Bill Casselman’s Euclid: thanks to ClayMath

The purpose is only to share and spread the awareness of availability of this second master piece on Euclid. Thanks to Clay Math Organization for serving students world wide, and thanks to the generous Mr and Mrs Clayton. I hope my math olympiad students will enjoy this and enrich themselves mathematically.

Set theory: more basic problems to solve and clear and apply

I am producing the list of questions first so that the motivated reader can first read and try them …and can compare with my answers by scrolling down only much later; here we go:

  1. If \{ A_{i}\} and \{ B_{j}\} are two classes of sets such that \{ A_{i}\} \subseteq \{ B_{j}\}, then prove that \bigcup_{i}A_{i} \subseteq \{ B_{j}\} and \bigcap_{j} B_{j} \subseteq \bigcap_{i}A_{i}.
  2. The difference between two sets A and B, denoted by A-B, is the set of elements in A and not in B; thus, A - B = A \bigcap B^{'}. Prove the following simple properties: (a) A-B = A-(A \bigcap B) = (A \bigcup B)-B; (b) (A-B)-C = A-(B \bigcup C); (c) A - (B-C) = (A-B) \bigcup (A \bigcap C); (d) (A \bigcup B) - C = (A-C) \bigcup (B-C); (e) A - (B \bigcup C) = (A-B) \bigcap (A-C)
  3. The symmetric difference of two sets A and B, denoted by A \triangle B, is defined by A \triangle B = (A-B) \bigcup (B-A); it is thus the union of their differences in opposite orders. Prove the following : (a) Symmetric difference is associative : A \triangle (B \triangle C) = (A \triangle B) \triangle C (b) A \triangle \phi=A (c) A \triangle A = \phi (c) Symmetric difference is commutative: A \triangle B = B \triangle A (d) Some sort of distributive rule also holds: A \bigcap (B \triangle C) = (A \bigcap B) \triangle (A \bigcap C)
  4. A ring of sets is a non-empty class A of sets such that if A and B are in A, then A \triangle B and A \bigcap B are also in A. Show that A must also contain the empty set, A \bigcup B, and A-B. Show that if a non-empty class of sets contains the union and difference of sets any pair of its sets, then it is a ring of sets. Prove that a Boolean algebra of sets is a ring of sets.
  5. Show that the class of all finite subsets (including the empty set) of an infinite set is a ring of sets but is not a Boolean algebra of sets.
  6. Show that the class of all finite unions of closed-open intervals on the real line is a ring of sets but is not a Boolean algebra of sets.
  7. Assuming that the universal set U is non-empty, show that Boolean algebras of sets can be described as a ring of sets which contain U.

I will put up my solutions as soon as I can.


Nalin Pithwa.

Number theory: let’s learn it the Nash way !

Reference: A Beautiful Mind by Sylvia Nasar.

Comment: This is approach is quite similar to what Prof. Joseph Silverman explains in his text, “A Friendly Introduction to Number Theory.”

Peter Sarnak, a brash thirty-five-year-old number theorist whose primary interest is the Riemann Hypothesis, joined the Princeton faculty in the fall of 1990. He had just given a seminar. The tall, thin, white-haired man who had been sitting in the back asked for a copy of Sarnak’s paper after the crowd had dispersed.

Sarnak, who had been a student of Paul Cohen’s at Stanford, knew Nash by reputation as well as by sight, naturally. Having been told many times Nash was completely mad, he wanted to be kind. He promised to send Nash the paper. A few days later, at tea-time, Nash approached him again. He had a few questions, he said, avoiding looking Sarnak in the face. At first, Sarnak just listened politely. But within a few minutes, Sarnak found himself having to concentrate quite hard. Later, as he turned the conversation over in his mind, he felt rather astonished. Nash had spotted a real problem in one of Sarnak’s arguments. What’s more, he also suggested a way around it. “The way he views things is very different from other people,” Sarnak said later. ‘He comes up with instant insights I don’t know I would ever get to. Very, very outstanding insights. Very unusual insights.”

They talked from time to time. After each conversation, Nash would disappear for a few days and then return with a sheaf of computer printouts. Nash was obviously very, very good with the computer. He would think up some miniature problem, usually very ingeniously, and then play with it. If something worked on a small scale, in his head, Sarnak realized, Nash would go to the computer to try to find out if it was “also true the next few hundred thousand times.”

{What really bowled Sarnak over, though, was that Nash seemed perfectly rational, a far cry from the supposedly demented man he had heard other mathematicians describe. Sarnak was more than a little outraged. Here was this giant and he had been all but forgotten by the mathematics profession. And the justification for the neglect was obviously no longer valid, if it had ever been.}


Nalin Pithwa

PS: For RMO and INMO (of Homi Bhabha Science Foundation/TIFR), it helps a lot to use the following: (it can be used with the above mentioned text of Joseph Silverman also): TI nSpire CAS CX graphing calculator.

You and your research or you and your studies for competitive math exams

You and your research ( You and your studies) : By Richard Hamming, AT and T, Bell Labs mathematician;

U.S. Team Takes First in 2016 IMO (International Mathematical Olympiad)

Friday, July 15 2016:

The U.S. team finished first with 214 points at the 57th International Mathematical Olympiad (IMO) in Hong Kong. All six members of the team– Ankan Bhattacharya (International Academy East, Troy, Michigan), Michael Kural (Greenwich High School, Riverside, Connecticut), Allen Liu (Penfield Senior High School, Penfield, New York), Junyao Peng (Princeton International School of Mathematics and Science, Princeton, New Jersey), Ashwin Sah (Jesuit High School, Portland, Oregon), and Yuan Yao (Phillips Exeter Academy, Exeter, New Hampshire)– earned gold medals. The team from the Republic of Korea earned 207 points and China finished third with 204 points. Three of the six U.S. team members are former contestants in Who Wants to Be a Mathematician (WWTBAM): Ankan Bhattacharya (2016 national champ), Michael Kural (2015 national contestant and a contestant at the Western Connecticut State University game), and Ashwin Sah (2014 winner at Oregon State University). All the participants on the U.S. team were selected through a series of competitions organized by the Mathematical Association of America (MAA), culminating with the USA Mathematical Olympiad. The U.S. team leader was Po-Shen Loh of Carnegie Mellon University. See results of the 2016 IMO, which had more than 100 countries participating. The 2017 IMO will be July 12-24 in Rio de Janeiro, Brazil.

Three cheers to the American team of 2016 for cracking one of the most daunting, real mathematics challenge!!
Nalin Pithwa
The International Mathematical Olympiad (IMO) is the World Championship Mathematics Competition for High School students and is held annually in a different country. The first IMO was held in 1959 in Romania, with 7 countries participating. It has gradually expanded to over 100 countries from 5 continents. The IMO Advisory Board ensures that the competition takes place each year and that each host country observes the regulations and traditions of the IMO.
Note: Some good source of problems for this examination  are the following:
1. The IMO Compendium (A Collection of Problems suggested for the International Mathematical Olympiads: 1956-2004) by Dusan Djukic, Vladimir Jankovic, Ivan Matic, Nikola Petrovic
2. International Mathematical Olympiad Volume I (1959-1975) by Istvan Reiman
3. International Mathematical Olympiad Volume II (1976-1990) by Istvan Reiman
4. International Mathematical Olympiad Volume III (1991-2004) by Istvan Reiman
5. Problem Solving Strategies by Arthur Engel
6. The USSR Olympiad Problem Book (Selected Problems and Theorems of Elementary Mathematics) by D. O. Shklarsky, N. N. Chentzov, and I. M. Yaglom
All the above books are available generally in Amazon India and Flipkart for cash-on-delivery.
The way for an Indian student to reach this IMO contest is to appear first for the Regional Mathematics Olympiad (RMO) conducted by TIFR/Homibhabha as explained further here: 
In India, the RMO will be held on Oct 9 2016. There will be no Pre-RMO this year.
This real IMO is not be confused with IMO conducted by SOF World, which is, Science Olympiad Foundation!

Sources of problems for the RMO, INMO, IMO

  1. IMAR Tests: These are a series of tests organized biannually by the Mathematics Institute of the Romanian Academy (IMAR). All students are invited, particularly those who  are interested in taking the IMO Team Selection Tests in the following year.
  2. 77 de Ture: This is a collection of 77 mathematics problems gathered from IMO team leaders from around the world. The Romanian IMO 2004 team used these problems as practice.
  3. MOSP: The US IMO team Math Olympiad Summer Program.
  4. JMBO: Junior Balkan Mathematics Olympiad.
  5. American Mathematical Monthly or AMM: This is probably the most important American mathematics periodical.
  6. Team Contest: These problems were given at a team contest organized in 2004 by several Romanian college students who were IMO veterans for their younger peers.
  7. Mathlinks Contest: This is the contest organized yearly (or biannually) by the Mathlinks forum.
  8. Romanian-Hungarian Training Camp: This is the yearly common IMO training of the Romanian and Hungarian teams. The camp lasts for one week and is organized alternatively by the two countries.
  9. William Lowell Putnam: This is the most important math competition organized in the USA for undergraduate students.

If you know some more sources, please share with us.


Nalin Pithwa







Announcement: A Full Scholarship Program

We are Mathematics Hothouse, Bangalore, We are pleased to announce that henceforth, every academic year, we will be admitting 5 students with full scholarship or 100% discount, from any part of India, who are talented, deserving or needy, to our program for RMO and INMO coaching. The coaching will be via on-line, live, video interactive Skype sessions mimicking traditional classroom or just classroom coaching or even correspondence course.

If you wish to apply, please write to


Nalin Pithwa