BR Maths Corner-1

[Vayutuvan]

Its an advance for AI in the math domain.
Here's more -
Google and OpenAI get 2025 IMO gold

@vera_k, your comments on Tao's post on Mathstodon, SVP.

[Vayutuvan]

https://terrytao.wordpress.com/2009/07/ ... h-project/

From Terence Tao's blog 2009

The International Mathematical Olympiad (IMO) consists of a set of six problems, to be solved in two sessions of four and a half hours each. Traditionally, the last problem (Problem 6) is significantly harder than the others. Problem 6 of the 2009 IMO, which was given out last Wednesday, reads as follows:

Problem 6. Let a_1, a_2, \ldots, a_n be distinct positive integers and let M be a set of n-1 positive integers not containing s = a_1 +a_2 +\ldots+a_n. A grasshopper is to jump along the real axis, starting at the point 0 and making n jumps to the right with lengths a_1, a_2, \ldots , a_n in some order. Prove that the order can be chosen in such a way that the grasshopper never lands on any point in M.

Of the 500-odd participants in the Olympiad, only a half-dozen or so managed to solve this problem completely (I don’t have precise statistics yet). I myself worked it out about seven hours after first hearing about the problem, though I was preoccupied with other things for most of that time period.

I thought that this problem might make a nice “mini-Polymath” project to be solved collaboratively; it is significantly simpler than an unsolved research problem (in particular, being an IMO problem, it is already known that there is a solution, which uses only elementary methods), and the problem is receptive to the incremental, one-trivial-observation-at-a-time polymath approach. So I would like to invite people to try solving the problem collaboratively on this blog, by posting one’s own comments, thoughts, and partial progress on the problem here.

To keep with the spirit of the polymath approach, I would however like to impose some ground rules:
...

[A_Gupta]

@Amber G., can you tell us about the Putnam prize in math? You might have said something previously, but I am not aware of it.
Thanks!

[Amber G.]

@Amber G., can you tell us about the Putnam prize in math? You might have said something previously, but I am not aware of it.
Thanks!

Yes, I’ve mentioned the Putnam here before — in fact, we've even discussed a few of its problems on this forum. ( eg <here - and few posts afterwards >

For a quick intro:

The William Lowell Putnam Mathematical Competition (usually just called the Putnam) is one of the most prestigious undergraduate math competitions in North America. It’s been held annually since 1938 and is known for its brutal difficulty. Students take a 6-hour exam (split into two 3-hour sessions) consisting of 12 proof-based problems. Each problem is worth 10 points — so a perfect score is 120.

To give a sense of scale:

- The median score is often around 0 or 1. (Yes, really.)
- Scores above 10 usually put you in the top 25%.
- The top 5 teams are almost always schools like MIT, Harvard, Princeton, Duke, and Caltech.
- The top 5 individuals get the title of Putnam Fellow and often go on to illustrious careers in math or academia.

It’s not just a test of knowledge — it’s about creativity, deep thinking, and mathematical maturity. Many of the problems are deceptively simple-looking but require clever insights.

Just a small tidbit: about 20 years ago, my son was actually recruited to join a college math team (yes, they do that — not just for sports!). The college he selected and ended up joining - his team won first place in the Putnam that year. (and did fairly well in later years) (

-- For just some fun perspective .. One problem from Putnam - simple looking :

In the Gregorian calendar (which we normally use_:

- A common year contains 365 days; a leap year contains 366.
- Years not divisible by 4 are common years.
- Years divisible by 100 but not by 400 are common years.
- Years divisible by 400 are leap years.

What is the probability that Christmas (December 25) falls on a Wednesday?

(Hint: Easy looking problem but it is *not* 1/7

[A_Gupta]

A cycle of 400 years in the Gregorian calendar is an exact number of 7 day weeks.
Given that December 25, 2001 was a Tuesday, my answer is 58/400 = 0.145.

[Amber G.]

^^^ Of course.. and the probability that it will fall on Thursday is less than that of Wednesday.
(This is similar to one math problem I based on this in other math contest - A problem similar to this has come in JEE too )

Other one, one of my favorite (also came in one JEE exam) is this beautiful and famous integral, one from the earlier Putnam..

[Amber G.]

Meanwhile sharing some other important news:

He’s the ‘Mozart’ of Math and Trump Killed His Funding

(Again, - as usual - Choosing not to engage with the transparent and juvenile attempts of trolling and thread hijacking)

Summary:

- Terence Tao, a world-renowned mathematician at UCLA, is widely considered one of the most brilliant minds alive—nicknamed the “Mozart of Math.”

From doing calculus at age 7, winning the IMO at 13, Ph.D. at Princeton at 19), Tao has made major contributions to cryptography, imaging (like MRIs), and number theory, (among other things he proved 50 years old Mehta-Dyson-Wigner conjecture..) ..has received numerous honors, including the Fields Medal ..—he’s also known for his humility, mentorship, and collaborative spirit ( his blogs - many time(s) referenced here in Brf- public lectures/articles on all types of subjects are *extremely* good(

The Trump administration has cut funding for STEM research, and Tao’s work is among the casualties.

The defunding has shaken Tao’s commitment to staying in the U.S. academic system. He says the risks are now "existential," and he can no longer make long-term plans at UCLA.

This is a larger decline in U.S. support for higher education and science, potentially driving away even the best minds.. all due to Trump.

[A_Gupta]

^^^ Of course.. and the probability that it will fall on Thursday is less than that of Wednesday.
(This is similar to one math problem I based on this in other math contest - A problem similar to this has come in JEE too )

Other one, one of my favorite (also came in one JEE exam) is this beautiful and famous integral, one from the earlier Putnam..

Beautiful! can be evaluated by elementary methods (tedious - express as polynomial + (something)/(1+x^2) and integrate).

My question is - how can one discover such an integral?

Added -
1. I want a positive definite integrand.
2. I use Integral of 1/(1+x^2) from 0 to 1 is Pi/4
3. I possibly want a 6th degree term in the integrand to get a factor of 1/7 on integration.

What else?

[Vayutuvan]

My question is - how can one discover such an integral?
...

The history is here. The recent oldest discovery seems to be by Mahler.

Look at this paper by Lucas.

https://paperzz.com/doc/8519004/integra ... %3E-%CF%80

or PDF at ResearchGate page of Lucas https://www.researchgate.net/publicatio ... t_355113_p

Evaluation of the integral itself easy but tedious using partial fraction expansion.

[A_Gupta]

[Vayutuvan]

https://www.lodhagroup.com/press-releas ... -in-mumbai

Lodha Foundation unveils Mathematical Sciences Institute (LMSI) in Mumbai
August 18, 2025

With an aim to provide a collaborative and meritocratic environment for mathematics researchers from across the globe, Lodha Mathematical Sciences Institute (LMSI) was launched on August 17 in Mumbai. The first ever privately funded mathematics research institute in the country is backed by Lodha Foundation, which received appx Rs 20,000 crore endowment from the Lodha family last year.

“It's very important for us as a company to make sure that our success contributes in a wholesome manner to India's development. In that context, the family decided last year to donate about one-fifth of the ownership of the company to the Lodha Foundation. We have realised that for an nation to become strong, we ought to have original thinking and promote innovation. This is what the Lodha Institute of Mathematical Sciences is focused on. We are undoubtedly committed to making sure that in the long term, this becomes one of the leading institutions in the world,” said Abhishek Lodha, CEO and MD of Lodha Developers.

The LMSI will encourage intellectual inquiry and mathematical research, both in its fundamental aspects as well as in its applications. The institute aims to be a global leader in mathematical research, both in its fundamental aspects as well as in its applications. The institute will serve as a beacon of mathematical excellence, attracting the best minds from around the world and fostering a culture of intellectual curiosity and discovery.

Ashish Kumar Singh, chief mentor at Lodha Foundation said, “The foundation of almost all technological progress today is mathematical, whether it be cryptography, cybersecurity, warfare, and definitely artificial intelligence. At LMSI, we are aiming to get the brightest researchers to come together and pursue research in cutting-edge frontier mathematics so that the frontiers of mathematics are pushed and new knowledge is created.”

The LMSI will be headed by Dr V Kumar Murty, a renowned mathematician, known for his significant contributions to number theory and arithmetic geometry. A Ph.D. from Harvard University, Dr Murty has served as the Director of the Fields Institute and as a professor at the University of Toronto. With several international accolades to his credit, Dr Murty’s research spans various areas, including analytic number theory, algebraic number theory, information security, and arithmetic algebraic geometry.

“We know that in the vast history of India, there have been many great mathematicians who contributed fundamental ideas that have shaped the world that we live in today. To be a part of that tradition and to take it forward to the next level, invoking not only the expertise within India, but even that which is available around the world, is truly critical and inspiring. The LMSI aims to address fundamental problems that will help India becoming a developed nation and a global leader by 2047,” said Dr Murty.

The LMSI is backed by a scientific advisory panel of leading researchers, who will oversee the scientific programs of the Institute and help to align them with the main international trends. This includes Dr Manjul Bhargava (Princeton University), Dr Vikraman Balaji (Chennai Mathematical Institute), Dr Sourav Chatterjee (Stanford University), Dr Ravi Vakil (Stanford University and President-Elect of AMS), Dr Yakov Eliashberg (Stanford University), Dr Alexander Lubotzky (Weizmann Institute of Science), Dr Kavita Ramanan (Brown University), Dr François Labourie (Université de Nice Sophia Antipolis), Dr Siddhartha Mishra (ETH Zurich), Dr Mahan Mj (TIFR), Dr Ngô Bảo Châu (University of Chicago), Dr Parimala Raman (Emory University).

The Institute recognises that it can contribute significantly to solving complex challenges in areas such as infrastructure, healthcare, and economics. These challenges that can be addressed through advanced mathematical approaches, including data science, machine learning, operations research, and optimization. As an institute rooted in India and focused on both fundamental research and applications, it can make meaningful contributions, both in the short term as well as the long term, nationally and globally. To realise this vision, the Lodha Mathematical Sciences Institute will empower faculty with the freedom to pursue their research interests without administrative constraints.

There are (at least) two Fields Medalists in the advisory board - Manjul Bhargava and Ngô Bảo Châu.