years ago

I felt it quite easy

- 5 years ago

but ., we go it wrong!

- 5 years ago

yep

- 5 years ago

Really awesome !

- 5 years ago

Great technique bro!

- 5 years ago

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

- 5 years ago

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.

- 5 years ago

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

- 4 years, 12 months ago

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

- 5 years ago

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

- 5 years ago

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

- 5 years ago