Russelle’s Perfect Square

Our featured problem today is from one of the Philippines Representatives in 2011 International Math Olympiad , Russelle Guadalupe. He is currently studying at the University of the Phillipines – Diliman, taking up BS Mathematics.  He also won several championships in local math challenges.



Here are some of the of his achievements in his Mathematics career.

2009 MTAP Math Challenge Individual Cat B( 2nd Place Sectoral, 8th Place Regional )

2009 MTAP Math Challenge Team Orals Cat B ( 3rd Place Sectoral, 2nd Place Regional )

2010 MTAP Math Challenge Individual Cat B( 1st Place Sectoral, 7th Place Regional )

2010 MTAP Math Challenge Team Orals Cat B ( 3rd Place Sectoral )

2011 MTAP Math Challenge Individual ( 2nd Place Sectoral, 4th Place Regional )

2011 MTAP Math Challenge Team Orals ( 1st Place Sectoral, 2nd Place Regional )

13th Phil Math Olympiad Area Stage ( 3rd Place )

13th Phil Math Olympiad Nat’l Stage ( Finalist )

2011 International Math Olympiad ( Participant)

2012 MTAP Math Challenge Individual ( 1st Place Sectoral, 1st Place Regional )

2012 MTAP Math Challenge Team Orals ( 1st Place Sectoral )

2012 1st Raffles Invitational Math Olympiad (RIMO) ( Bronze Medal )

2011 – Canadian Mathematics Contest Senior Category ( Medalist )

2012 – Euclid Mathematics Contest ( Top 25% of all contestants )

2011 – 6th Philippine Sudoku Super Challenge Elims ( Finalist )

2011 – Sharp MTG Math Trail and Problem Solving Competition ( 3rd Runner Up )

2013 – International Regions Math League ( Top Scorer )

Now, here is a breathtaking problem from Russelle himself. Hope you will learn from this.


Let f(n)=\sum_{i=1}^n\sum_{j=1}^{n} (i-j)^2. Determine the sum of all positive integers n\leq 1000 for which f(n) is a perfect square.

( 2014 EMC Online Test Problem 14)


First, we need to find a closed-form expression for f(n)=\sum_{i=1}^n\sum_{j=1}^{n} (i-j)^2 by summing each term involving j and treating terms involving i as a constant. Thus, we have

f(n)=\sum_{i=1}^n\sum_{j=1}^{n} (i-j)^2=\sum_{i=1}^n\sum_{j=1}^{n} (i^2-2ij+j^2)=\sum_{i=1}^n[i^2\sum_{j=1}^n 1-2i\sum_{j=1}^n j+\sum_{j=1}^n j^2]

=\sum_{i=1}^n[ni^2-2i\cdot \displaystyle\frac{n(n+1)}{2}+\displaystyle\frac{n(n+1)(2n+1)}{6}

=n\sum_{i=1}^n i^2-n(n+1)\sum_{i=1}^n i+\displaystyle\frac{n(n+1)(2n+1)}{6}\sum_{i=1}^n 1

=n\cdot \displaystyle\frac{n(n+1)(2n+1)}{6}-n(n+1)\cdot \displaystyle\frac{n(n+1)}{2}+\displaystyle\frac{n(n+1)(2n+1)}{6}\cdot n

=n^2(n+1)[ \displaystyle\frac{2n+1}{3}-\displaystyle\frac{n+1}{2}]


Thus, f(n)=\displaystyle\frac{n^2(n^2-1)}{6}. We then find all positive integers N such that f(N)=t^2 for some nonnegative integers

t. This is equivalent to N^2(N^2-1)=6t^2 or (2N^2-1)^2=24t^2+1. Letting x=2N^2-1. We have x^2-24t^2=1, a Pell’s equation. The general solution for this equation is given by

x_k=\displaystyle\frac{(x_1+t_1\sqrt{24})^k+(x_1-t_1\sqrt{24})^k}{2}, t_k=\displaystyle\frac{(x_1+t_1\sqrt{24})^k-(x_1-t_1\sqrt{24})^k}{2\sqrt{24}}


For all integers k\geq 0, where (x_1,t_1) is the initial solution. Clearly, by testing values with t_1>0, we have (x_1,t_1)=(5,1). Thus, we can then make a sequence \{N_k\}_{k\geq 0} of values of N based from the solutions of the given Pell’s equation. Note that






Thus, we have N_k=\displaystyle\frac{(\sqrt{3}+\sqrt{2})^k+(\sqrt{3}-\sqrt{2})^k}{2}. Since

(\sqrt{3}+\sqrt{2})^k=\sum_{p=0}^k {k\choose p}3^{\frac{p}{2}}\cdot 2^{\frac{k-p}{2}}(\sqrt{3}-\sqrt{2})^k=\sum_{p=0}^k {k\choose p}3^{\frac{p}{2}}\cdot 2^{\frac{k-p}{2}}(-1)^p

we have N_k=\sum_{m=0}^{\lfloor \displaystyle\frac{k}{2}\rfloor} {k\choose 2m}3^m\cdot 2^{\frac{k}{2}-m},  which follows that  N_k  is an integer when k  is even. Thus, it suffices to check the values of  N_k  for  k\leq \log_{\sqrt{3}+\sqrt{2}} 200<7 (since N_k\leq 1000) and \sqrt{3}+\sqrt{2}>1.7+1.4=3.1>3  implies  (\sqrt{3}+\sqrt{2})^7>2187>2000.) Thus, k\in \{0,2,6\}, so by brute force calculation we get N_0=1,N_2=5,N_4=49 and N_6=485. Therefore, the sum of all integers N\leq 1000 such that f(N) is a perfect square is 1+5+49+485=\boxed{540}.



Blogger and a Math enthusiast. Has no interest in Mathematics until MMC came. Aside from doing math, he also loves to travel and watch movies.

Latest posts by Dan (see all)

You may also like...

75 Responses

  1. Darrin says:

    Fantastic items from you, man. I have be aware your stuff prior to and you are simply too great. I really like what you have obtained right here, really like what you are stating and the best way in which you assert it. You make it entertaining and you still care for to keep it sensible. I cant wait to learn far more from you. This is actually a great site.

  2. There’s noticeably a bundle to learn about this. I assume you made sure nice points in features also.

  3. Excellent beat ! I wish to apprentice while you amend your website, how can i subscribe for a weblog site? The account aided me a appropriate deal. I have been tiny bit acquainted of this your broadcast offered vibrant transparent concept

  4. this site says:

    bKMQrf Well I sincerely enjoyed reading it. This subject provided by you is very useful for good planning.

  5. when it comes to tv fashion shows, i really love Project Runway because it shows some new talents in the fashion industry

  6. I cannot thank you enough for the article post.Really looking forward to read more. Fantastic.

  7. dramacools says:

    It as very easy to find out any matter on web as compared to textbooks, as I found this piece of writing at this web page.

  8. Very neat blog.Really looking forward to read more. Really Great.

  9. sofa pillows says:

    Wow, great article.Really looking forward to read more.

  10. Nice Post. It as really a very good article. I noticed all your important points. Thanks.

  11. Great blog post. Really Cool.

  12. I really liked your article.Much thanks again. Awesome.

  13. Jasco says:

    Thanks again for the post.Thanks Again. Really Cool.

  14. Relatedjust beneath, are numerous totally not related sites to ours, however, they are surely worth going over

  15. Thanks for another great article. Where else could anybody get that kind of info in such an ideal method of writing? I have a presentation subsequent week, and I am at the search for such info.

  16. mavic says:

    one is sharing information, that as truly good, keep up writing.

  17. sex spiele says:

    Precisely what I was looking for, thanks for posting.

  18. I was suggested this web site by my cousin. I am not sure whether this post is written by him as nobody else know such detailed about my problem. You are amazing! Thanks!

  19. apiary says:

    Woah! I am really loving the template/theme of this blog. It as simple, yet effective.

  20. UoN says:

    Thankyou for all your efforts that you have put in this. very interesting information.

  21. bbw models says:

    It is best to participate in a contest for among the finest blogs on the web. I all recommend this web site!

  22. This is really interesting, You certainly are a very qualified blogger. I possess joined your rss and enjoy seeking more of one as fantastic post. Also, I have got shared your blog in my myspace!

  23. Cheap jacket says:

    This very blog is definitely cool and diverting. I have chosen many interesting advices out of it. I ad love to come back every once in a while. Thanks a bunch!

  24. Really enjoyed this blog article. Fantastic.

  25. e-liquid says:

    Perhaps you can write next articles relating to this article.

  26. Terrific work! That is the type of info that are supposed to be shared around the web. Shame on Google for now not positioning this submit upper! Come on over and discuss with my web site. Thanks =)

  27. amiclubwear says:

    I really liked your blog. Awesome.

  28. STForex says:

    Pretty! This was an extremely wonderful post. Thank you for supplying this info.

  29. Angel says:

    I think you have noted some very interesting points , thanks for the post.

  30. I appreciate you sharing this blog post. Awesome.

  31. Really appreciate you sharing this blog.Really looking forward to read more. Keep writing.

  32. click site says:

    Wow, wonderful blog layout! How long have you been blogging for? you make blogging look easy. The overall look of your site is fantastic, let alone the content!. Thanks For Your article about sex.

  33. home page says:

    Awesome post.Thanks Again. Keep writing.

  34. to be capable of get these phones add alone is usually to pay for

  35. Wow, great post. Really Great.

  36. Local says:

    What a awesome blog this is. Look forward to seeing this again tomorrow.

  37. Photography says:

    I cannot thank you enough for the blog article. Cool.

  38. teachers says:

    Very informative article.Really looking forward to read more. Will read on…

  39. pretty useful stuff, overall I think this is really worth a bookmark, thanks

  40. It as hard to come by well-informed people about this subject, however, you sound like you know what you are talking about! Thanks

  41. HVAC says:

    I really liked your blog article.Really looking forward to read more. Cool.

  42. It’а†s really a great and helpful piece of info. I’а†m glad that you just shared this helpful info with us. Please keep us up to date like this. Thanks for sharing.

  43. I think this is a real great blog post.Really thank you! Will read on

  44. law firm says:

    Very fantastic information can be found on site.

  45. Well I really liked studying it. This post offered by you is very useful for proper planning.

  46. I value the blog post.Really thank you! Awesome.

  47. This web site certainly has all of the information I needed about this subject and didn at know who to ask.

  48. Thanks-a-mundo for the article post.Really thank you! Fantastic.

  49. There is perceptibly a bunch to know about this. I suppose you made certain nice points in features also.

  50. Appreciate you sharing, great blog post.Much thanks again.

  51. Thank you for your post.Much thanks again. Fantastic.

  52. Thanks-a-mundo for the blog.Much thanks again. Really Cool.

  53. very nice post, i actually love this web site, carry on it

  54. amazon seo says:

    I appreciate you sharing this article post. Want more.

  55. Spot on with this write-up, I truly think this website needs rather more consideration. I?ll probably be again to read rather more, thanks for that info.

  56. small patio says:

    I truly appreciate this blog.Really looking forward to read more. Great.

  57. Jackeline says:

    Remarkable issues here. I am very glad to peer your post. Thank you a lot and I am looking forward to touch you. Will you kindly drop me a mail?

  58. usa jobs says:

    Great, thanks for sharing this article post. Fantastic.

  59. Tai says:

    Usually I do not read post on blogs, however I wish to say that this write-up very forced me to take a look at and do it! Your writing style has been surprised me. Thank you, very nice article.

  60. I value the article.Really looking forward to read more. Cool.

  61. Mei says:

    Your content is excellent but with pics and videos, this site could definitely be one of the most beneficial

  62. Candidates says:

    I cannot thank you enough for the article post.Really looking forward to read more. Want more.

  63. Lavone says:

    very good submit, i certainly love this website, keep on it

  64. Wow, great article post. Really Great.

  65. This is a topic which is near to my heart Best wishes! Exactly where are your contact details though?

  66. Thanks again for the article.Really thank you! Will read on…

  67. Music says:

    Really informative blog article.Thanks Again. Fantastic.

  68. this wonderful read!! I definitely really liked every little

  69. This is one awesome blog.Really thank you! Really Cool.

  70. #mme says:

    A round of applause for your post. Cool.

  71. Very interesting subject , thanks for posting. Not by age but by capacity is wisdom acquired. by Titus Maccius Plautus.

  72. to be using? I am having some small security problems with

  73. This is a topic which is near to my heart Take care! Exactly where are your contact details though?

  74. I went over this web site and I conceive you have a lot of great information, saved to bookmarks (:.

  75. Really appreciate you sharing this article.Thanks Again. Awesome.

Leave a Reply

Your email address will not be published.