Notre Dame Math Club

NDMC Math Cup 2026 has come to an end, but every problem still has a story to tell.
Revisit the complete collection of questions and their solutions, and rediscover the beauty of mathematics.

"Mathematics is not a prison of rules; it is an endless sky of free thought, where the greatest breakthroughs often begin with a change in perspective."

In calculus, we often encounter integrals that refuse to yield to familiar techniques. When substitution and integration by parts reach their limits, solving the problem demands more than computational skill. It demands a different way of thinking. It is precisely at this point that Richard Feynman's celebrated technique of differentiating under the integral sign reveals its true elegance.

One of the brilliant members of our club, 𝐌𝐮𝐧𝐭𝐚𝐬𝐢𝐫 𝐊𝐡𝐚𝐧 𝐑𝐚𝐟𝐢 (𝟏𝟐𝟕𝟏𝟑𝟎𝟑𝟒), presents 𝐅𝐞𝐲𝐧𝐦𝐚𝐧 𝐓𝐞𝐜𝐡𝐧𝐢𝐪𝐮𝐞 𝐨𝐟 𝐈𝐧𝐭𝐞𝐠𝐫𝐚𝐭𝐢𝐨𝐧, an article that explores this remarkable idea in depth.

Through the introduction of a simple parameter, Rafi demonstrates how seemingly intractable integrals can be transformed into far more manageable differentiation problems. From the 𝐋𝐞𝐢𝐛𝐧𝐢𝐳 𝐈𝐧𝐭𝐞𝐠𝐫𝐚𝐥 𝐑𝐮𝐥𝐞 to the 𝐃𝐢𝐫𝐢𝐜𝐡𝐥𝐞𝐭 𝐈𝐧𝐭𝐞𝐠𝐫𝐚𝐥, 𝐆𝐚𝐮𝐬𝐬𝐢𝐚𝐧 𝐈𝐧𝐭𝐞𝐠𝐫𝐚𝐥, 𝐁𝐚𝐬𝐞𝐥 𝐏𝐫𝐨𝐛𝐥𝐞𝐦, and several elegant applications, the article showcases how a shift in perspective can unlock solutions that once appeared impossible.

Whether you are preparing for mathematical olympiads or simply fascinated by the beauty of advanced calculus, this article offers a fresh way of looking at integration. After all, as Feynman himself remarked, his reputation for solving difficult integrals came not from extraordinary calculation, but from having "a different box of tools" than everyone else.

অসমতা—গণিতের এমন এক শাখা, যেখানে একটি মাত্র চিহ্ন ‘>’ বা ‘

𝗕𝗘𝗥𝗧𝗥𝗔𝗡𝗗 𝗥𝗨𝗦𝗦𝗘𝗟𝗟’𝗦 𝗣𝗔𝗥𝗔𝗗𝗢𝗫
By 𝐒𝐚𝐝𝐦𝐚𝐧 𝐀𝐛𝐝𝐮𝐥𝐥𝐚𝐡 𝐒𝐡𝐚𝐤𝐢𝐤 (𝟏𝟐𝟕𝟏𝟎𝟎𝟕𝟏)

Let’s start off with a related riddle: There is a barber in a village who cuts the hair of the people who don't cut their hair themselves. 𝐓𝐡𝐞 𝐪𝐮𝐞𝐬𝐭𝐢𝐨𝐧 𝐢𝐬 𝐰𝐡𝐨 𝐰𝐢𝐥𝐥 𝐜𝐮𝐭 𝐭𝐡𝐞 𝐛𝐚𝐫𝐛𝐞𝐫’𝐬 𝐡𝐚𝐢𝐫?

A riddle which seems pretty simple, but when you think about it thoroughly, you will understand its depth. It’s more like a loop from which you can’t escape.

So if you try to solve it, you will be able to identify the contradiction. If the barber tries to cut his own hair, he will violate the rule of not cutting the hair of those who cut their own hair, since he is doing exactly the same thing. Again if he stops doing it, then he has to cut his own hair because he cuts hair of those who don’t do it themselves (Imagine it yourself–it’ll become clearer)

But what’s the relation between this problem and Advanced Mathematics? Let’s connect this problem with Mathematics. This connection was first identified and analyzed by 𝐁𝐞𝐫𝐭𝐫𝐚𝐧𝐝 𝐑𝐮𝐬𝐬𝐞𝐥𝐥. We don’t often see these kinds of problems being related to Mathematics.

It’s connected with sets. A set is a collection of elements which can contain anything. For example:
P={Paradox,Hypothesis,Proof}
Q={π,e,φ}

It is also possible for sets to contain other sets.
A={{Paradox,Hypothesis,Proof},{π,e,φ}}

So, set A contains both set P and set Q.

There are some sets that contain themselves and some don’t. For example:
C={P,Q}
D={P,Q,D}

Here, set C does not contain itself, but set D does.
Now, the set of all sets that contain itself is like this:
T={.....,T,.....}

The paradox mainly starts from here. Let’s consider a set of all the sets that don’t contain themselves.
R={x;x∉x}

Now, the question is: 𝐝𝐨𝐞𝐬 𝐭𝐡𝐢𝐬 𝐬𝐞𝐭 𝐑 𝐜𝐨𝐧𝐭𝐚𝐢𝐧 𝐢𝐭𝐬𝐞𝐥𝐟?

If this set doesn't contain itself, then it’s a part of the sets that don’t contain themselves and should be a part of the set R. On the contrary, if it becomes a part of set R, then it contains itself, which contradicts the condition. So, it shouldn’t be a part of set R.

Sounds weird, right? But if you think about it, this is identical to the barber’s problem. These types of problems showcased in 𝐑𝐮𝐬𝐬𝐞𝐥'𝐬 𝐩𝐚𝐫𝐚𝐝𝐨𝐱 may not have direct solutions, but they help us think deeper and catch a clear concept of such contradictory topics.

পরিসংখ্যান গণিতের একটি চমৎকার শাখা। তবে স্থূল দৃষ্টিতে একে কেবলই নিরস সংখ্যাতত্ত্ব আর সমীকরণের দুর্ভেদ্য গোলকধাঁধা মনে হওয়াই স্বাভাবিক। না বুঝে মুখস্থ করার কারণে আমরা প্রায়শই বিস্মৃত হই, এই শুষ্ক সূত্রাবলির অন্তরালেও স্পন্দিত হচ্ছে এক অনবদ্য গাণিতিক কাব্য।

মোহাম্মদ সাকিব শাহরিয়ার (১২৬১৫০৯৪) তাঁর "Making Sense of Statistics: ফর্মুলার পেছনের গল্প" লেখায় এই সূত্রগুলোর পেছনের রহস্য তুলে ধরে এক নতুন দিগন্তের উন্মোচন করেছেন। লেখকের সহজ-সরল বর্ণনায় গড়, মধ্যক বা প্রচুরকের মতো আপাত পরিচিত বিষয়গুলো কেবল আর কাঠখোট্টা সমীকরণের মাঝে আটকে থাকেনি। এমনকি ডেটার বিস্তৃতি মাপার ক্ষেত্রে ‘ভ্যারিয়েন্স’ বা ‘স্ট্যান্ডার্ড ডেভিয়েশন’-এর মতো বিষয়গুলো সহ পরিসংখ্যানগত ধারণা এক গাণিতিক রূপ পেয়ে একীভূত হয়েছে।

আপনি যদি পরিসংখ্যানের এই সংখ্যাতাত্ত্বিক খোলস ভেদ করে এর অন্তরালের ধ্রুপদী আখ্যানটিকে অন্বেষণে আগ্রহী হন, অথবা যুক্তির মাধ্যমে আপাত পরিচিত বিষয়গুলো বুঝতে চান, তবে এই লেখাটি আপনার জন্যই।

“𝘔𝘢𝘵𝘩𝘦𝘮𝘢𝘵𝘪𝘤𝘴 𝘪𝘴 𝘯𝘰𝘵 𝘢𝘣𝘰𝘶𝘵 𝘯𝘶𝘮𝘣𝘦𝘳𝘴, 𝘦𝘲𝘶𝘢𝘵𝘪𝘰𝘯𝘴, 𝘤𝘰𝘮𝘱𝘶𝘵𝘢𝘵𝘪𝘰𝘯𝘴, 𝘰𝘳 𝘢𝘭𝘨𝘰𝘳𝘪𝘵𝘩𝘮𝘴: 𝘪𝘵 𝘪𝘴 𝘢𝘣𝘰𝘶𝘵 𝘶𝘯𝘥𝘦𝘳𝘴𝘵𝘢𝘯𝘥𝘪𝘯𝘨.”
— 𝐖𝐢𝐥𝐥𝐢𝐚𝐦 𝐏𝐚𝐮𝐥 𝐓𝐡𝐮𝐫𝐬𝐭𝐨𝐧

Widely regarded as the highest honor in mathematics, the Fields Medal is often described as the "Nobel Prize of Mathematics." Awarded every four years to exceptional mathematicians under the age of 40, it celebrates groundbreaking discoveries and extraordinary contributions to the mathematical world.

𝐍𝐨𝐭𝐫𝐞 𝐃𝐚𝐦𝐞 𝐌𝐚𝐭𝐡 𝐂𝐥𝐮𝐛 is excited to honor the four brilliant mathematicians who recently received Fields Medals.

They include 𝐉𝐚𝐜𝐨𝐛 𝐓𝐬𝐢𝐦𝐞𝐫𝐦𝐚𝐧, who has made great progress in the area of complex geometry, 𝐇𝐨𝐧𝐠 𝐖𝐚𝐧𝐠, who was able to solve one of the hardest math puzzles in 3D and proved the impressiveness of math, 𝐉𝐨𝐡𝐧 𝐏𝐚𝐫𝐝𝐨𝐧 from the USA, who has changed the direction of development in knot theory, and 𝐘𝐮 𝐃𝐞𝐧𝐠, who has solved one of the greatest problems in the fluid mechanics area.

All of that reminds us that math is not only hard formulas on the board, it is also curiosity, patience, and finding beauty in the ideas.

"𝐓𝐡𝐞 𝐬𝐦𝐚𝐥𝐥𝐞𝐬𝐭 𝐩𝐢𝐞𝐜𝐞𝐬 𝐨𝐟𝐭𝐞𝐧 𝐜𝐚𝐫𝐫𝐲 𝐭𝐡𝐞 𝐠𝐫𝐞𝐚𝐭𝐞𝐬𝐭 𝐭𝐫𝐮𝐭𝐡𝐬"

Sometimes, the smallest things come together to make something much bigger than themselves, and integration is one of the finest examples of that concept. This idea was the core of the recent session of Notre Dame Math Club, “𝐈𝐧𝐭𝐞𝐠𝐫𝐚𝐭𝐢𝐨𝐧: 𝐖𝐡𝐞𝐫𝐞 𝐋𝐢𝐦𝐢𝐭𝐬 𝐌𝐞𝐞𝐭 𝐈𝐧𝐟𝐢𝐧𝐢𝐭𝐲,” conducted by 𝐌𝐝. 𝐄𝐫𝐟𝐚𝐧𝐮𝐫 𝐑𝐚𝐡𝐦𝐚𝐧 sir. Instead of teaching integration as a collection of formulas to memorise, he took participants through the basic ideas of integration that allow them to develop an intuition for how limits and infinitesimal pieces can be combined to describe a whole.

To help participants re-engage with the discussion and explore further, we are sharing the 𝐬𝐞𝐬𝐬𝐢𝐨𝐧 𝐡𝐚𝐧𝐝𝐨𝐮𝐭 below.

The FIFA World Cup 2026 has come to an end, and with it, the 𝐍𝐃𝐌𝐂 𝐌𝐚𝐭𝐡 𝐂𝐮𝐩. While Spanish and Argentine footballers battled for the trophy on the pitch, our Math Athletes were plotting points on the Cartesian plane, connecting them, bisecting lines, and determining their points of intersection.

The results of the final round of the NDMC Math Cup are now out. Congratulations to everyone who scored goals on the mathematical pitch, and to those who couldn't score but still played their part in promoting enthusiasm for mathematics, shifting the curve one step closer to the y-axis.

The 𝐓𝐨𝐩 𝟑 performers of this round will receive certificates.
The 𝐓𝐨𝐩 𝟓 of the NDMC Math Cup will be announced soon!

বহুপদী নামটি যতটা পরিচিত, তার ভেতরের জগৎ ততটাই বিস্ময়কর। কখনো এটি অগণিত সমস্যার সহজ সমাধান, কখনো আবার যুক্তি, সৌন্দর্য ও সৃজনশীলতার এক অনন্য ভাষা। গণিতের এই চিরন্তন ধারণাকে নতুন দৃষ্টিতে দেখার এক সুন্দর প্রয়াস হলো "বহুপদীর কারসাজি", যেখানে পরিচিত সূত্রের আড়াল থেকে ধীরে ধীরে উন্মোচিত হয় বহুপদীর বিস্ময়কর রূপ।

এই অসাধারণ লেখাটি রচনা করেছেন স্নেহাশীষ চৌধুরী (১২৬০২০১৯)। তাঁর লেখনী পাঠককে শুধু বহুপদী শেখায় না, বরং গণিতের সৌন্দর্যকে নতুনভাবে অনুভব করতেও উদ্বুদ্ধ করে।

When passion meets pure logic, history naturally rewrites itself. 𝐍𝐨𝐭𝐫𝐞 𝐃𝐚𝐦𝐞 𝐌𝐚𝐭𝐡 𝐂𝐥𝐮𝐛 is overwhelmed with joy to applaud Team Bangladesh for their incredible triumph at the 𝟔𝟕𝐭𝐡 𝐈𝐧𝐭𝐞𝐫𝐧𝐚𝐭𝐢𝐨𝐧𝐚𝐥 𝐌𝐚𝐭𝐡𝐞𝐦𝐚𝐭𝐢𝐜𝐚𝐥 𝐎𝐥𝐲𝐦𝐩𝐢𝐚𝐝 𝟐𝟎𝟐𝟔!

What an incredible display of skill and perseverance! All six members of the team brought 𝟏 𝐒𝐢𝐥𝐯𝐞𝐫 and 𝟓 𝐁𝐫𝐨𝐧𝐳𝐞. The team accumulated 121 points to move Bangladesh up to 𝟑𝟗𝐭𝐡 position out of the 117 competing countries.

For 𝐌𝐝. 𝐑𝐚𝐢𝐡𝐚𝐧 𝐒𝐢𝐝𝐝𝐢𝐪𝐮𝐞𝐞, 𝐓𝐚𝐡𝐬𝐢𝐧 𝐊𝐡𝐚𝐧, 𝐌. 𝐙𝐚𝐦𝐢𝐮𝐥 𝐇𝐨𝐬𝐬𝐚𝐢𝐧, 𝐉𝐚𝐰𝐚𝐝 𝐇𝐚𝐦𝐞𝐞𝐦 𝐂𝐡𝐨𝐰𝐝𝐡𝐮𝐫𝐲, 𝐌𝐨𝐧𝐚𝐦𝐲 𝐙𝐚𝐦𝐚𝐧 𝐚𝐧𝐝 𝐌𝐨𝐡𝐚𝐦𝐦𝐞𝐝 𝐌𝐚𝐫𝐳𝐮𝐪 𝐑𝐚𝐡𝐦𝐚𝐧, these two days in Shanghai were not only about passing difficult exams. It was about all of that endless curiosity and years spent practicing in silence forging raw magic on the page. They answered some of the toughest questions in the world and did so with sheer grace.

Winning medals across the entire team, these six stars have given us all a reason to stand proud. A massive round of applause for 𝐓𝐞𝐚𝐦 𝐁𝐚𝐧𝐠𝐥𝐚𝐝𝐞𝐬𝐡! You’ve set a brilliant new standard, and we know this is only the beginning of your amazing stories.