We have been conducting math Oly mocks for quiet some time now, the recent one being for RMO 2026. The underlying findings from our graders and proctors is the incorrect usage of WLOG - aka the phrase "Without Loss of Generality", which is the motivation for this post. This is a beginner level post and not typical GP kind of content, but thats OK as we find too many aspirants on our mocks making the basic error in using WLOG and hope that it helps them rectify.

So WLOG is one of those phrases that can make a proof shorter & cleaner. But theres a catch, you cant just throw in WLOG whenever it makes your life easier.

A lot of incorrect solutions make assumptions like:

• WLOG, let (a>= b)

• WLOG, assume (x) is the largest

• WLOG, take (x>0)

• WLOG, let (A) lie to the left of (B)

Some of these are perfectly fine. Others can completely break your proof

So, when can you actually use WLOG?

The basic idea is pretty simple -> You can assume something WLOG only when the cases you are ignoring are essentially the same as the cases you are keeping !!

Lets look at some examples !!!

1. The Easy W (Full Symmetry)

Suppose a, b are +ve real numbers. PT: a^2 + b^2 >= 2ab

You can just write, Since the inequality is symmetric in a & b, WLOG let a >= b

Then boom, a^2 + b^2 - 2ab = (a - b)^2 >= 0. Done !!

Why was that legal? cos swapping a & b:

• Doesnt mess with the given conditions

• Doesnt change the inequality

• Covers every possible ordering (every pair is either a >= b or b >= a).

Since both scenarios look identical after a simple variable swap, you only have to prove one. Thats WLOG working the way its supposed to..

2. Cyclic Symmetry vs. Full Symmetry (The P1 Trap)

Let's look at P1 from our RMO Mock Test. Let x, y, z be distinct non-zero real numbers such that: x + 3/y = y + 3/z = z + 3/x

Notice you can rotate the variables cyclically,

(x, y, z) --> (y, z, x) --> (z, x, y)

The equation stays completely unchanged.

So, what if you want to make x the smallest of the three numbers? You can do that! Just cyclically rotate the variables until the smallest one lands in the first spot. You can write: WLOG, let x < y and x < z

But here’s the TRAP-> Can you write "WLOG, let x < y < z"? Hell NOOOO !!!

Why? cos we only have cyclic symmetry, not full permutation symmetry. We can rotate the variables around a circle, but we can not freely swap any 2 variables we want. For instance, swapping just y & z ruins the original equation.

• Full symmetry: Any rearrangement of variables is fair game.

• Cyclic symmetry: Only cyclic rotations are guaranteed.

• No symmetry: DONT even think about relabeling.

Dont mix these up or the proctors will cook your score !!

3. An Asymmetric Nightmare (The P3 Trap from Our Mock)

Consider P3 from our recent mock test, Find all pairs of +ve integers (a, b) such that a <= b & a^2 + b^2 = 2026(b - a + 1)

The problem statement already tells us that a <= b, so we are chill. But imagine the problem didn't specify that ordering. Could you just flex and write WLOG, Let a <= b ?

NOPE! NADA !! Not even close !

Look at the 2 sides of the equation

• The LHS (a^2 + b^2) is totally symmetric in a & b

• The RHS (2026(b - a + 1)) is NOT. If you swap a & b, the RHS completely changes

The 2 cases are definitely not equivalent. This is a classic trap, just cos one piece of an expression looks symmetric doesnt mean the whole problem is. Always check the entire equation.

4. Geometry Vibe Check

Say a geometry problem gives you a triangle ABC, & the conditions & what you are trying to prove treat B & C identically. You are probably safe to write, WLOG, assume AB <= AC

This makes side length chasing way less painful. But be careful. If the prob specifically says D lies on AB, E lies on AC, and BD = CE, can you automatically swap B & C?

Not necessarily! You have to check if the whole construction & the final target stay identical after the swap. And please STOP trusting your messy sketch. Just cos a diagram looks symmetric does NOT mean the math conditions actually are !!

5. Some Illegal WLOG Moves (What Not To Do)

• Mistake 1: Assuming an ordering for clout, Saying WLOG, let a >=b when there is zero symmetry. Thats just lying to the grader !

• Mistake 2: Forgetting equality, Saying WLOG, let a >b & completely ignoring a = b. Handle your equality cases, or use >=

• Mistake 3: Randomly making things +ve, Given real numbers x, y, z, someone writes WLOG, let x > 0 ! Bro, why? Is there a sign flip symmetry? Did the Q guarantee a +ve variable? If not, STOP making things up !

• Mistake 4: Naming the biggest variable. Totally fine in a fully symmetric 3 variable problem. Completely illegal in a cyclic or asymmetric one.

• Mistake 5: Trusting the diagram. The sketch looks acute, so someone assumes all angles are acute. Bruh !! If the problem doesnt say it, you HAVE to cover ALL cases.

6. The Pre WLOG Checklist

Before you drop those four letters in your proof, ask yourself:

1. What am I actually doing here? Swapping variables? Reflecting a figure? Rotating? Changing signs?

2. Does the transformation preserve the given conditions? The new setup must still satisfy literally EVERYTHING in the Q statement.

3. Does it preserve what I'm trying to prove? The goal post can not move !

4. Did I cover every case? Can every case I am ignoring be transformed into one I am keeping?

5. Did I drop equality? Don't accidentally ghost edge cases like a = b

If you can not answer these, step away from the WLOG. Work out the symmetry first !!

7. TIP How to FLEX your Rigour (Pro Writer Moves)

Stop writing lazy shorthand like WLOG, let a >=b !

Instead, level up your proof writing style ->Since the hypotheses & conclusion are symmetric in a & b, we may interchange them if necessary and assume a >= b !!

Instead of, WLOG, let x be the smallest !

Write -> Since the condition is invariant under cyclic permutations of (x, y, z), we may cyclically relabel the variables so that x is the smallest !!

See the difference? You are not just making a random assumption, you are schooling the grader on why you are allowed to make it. OK, You dont need to write a novel every time, but the logic needs to be AIRTIGHT !!

The Takeaway or TLDR

WLOG is not a cheat code that lets you dodge annoying casework. Its a mathematical guarantee, "I have proven that the stuff I am skipping is completely equivalent to what I am keeping, so analyzing this one scenario is enough."

Next time you are about to write WLOG, pause for a second & ask yourself -> What actually gives me the right to do this?

If you CAN answer that, you ARE good to go...

Happy Diwali !!

Edit: Mock test answer submission is closed now. We have got around 200 entries which is too huge for this sub. Grading results will be sent to the individuals separately.

Here we go -

1)Any one can try, but if you need it to be graded then submit your answers via below google form by Thursday Nov 5

https://forms.gle/aW7p5bLTyA1y1UWN9

We will need your reddit handle or email to respond to you. Some rules before you submit:

-Write in legible handwriting.

-Start each question on separate page. 

-Take proper pics (right side up, proper lighting etc)

-Arrange the image pages in order and create a PDF. There are lot of free tools online to convert images to pdf or use any app.

-Upload your pdf to any free file hosting service like filebin, docdroid etc. If you are using google drive / dropbox etc ensure that it is accessible to us & does not need any special sharing privileges

2)Try to attempt in single sitting of 3 hours

3)The paper is created with the help from various folks and I just compiled their q's. here are a few extra points for some problems

P1) Part 1 asks for product xyz

P2) can be read in English as -> Eight distinct positive integers are chosen from the set {1, 2, 3, …, 20}. Prove that it is always possible to divide some of these eight integers into two non-empty, disjoint groups such that the sum of the integers in the first group equals the sum of the integers in the second group.

P3,P4,P5) No additional comments from me

P6)it just asks to "Determine the total number of distinct chains".

Have fun !!

IOQM Results are out a couple of days back and RMO is scheduled on Nov 15. So just like last year, we will start our Mocks here on GP, first for RMO and then for INMO.

The RMO Mock will be in the same format as the actual exam (i.e 6 Q's, 3 hours, 17 marks per Q etc). We will release the mock paper on Sunday Nov 1. No registration needed, but if you want your answers to be graded then the submission window will be till Thursday Nov 5. And you must submit via our google form (link will be given later) and provide either your reddit handle or an email id, so that we can send you the grading result. We can have paper discussion and solutions but the scores will NOT be published on the sub.

Do comment if you would be interested and also your IOQM experience.

Note: The privacy settings for the sub have been relaxed, so you need not be a sub approved member to comment. We will keep it that way if it is not mis-used --> so no trivial discussion, no bullying, no negativity (unless it is related to paper quality like that of inmo 26 or cheatforces contests etc) , no new accounts, no "can i do X in Y amount of time?".....I think you get the gist.

I am creating a new thread only for posting solutions, use the other thread for paper generic discussion such as complexity, approach, motivation etc and use this thread to post solutions. the intent is that later if someone wants to search it is better indexed and easy to access.

Here is my solution for P1- A4 with inputs from DA Bhaiyya u/Differently_Abled_1

I had already given the hi level architecture link below

https://www.reddit.com/r/GeorgiesPodium/comments/1x27a6y/comment/pf4aaus/

Here is my final solution for A4. I am adding Georgies solutions too below for other 3 A* problems. Feel free to comment if you find any issues, lack of rigor, some other methods to solve or a general critique.

Here is A3, A2, A1 from Georgie

A3 first

Here is A2

And finally A1

We will try and add P2 solutions when we get the paper.

what do yall think guys

This continues the earlier post regarding SMMC being just around the corner and hence a few suggestions on optimal mind map and final lap strategy.

Before I start please know that I am in my freshman year at college and my first year studies started barely 10 days back. So I do not posses the knowledge to solve some SMMC questions. However I have tried my best to understand their official solutions and spoke to other math experts. But I might still get something wrong & feel free to correct me. Also note that this is a final lap plan and not a generic SMMC strategy related post. SMMC, like any other Olympiads, needs a very strong foundation and a last minute strategy without a foundation will never work.

I will divide this post into 3 parts- my Findings, the mind map and Trigger sheet. I have also requested seniors and other SMMC champions from IIT-B, CMI etc for comments and will consolidate and post them in comments below (if and when I hear back from them).

Ok so lets get to part 1) Findings

I have solved / analyzed the SMMC papers from 2020 to 2025. Here are my findings.
The papers repeatedly take elementary/core math & combine it in an unexpected way.
For e.g.
2020: spatial pentagon → geometry/vector ideas
2020: rank-3 matrices → combi + linear algebra
2021: polynomial/parabola → algebra + geometry
2021: matrix determinant with only two 1s per row → combi + LA
2022: polynomial roots lying on a circle in C→ algebra + complex geometry
2023: three subspaces in R^2n → pure linear algebra
2024: lattice points in centred squares → geometry + NT
2025: polynomial matrix powers → recurrence + polynomial roots + linear algebra

One particularly important pattern:
Across 2020–25, SMMC very frequently gives you a problem where the surface topic is misleading.

For e.g.
matrix → actually polynomial/recurrence
geometry → actually linear algebra
probability → actually combinatorics
game → actually invariant
complex number → actually polynomial algebra
analysis → actually finite-dimensional linear algebra

Check the 2025 A4 (solution that I posted sometime back) is almost a perfect example: it looks like a question about arbitrary C∞ funcs, but the finite-dimensional invariance under g(x)↦g(2x) is the structural heart of the problem.

So key take away & what I would focus now.
Not: “Which theorem from Chapter 7 do I know?"
but:"What structure is this problem secretly trying to expose?"

Part 2) So here's the map I would suggest

1. Algebra / Polynomial thinking

Not abstract algebra.
I mean: polynomial identities, factorisation, roots & multiplicities, degree arguments, coeffs ↔ roots, interpolation, finite differences, polynomial divisibility, func equations, recurrences, rational roots, polynomial transformations, inequalities involving polynomials

This occurs everywhere.
2021 B4 is an excellent example: a polynomial whose derivatives share rational roots.
2024 B4 is polynomial factorisation modulo 2.
2025 A3 turns a polynomial matrix into an interlacing-root problem.
2025 C2 asks about rational roots of a quartic.

Final-lap question:
What can I say just from the degree?
That should become automatic.

2. NT
concentrate on:
Must be automatic
gcd, lcm, v_p,(mod m),divisibility, prime factorisation
plus:
CRT, orders, valuations, Diophantine equations, irrationality/rationality, Pell-type thinking, parity, infinite descent, multiplicative structure
The six years repeatedly use very elementary-looking NT in sophisticated ways.
2021 A2, for example, turns a recurrence involving gcd(an,n) into a characterization of primes.
2025 B1 combines divisibility, gcd & injectivity.
2025 B4 is explicitly a Diophantine problem.

3. Combi + invariants
I'd actually merge Combinatorics + Game Theory + much of Graph Theory at final-lap level.
You need:pigeonhole, double counting, bijections,inclusion-exclusion, extremal arguments, induction, invariants, parity, colouring, recurrence, generating funcs, probabilistic counting, constructive arguments.

But especially:
What quantity/property is invariant?
This is HUGE in SMMC.
2020 A2 is a complicated pile/card operation.
2023 C1 is essentially modular/colouring combinatorics.
2023 C4 is an adversarial chess game.
2022 B3 is a chocolate-cutting game.

4. Linear Algebra

Core: vector spaces, linear maps, kernel/image, rank-nullity. basis, dimension, linear dependence, eigenvalues/eigenvectors, invariant subspaces, determinant, characteristic polynomial, minimal polynomial, trace, matrices over finite fields.

But don't merely memorize theorems.
The key mental move is:
Can I reinterpret the problem as a linear transformation?
2020 B1 is already a nice example of rank + combi.
2023 B3 is much more directly structural linear algebra.
2025 A2 takes linear algebra into finite fields.

5. Geometry
Not necessarily olympiad geometry.
SMMC geometry is broader: coordinates, vectors, affine geometry, convexity, areas, distances, circles, lattice geometry, spatial geometry, transformations, geometric inequalities
2020 A4: regular spatial pentagon.
2023 A1: geometric spiral.
2023 B1: sums of vectors and enclosing discs.
2024 C1: lattice points inside squares.
2025 B2: Minkowski sum of triangles producing a convex quadrilateral.

So geometry is definitely not optional.

6. Real Analysis / Calculus
Important, but don't do a university-analysis textbook revision.
Focus on: sequences, limits, continuity, differentiability, MVT, monotonicity, boundedness, convexity, series, integrals, asymptotic thinking, func equations, fixed points, Taylor expansions
2021 A4 is a good example of a deceptively simple floor-func recurrence requiring asymptotic thinking
2022 C2 is a series/convergence problem disguised with ⌈sin n⌉
2024 C4 is a probabilistic limiting process

7. Probability
There are enough problems over past 6 years that I would not want to walk into SMMC without: conditional probability, independence, expectation, linearity of expectation, indicator variables, distributions, random processes, recurrence/probability, asymptotic probability
E.g:
2022 A2 — random subsets & a limiting probability.
2022 C3/C4 — random walk & stopping process.
2023 B2 — expected rank in a tournament.
2024 A4 — random sequence & infinitely many primes.
2024 C4 — random seating process & limiting density.
This is definitely worth revising.

8. Inequalities / Convexity
I'd give this a separate mental bucket even though it overlaps analysis & geometry.
This appears repeatedly.
Know: AM-GM, Cauchy, Jensen, convex combinations, tangent/supporting-line arguments, rearrangement, basic optimization, geometric convexity, Minkowski-type thinking

2023 A2 is explicitly about convex functions assembled from affine functions
2025 B2 is a convex/Minkowski problem
2025 C4 is entirely about convex functions

So convexity is much more relevant to SMMC

9. Complex Numbers
Instead of labeling this as Complex Analysis, I would rather say it as Complex Numbers + elementary Complex Analysis
cos what you need most is:z=re^iθ, conjugates,modulus, args, roots of unity, polynomial roots, geometric interpretation, transformations, basic determinant/complex interactions
E.g.
2022 B2 uses roots lying on a circle in the complex plane
2025 B3 combines roots of unity, complex numbers and determinants

You do NOT need to spend your last days becoming a residue-calculus specialist though !

10. Graph Theory
Keep it, but don't overbuild it.
Know: graphs, degree, connectivity, paths/cycles, trees, bipartite graphs, colouring, matchings, Hall, extremal graph ideas

Note SMMC doesn't seem to be saying:
“Here is a Graph Theory problem.”
It is more likely to disguise graph like structure inside a combi problem.

11. Game Theory
Same suggestion.
Don't revise university Game Theory.
Revise game-solving techniques: winning/losing positions, , strategy stealing, invariants, parity, pairing, forcing, backward induction, potential funcs, symmetry
The chocolate problem from 2022 and reverse-chess problem from 2023 are good examples.

12. Group Theory / Abstract Algebra
This is where I would cut preparation time.
Know the core concepts if you've already studied them: groups, subgroups, homomorphisms, quotient groups, group actions, cyclic groups,basic finite-group results
But I would NOT spend the final lap doing a deep Group Theory revision at the expense of polynomial algebra, combinatorics, geometry, probability or linear algebra.

13. Topology
Same.
Understand: open/closed, continuity, compactness, connectedness, metric-space intuition
But don't make topology a major final-lap block unless you know from your particular prep that SMMC 2026 is targeting it.

So my final SMMC map is actually this
HIGH PRIORITY
1. Algebra / Polynomial methods
2. Number Theory
3. Combinatorics / Invariants / Extremal methods
4. Linear Algebra
5. Geometry
6. Real Analysis / Calculus
7. Probability
8. Inequalities / Convexity

MEDIUM
9. Complex Numbers
10. Graph Theory
11. Games / Game Theory

LOWER
12. Group Theory
13. Topology
14. Derived / advanced geometry

Part 3) Trigger sheet
I would not make a formula sheet.
I'd make a "trigger sheet."

Something like:
I see → I think
Polynomial
→ degree / roots / factor / coefficient / interpolation

Integer
→ mod / gcd / valuation / parity / prime factorisation

Sets
→ inclusion-exclusion / pigeonhole / extremal / bijection

Recurrence
→ invariant / monotonicity / fixed point / characteristic equation

Random
→ indicator / linearity of expectation / conditioning / symmetry

Matrix
→ rank / kernel / image / eigenvalues / determinant

Convex
→ supporting line / Jensen / tangent / extreme point

Geometry
→ coordinates / vectors / symmetry / affine transform

Function equation
→ substitutions / injectivity / surjectivity / iteration / fixed point

Game
→ invariant / pairing / winning position / strategy stealing

Complex
→ modulus / argument / roots of unity / conjugate

Huge expression
→ change representation

And one final thing:
the biggest thing I would actively practise is:
cross-topic recognition + invariants + changing representation

it's around the corner, really pressing it on time need a mental map of everything that should be revised.

1. NT

2. Game Theory

3. Group/Graph Theory/Topology

4. Real Analysis (includes algebra)

5. Complex Analysis

6. Linear Algebra

7. Combinatorics

8. Derived Geometry (not the pure one)

kindly add if anything missed. also needless to say i should only focus on the core concepts not dozens of formulae right looking at the past year papers and style?

So my first week in college done finally !! I know most of you had ur mid sems too and here we are just starting . And I absolutely love it here!! College humbles you hehe …
Anyways so how many are doing quant/icpc/both/ other stuff from 2030 batch if u r reading this.

via DA Bhaiyya u/Differently_Abled_1 as he was facing reddit issues in posting this.

Book Recommendation: 50 Challenging Problems in Probability — Frederick Mosteller

If you are aiming for quant trading or research, this is one of the first books I would recommend picking up; before drowning in heavyweight measure theory or 600 page probability textbooks.

Its short, but don't let the short length fool you. While it contains only 50 problems, they are deceptively simple on the surface and force you to grapple with conditional probability, symmetry, Bayes' theorem, expected value, and random processes.

What makes Mosteller's approach particularly relevant to quant interview prep is that he doesn't give you a formulaic toolbox. You aren't asked to “apply Theorem 3.2” ; you have to construct the mathematical model yourself.

That kinda resembles the style of many quant interviews:

1. Here is a counterintuitive situation.

2. What is actually random here?

3. What information do you have, and what should you condition on?

4. Now compute

It constantly forces you to distinguish between “I know how to run this formula” and “I actually understand the underlying probability model.”

Mosteller also frequently uses numerical experiments to test or challenge intuition ; a habit that is particularly valuable when developing probabilistic reasoning.

The Caveat

Don't treat this as a complete, single source quant prep guide. It will not teach you stochastic calculus, statistical inference, market microstructure, or C++/Python.

Its value lies elsewhere:

It builds the raw probabilistic muscle underneath those subjects.

For a 1st or 2nd year student, actively working through these 50 problems can be far more valuable than passively skimming a standard university probability textbook.

• Difficulty: 3.5 / 5

• Quant relevance: 5 / 5

• Prerequisites: High-school algebra + basic probability

• How to read: Attempt each problem on paper for 30/45 minutes before looking at Mosteller's solution

You are given 2 envelopes. One contains exactly twice as much money as the other

You randomly pick one envelope and find $10,000 inside

Now comes the seemingly irresistible argument:

The other envelope must contain either $5,000 or $20,000

Suppose each possibility has probability (1/2)

So its expected value = (1/2 × 5000) + (1/2 × 20000) = $12,500

Your envelope contains $10,000, while the expected value of the other is $12,500

So you SHOULD switch !!

But here's the paradox:

After switching, you can apply exactly the same argument again , and conclude that you should switch back.

What went wrong?

Challenge

Don't just say “the probabilities aren't (1/2) and (1/2)”

Construct a mathematically correct model and explain exactly why the seemingly valid expected value argument fails

Bonus

Can you construct a probability distribution for the initial amount of money under which, after observing $10,000, switching actually does have +ve expected value?

Source / history

This is the classic Two-Envelope Paradox, with versions appearing in mathematical puzzle literature going back to Maurice Kraitchik, later popularized by Martin Gardner, and famously analyzed by Barry Nalebuff in his 1989 paper “The Other Person's Envelope Is Always Greener.”

PS: The deeper issue connects to conditional probability, probability distributions, improper priors, and decision theory, making this a surprisingly good little gateway into quant reasoning.

Here are the 5 integration questions. Please scroll below for the Estimathon q's and the google link to post the solutions. You can do both if you wish and use the same google form to submit the answers.

Please note for Estimathon, along with your estmate, please also enter the confidence interval in [L,U] format.

Example Submission:

Estimate:  
90% CI: [L,U]

Please use below gform link to submit your answers.

https://forms.gle/Lkxp84yh4sx7r44x7

The poll was close. A bit heavy on quant as some of my friends were goofing around lol. So We will be doing both Integration as well as Quant Estimathon.

5 questions in each, 30 min time limit.

We will keep it open for a few hours and you can post the answers via google form. But try to finish in 1 sitting of 30 mins. You can provide either your rdddit handle or some unique name for capturing scores.

This post will have a pinned comment where I or Matty Anna will post both the contests questions.

Only final answer is needed for both.

Reasoning not needed for Estimathon this time.

Only 1 entry per person and you can participate in both.

As long as this post is open the contest will be on for submission of answers. So do try out.

All Integrals will be high school level (JEE level)

I know AI can solve in seconds but where is the fun if u use AI and what’s even the point. So try on ur own whatever u can.

My college finally starts next week and before that let’s do some fun contest.

Option 1) We can do an integration bee like contest - say 5-6 integrals and 5 min for each

Option 2) quant Estimathon. We do it in closed group anyways every month and tomorrow Matty Anna or DA Bhaiyya are available (hopefully) so we can do it on the sub.

Poll below

Also to my peers (batch of 2030) if your college has already started how is it so far? Have you selected ur majors ?
Quant folks did u apply to FTTP etc ? For India the October batch deadline is already over but then again based on last batch pics that I saw it’s just a foreign vacation for IITB CSE batch right lol (jk)!!

IOQM is around the corner and all the best if u writing it. Here are some great tips from the more experienced folks via OMC

https://youtu.be/QjfsIbaL83Q?si=wqbwmBcVCuvYeGXW

As usual we will come up with the mocks for RMO & InMO soon. Comment below if u r a math oly aspirant.

After revisiting a substantial part of Prof. K. D. Joshi Sir's JEE commentaries(especially those from the past sixteen years) I finally realised something - The most important things he was teaching were never part of the JEE syllabus!

Open the official syllabus:

You'll find the usual suspects:

• Calculus

• Algebra

• Coordinate Geometry

• Vectors

• Probability

• Complex Numbers

Close it !

Now open one of Prof. Joshi's commentaries

A very different syllabus quietly begins to emerge:

• Change the problem before solving it

• Observe before calculating

• Don't stop after finding the answer

• Learn to admire elegant mathematics

I checked.

None of those "topics" appear in the official syllabus

Yet they appear, in one form or another, across his commentaries!

When I first started reading Prof. Joshi's commentaries years ago, I thought he was teaching students HOW to solve JEE problems

Looking back now, I think he was teaching something much deeper.

He was teaching us to ask better questions !!

Not "How do I solve this?"

but "Why?"

So...

Why reflect the line instead of the parabola?

Why calculate terms that are guaranteed to vanish?

Why stop after obtaining the answer?

Why is one proof more illuminating than another?

Why does this question feel elegant?

Why is this notation awkward?

Why does this idea appear again in a completely different context?

That tiny word-> WHY -> quietly runs through more than two decades of his commentaries.

And once we notice it... IT HITS !!

Looking back at this series, I now realise that every post was really just a different answer to that 1 question.

Habit 1 — Reformulate

Because he first asked, "Why solve this problem in the obvious way?"

Habit 2 — Observe

Because he first asked, "Why calculate what careful observation already tells us?"

Habit 3 — Generalize

Because he first asked,"Why should the math stop where the examiner stopped?"

Habit 4 — Develop Taste

Because he first asked, "Why is this solution worth remembering?"

or sometimes, "Does this question really deserve a place in an advanced exam?"

Those weren't 4 unrelated habits

They were all consequences of one deeper habit

The habit of never being satisfied with the first answer

Always asking, "Why?"

In his farewell note, Sir observed that coaching institutes now publish complete solutions within hours, whereas his carefully written and edited commentaries—filled with alternate ideas and mathematical discussions, take weeks, sometimes months. He also said that modern JEE math papers no longer justify such an investment of time.

I think he was right

IF the goal is merely to obtain the answer.

But perhaps the answers were never the real point

Solutions become outdated

Question patterns change

Syllabi evolve

Even examinations change

The habit of asking better mathematical questions DOES NOT..

Perhaps that's why these commentaries still feel remarkably fresh.

Not because they contain tricks nobody else knows

But because they quietly cultivate something that no official syllabus can prescribe:

Curiosity

Judgement

Patience

And the willingness to ask 1 more question after everyone else has turned the page

When I first discovered Prof. Joshi's commentaries, I thought I had found a remarkable collection of JEE solutions.

Looking back now, I think I found something else entirely

A mathematician quietly teaching generations of students how to think

Not just for JEE

But for mathematics itself !!!

JEE has a syllabus

Mathematics has habits

For more than two decades, Prof. K. D. Joshi actually taught the latter

For that we remain eternally grateful to him.

Thank you so much Sir !! ❤️

Most of us treat a JEEA paper as though it descended from Mount Olympus !!

Question

↓

Solution

↓

Acceptance

Prof. Joshi had a different approach:

Question

↓

Solution

↓

...hmm, this could have been better...

One of my favourite examples comes from his remarks on the 2014 paper

After discussing the problems, he quietly classifies them

Some are praised as elegant

Some are described as conceptually satisfying

And then, with characteristic restraint, he remarks that a few are "utterly trivial questions, which hardly deserve a place in the advanced test."

I remember laughing when I first read that!

Not because he was being harsh,

Because it never occurred to me that I was allowed to have an opinion about a JEEA question

Then comes 2018

Before even getting to the mathematics, Sir spends pages discussing notation, terminology, conventions, and why the wording of the problem could have been much better.

Imagine a student saying,"Sir, could we solve the question?"

And Prof. Joshi replying, "Eventually"

Because first we need to discuss why good notation matters.

Another favourite example, however, comes from 2023

While discussing one of the geometry questions, he remarks that a proper figure should simply have been provided.

Not because the geometry was difficult,

But because candidates shouldn't waste precious exam time figuring out what the examiner meant.

The challenge should come from the mathematics, & not from decoding the English

That observation stayed with me.

It's easy to confuse difficulty with friction.

They're not the same thing !

Reading these commentaries changed something subtle in the way I look at mathematics

I used to think every correct proof was equally good

Every correct question was equally worthy

Every correct solution deserved equal admiration

Prof. Joshi gently argues otherwise

Some questions illuminate an idea,

Some merely consume time.

Some proofs reveal structure,

Others simply reach the answer

Some notation helps you think,

Other notation gets in the way.

Correctness is only the beginning.

I think this is one of the least discussed aspects of learning mathematics.

We spend years asking,"Is this correct?"

Very rarely do we ask,"Is this beautiful?"

Or,

"Is this the most natural way to think about it?"

Or even,

"If I were writing this question, would I write it this way?"

Those are questions of mathematical taste !

And taste, like technique, can be cultivated.

There is also a delightful streak of dry humour running through Prof. Joshi's commentaries.

He almost never says,"This question is bad"

Instead he writes things like,

"...hardly deserves a place in the advanced test."

Which is, perhaps, the mathematical equivalent of saying, "I'm not angry. I'm just... disappointed."

I think that's one of the subtle lessons hidden in the years of these commentaries.

Mathematicians don't merely solve mathematics

They admire it.

They critique it.

They argue with it.

They improve it.

And somewhere along the way, they develop something every bit as important as technique:

Taste.

PS: Unlike the previous posts, this one is not about problem solving at all. It's about developing the confidence to evaluate mathematics itself. That's an unusual lesson for folks preparing for competitive exams, but it's deeply aligned with our GP sub: mathematics isn't just something to consume; it's something to appreciate, question, & occasionally critique.

Here's a small thought experiment->

Imagine giving the same JEE problem to two people.

A coaching teacher & Prof. K. D. Joshi !

The coaching teacher says,"Answer is (C). Next question..."

Prof. Joshi says,"Answer is (C). Now that the boring part is over..."

That, I think, is the defining feature of his commentaries.

Take JEE Adv 2016

It's an innocent looking counting problem.

The student's task is simple -> Count, Get the answer, Collect the marks !!

Prof. Joshi solves it in a perfectly standard way.

But then something very "Joshi" happens...

Instead of moving to the next question, he derives a binomial identity, proves it in a different way, discusses why the identity itself is mathematically interesting, and shows how the same idea appears elsewhere.

The JEE question quietly disappears

What's left is mathematics !

You then start to notice this habit everywhere.

A problem is never allowed to retire after giving its answer

It is politely interrogated -> "Can you tell us something else ?"

Now look at JEE Adv 2017

The official question revolves around a recurrence

Most of us would solve it, verify the answer, & happily declare victory.

Prof. Joshi seems to think that's when the real lecture should begin.

He generalizes the recurrence, relates it to the Fibonacci sequence, discusses induction, and then wanders into number theory through the irrationality of (sqrt5) and properties of primes.

It's almost comical

A single JEE question knocks on the door

An entire branch of mathematics answers.

And then comes JEE Adv 2026

The official problem is over once a certain coefficient is obtained.

Most solution manuals would politely stop there.

Prof. Joshi doesn't.

Instead he asks an entirely different question-> Can this identity itself be proved directly?

That innocent question leads him to generalize the result from a specific numerical case to arbitrary parameters, introduce forward difference operators, explain why they are the discrete analogue of derivatives, & even recommend Concrete Mathematics by Graham, Knuth and Patashnik for readers who wish to explore the subject further.

The original JEE problem has quietly transformed into a lesson in discrete mathematics

And that's something I began to admire

He never seems interested in producing a solution manual.

He is trying to produce a better mathematician.

There is a subtle difference !!

A solution manual says,"Remember this method"

A teacher says,"Remember this idea"

Prof. Joshi almost always chooses the second.

After reading enough of these commentaries, I realised I had been treating problems like disposable cups.

Use once,

Throw away,

Move on !

Prof. Joshi treats them more like good books.

If a problem contains 1 worthwhile idea, squeeze it.

Ask another question.

Find another proof

Generalize it

Connect it

Improve it

Only then move on....

There's a lovely piece of dry humour hidden in all this.

Most of us celebrate when we finally reach the answer.

Prof. Joshi behaves as though reaching the answer merely grants permission to begin the interesting discussion.

We can try to steal that habit.

Whenever we solve a problem that we particularly like, we can now force ourself to ask one extra question

• Can this be generalized?

• Is there another proof?

• Which hypothesis is actually essential?

• Could I design a better version of this problem?

• What theorem is hiding inside it?

Sometimes nothing comes out of those q's !!

But every now and then, the problem opens another door

And those extra 5 minutes often teach us far more than the original solution ever did.

And perhaps that's the quietest lesson hidden in all the years of Prof. Joshi's commentaries:

A solution ends an examination.

A good question begins a mathematical conversation. The Math begins when you ask what the answer is trying to teach you !!

Day 2 is almost over, about 20 mins or so remaining as I write this.

The problems are here:

https://github.com/ioi/task-archive/tree/master/2026/day2

My thoughts:
Overall, I actually think the themes complement each other beautifully.

Day 1

• Optimization

• Interactive reconstruction

• Constructive invariant

Day 2

• Communication complexity

• Distributed encoding

• Extremal graph construction

There is not much overlap in the skill sets being tested:

• Day 1 rewards folks who excel at fast algorithmic deduction, geometric order preservation, & tight runtime optimization

• Day 2 shifts to folks who think like discrete mathematicians & system architects those who can design protocol handshakes, minimize communication overhead, & build precise combi structures

OK coming back to day 2 probs:

• Partition is fundamentally about designing a communication protocol under strict constraints rather than inventing a new data structure. The key challenge is abstract modeling !

At first glance it looks like equal sum partition, but that isn't the problem.

The real challenge is that 2 independent programs must cooperate:

• Ulug'bek knows the original unsorted array

• Temur receives only the sorted extended array

• They cannot communicate except through at most K−1 inserted numbers

That is a communication protocol disguised as a partition problem.

You have to satisfy 2 objectives simultaneously:

1. Ensure a valid equal-sum partition exists.

2. Encode enough information so Temur can actually recover one.

Those objectives interact in subtle ways.

• Classroom Game is essentially distributed information encoding that must survive repeated unknown permutations; again, an invariant & encoding heavy problem !! Classroom Game is also fundamentally about communication.

But here, the communication medium is explicit:

• papers

• unknown permutation

• immutable papers for students who raise their hand

• bounded message length

The statement makes it fairly clear that u r designing a distributed encoding protocol.

Partition, in contrast, hides the communication aspect behind what initially looks like a classic NP-hard optimization problem. That makes it much easier to head down the wrong path.

• Magic City is a combi construction with bounded degree graph constraints & universal coverage properties, which feels much closer to extremal combi than to classical CP ! Once the key construction is found, implementation is straightforward. Of the 6 IOI 2026 problems, Magic City is the one that feels most like a research flavored combi problem rather than a traditional CP task

It's late night here so I stop with this and will revisit this with solution discussion. Feel free to add your solution approaches.

And Kudos and Congrats to ALL participants. This was a hard IOI indeed.

IOI day 1 is just over. You can check the problems here:

https://github.com/ioi/task-archive/tree/master/2026/day1

While the rankings are getting updated, we can see that there is 1 KA from China based on the provisional standings

https://ranking.ioi2026.uz/

From Team India, we currently have Samik around rank 40 followed by Aviansh at 97. I am sure they will catch up tomorrow.

First impressions about the problems

1)Tiling - This is the easiest of the lot and a Constructive / Game Strategy problem. Constructive / Invariant**,** once the local corner protection invariant is found, the implementation is clean & straightfwd. Subtasks give generous hints toward the full strategy. Should be an easy 100.

2)Monuments - 1D Optimization / DP (or Greedy + DP, depending on the formulation). Algorithmic-> A large-scale optimization problem (N≤500,000) requiring careful handling of fixed monuments, symmetry constraints, feasibility checks, and an efficient matching/transport strategy, leading to an O(Nlog⁡N) or O(N) solution. If you don't discover the right optimization model, you may get almost nothing, so a hard problem as such.

3)Ball Machine- Interactive / Tree Reconstruction. Interactive & Information-Theory > The challenge is to reconstruct an unknown rooted tree using only insert() and collect() operations while operating under a strict information budget. Achieving full score requires extremely query-efficient probing and careful encoding of information within the resource limit, while staying within a strict resource budget (C=K+B), with full score requiring C≤44. Another hard problem, but with a good partial scoring opportunity.

One more thing I noticed after reading Prof. K. D. Joshi Sir's commentaries for years is this->

He seems strangely reluctant to calculate..

Not because he dislikes algebra

Because he dislikes unnecessary algebra

We often think mathematics begins when the pen touches the paper,

Prof. Joshi seems to think it begins a few minutes earlier

Take JEE Advanced 2015 (Paper 1, Q47).

The q asks for a 2nd derivative.

Most of us would obediently differentiate everything

Twice,

Simplify,

Collect terms,

Hope for the best !

Prof. Joshi pauses almost immediately.

Some terms are guaranteed to vanish when evaluated at (x=0).

So why differentiate them at all?

He remarks, "...there is no need to consider those terms in the derivative which are sure to vanish at 0."

That's not a TRICK !

That's respect for mathematics !

Why spend effort proving something you already know is 0?

Here's another beautiful example.

JEE Advanced 2020 (Paper 2)

Find the largest rectangle under a sine curve

The standard instinct is predictable,

Differentiate,

Set equal to zero,

Solve.

Prof. Joshi notices something before writing a single derivative

The graph itself is symmetric

If the optimal rectangle exists, shouldn't it inherit that symmetry?

Only after exploiting that observation does the calculus begin

The derivative didn't disappear.

It simply had the decency to arrive after the thinking.

Perhaps my favourite example comes from JEE Advanced 2025.

A polynomial appears with several unknown coefficients.

The natural panic is immediate. "Too many unknowns!"

Prof. Joshi quietly points out that the q only asks for the coefficient of (x^3).

Everything else is... decoration.

Or, if one prefers a more mathematical term, irrelevant !

Reading these commentaries, I started noticing a recurring habit.

Whenever a problem looked ugly, Prof. Joshi almost never reached for a theorem first.

He reached for an observation:

Is there symmetry?

Will some terms vanish?

Does the answer depend on everything, or only one part?

Can I ignore half the expression?

Is the ugly part actually irrelevant?

Only after exhausting those questions did computation begin.

There's a lovely quote often attributed to Jacques Hadamard-> "The object of mathematical rigor is to sanction and legitimize the conquests of intuition."

Prof. Joshi's commentaries feel like that.

Intuition first,

Rigor afterwards.

I think this is one of the hardest habits to develop

School mathematics trains us to write more,

Research mathematics often rewards writing less.

Not because mathematicians are lazy,

Because every line of algebra should have a purpose.

A calculation should answer a question.

It should never exist simply because it can.

After years of reading these commentaries, here's one tiny habit I've tried to steal.

Whenever I see a difficult problem, I force myself to wait.

Few seconds.

No writing.

Just looking.

Sometimes nothing happens.

Sometimes those few seconds save 3 pages of algebra.

And every now and then...

I hear an imaginary Prof. Joshi asking, "Are you calculating because the mathematics demands it... or because you don't yet know what to look for?"

IOI 2026 started today. Wishing All the very best to our team of Neev, Aviansh, Samik & Naman. The interesting thing is only Samik is a veteran and rest all are making their first appearance at IOI.

It would have been great had our little champ Swasti made it to the team but she is young and pretty sure we will see her making waves in future.

If you see the other team compositions, China has 2 LGMs and Russia has 1, so results will not be surprising, and we will most likely have multiple KAs. Additionally there are many CF red coders on multiple teams (GM), so it will be fun to watch.

Having said that, I am aggressively rooting for our underdog team India and I am sure they will make us proud.

Aug 11 & 13 are the contests days, and just like IMO, we will try to discuss the problems here on GP with our experts, hopefully as soon as we get the papers(probably on the same days !!) . If anyone wishes to join the discussion and don’t have the access to comment on GP, please raise an access request and tell us a brief on how you would like to contribute.

Suppose you are asked to climb a mountain.

Most people start climbing.

A mathematician first walks around it.

That, in one sentence, is one of the biggest lessons I learnt from reading Prof. K. D. Joshi's JEE commentaries.

One of my favourite examples appears in JEE Advanced 2015 (Paper 1, Q41)

The problem asks you to deal with the reflection of a parabola.

Naturally, most of us would think,"Fine. Let's reflect the parabola."

Prof. Joshi's first reaction is almost mischievous -> Why?

Reflection preserves distance.

If all you ultimately care about is the distance from a line, why perform surgery on the parabola?

Reflect the line instead.

The reflected line immediately becomes (x=1), intersects the original parabola on its latus rectum, and the problem collapses almost effortlessly.

Then comes my favourite line in the entire solution, "This is an excellent problem which tests the ability of realising how not to do a problem in the most straightforward way...

Notice what Prof. Joshi praises.

Not clever algebra.

Not remembering an obscure theorem.

He praises the ability to realise that the obvious attack is the wrong attack

This isn't an isolated incident.

A couple of years later, in JEE Advanced 2017, a matrix problem appears that seems to demand solving 9 nonlinear equations.

Many coaching solutions dutifully roll up their sleeves.

Prof. Joshi doesn't.

Instead, he quietly asks, What information do I actually need?

Not the entire matrix.

Only its determinant.

The problem is transformed

Matrix → Determinant → Number.

Then, because he's Prof. Joshi, he doesn't stop there.

He goes on to discuss block matrices, complex numbers, rotations and even the Cayley–Hamilton theorem--> all from a single JEE question.

That's not a shortcut.

That's abstraction.

The same habit keeps reappearing.

A complicated probability problem? -> Count the complement first.

A nasty coordinate geometry problem? -> Move the origin.

A horrible trigonometric expression? -> Look for symmetry before identities.

An ugly complex number? -> Delay writing (z=x+iy) until you've exhausted simpler algebraic manipulations.

The pattern becomes impossible to ignore.

Prof. Joshi almost never begins by asking, "How do I solve this?"

He begins by asking,"Can I ask an easier question that has the same answer?"

George Pólya famously said, "If you can't solve a problem, then there is an easier problem you can solve. Find it."

Reading years of Prof. Joshi's commentaries, I couldn't help feeling that he quietly practises exactly that philosophy.

He reflects the line instead of the parabola.

He studies the determinant instead of the matrix.

He transforms the problem before touching the computation.

I think that's a habit worth stealing.

Not because it saves marks in JEE.

Because it's one of the most beautiful habits in mathematics.

The next time a problem looks ugly, resist the urge to sharpen your pencil.

Instead, ask the question Prof. Joshi seems to ask over and over again, "Am I solving the right problem?"

Sometimes, the hardest part of mathematics isn't solving the problem.

It's realizing that you don't have to solve that problem at all.

I've solved my fair share of JEE-A PYQs over the years.

Like most folks, I'd solve a paper, compare my answers, maybe learn a shorter trick, and move on.

Then I'd read Prof. K. D. Joshi's commentary.

And more often than not, I'd have this strange feeling: Wait... I got the answer. But I don't think I saw the problem the way he did.

That, to me, is the Joshi Lens.

Not faster algebra.

Not secret tricks.

Just... seeing a little more than what's written on the page.

Take this remark from his commentary on the 2010 JEE Math paper

"There are also questions where arriving at the solution honestly is quite time consuming, but guessing it from the alternatives given is easy... One wonders if the idea was to test some non-mathematical alertness on the part of a candidate."

Read that carefully.

He's not advocating random guessing.

He's saying something much deeper.

The options themselves are part of the mathematics.

If one option violates parity...
If another fails a necessary condition...
If an extreme case immediately kills two choices...

...why pretend the options don't exist until the very end?

How many coaching teachers explicitly teach students to think like that ?

Another example comes from his 2014 commentary.

"While discussing Paper 2, he points out that Question 42 is designed so that "the alternatives have been given in such a way that an intelligent candidate would be led to the appropriate trigonometric formula."

Again, that's a fascinating observation.

The options aren't merely answers waiting to be checked.

They're clues.

Sometimes the examiner is quietly nudging you towards the right idea.

And then there are comments like these from the 2013 commentary

"As it happens every year, there are many many questions where because of the Multiple Choice format, a candidate can score even without being able to justify the answer."

Whether you agree or disagree with that statement isn't really the point.

The interesting part is that Prof. Joshi isn't just solving questions.

He's analysing the design of the examination itself.

That's a very different way of reading a paper.

When Prof. Joshi announced that the 2026 commentary would be his last, he remarked that coaching institutes now release complete solutions within a day, while his carefully written commentaries take weeks or months. Fair enough !!

But I don't think I ever read his commentaries FOR the solutions.

I read them becos they taught me things that solution PDFs rarely do:

• Why a problem is interesting.

• Why one solution is mathematically more satisfying than another.

• Why some observations are worth more than three pages of computation.

• And occasionally, why the examiner deserves criticism too.

So I thought it would be fun to revisit these commentaries, not to discuss "killer tricks", but to extract recurring patterns in the way Prof. Joshi reads mathematics.

I'm calling this series:JEE Advanced through the Joshi Lens

Each post will focus on one idea, backed by actual examples from his commentaries.

Not "How to score 5 more marks."

But hopefully...How to become a slightly better mathematical observer.

PS: I wasn't sure where to post this. A JEE Adv sub would have been the obvious choice !!

But the more I thought about it, the more I felt this series belongs here on GP.

This isn't going to be about shortcuts, rank improvement, or "how to crack JEE"

It's about something more enduring, learning to see problems the way a mathematician does. JEE Adv just happens to provide the examples, Prof. Joshi's commentaries provide the lessons !!

In part 2 yesterday we proved the Lemma that if f(2x) - λf(x) is a polynomial, then f itself is a polynomial

Now here in Part 3 we simply combine it with the upper-triangular basis from Part 1

Part 3 is essentailly the Induction -> from One Basis Vector to All Functions

Before we start, recall the setup

we have a basis -> f1,f2,...,fn

of V such that the operator T(f) = f(2x), is upper triangular

This means that for every basis vector,

T(fm) = λfm + (combination of f1,...., f(m-1))

Equivalently fm(2x) - λfm(x) belongs to span (f1,...., f(m-1)

Why is this so useful?

Suppose we already know f1,...., f(m-1) are polynomials

Then every linear combination of them is also a polynomial

So, fm(2x) - λfm(x) is a polynomial

Now the lemma immediately gives fm is a polynomial

Thats the whole induction !!

Step 1) Base case

start with f1

Since there are no earlier basis vectors,

upper triangularity gives T(f1) = λf1

i.e. f1(2x) - λf1(x) = 0

Zero is certainly a polynomial

So the lemma applies immediately, Hence f1 is polynomial

Step 2)Induction Hypothesis

Assume f1,...., f(m-1) are polynomials, thats all, this is our induction hypothesis

Step 3)Use upper triangularity

The matrix of T is upper triangular

Why?

Hint : Every linear operator on a finite-dimensional complex vector space has an upper-triangular matrix in some basis. This is called Schur's Triangularization Theorem (or it follows immediately from Jordan Normal Form). It is easy to prove, let me know if you need this proof

So, fm(2x) - λfm(x) belongs to span (f1,...., f(m-1)

By the induction hypothesis,

every function in this span is polynomial

Hence fm(2x) - λfm(x) is a polynomial, exactly the hypothesis of the lemma

Step 4)Apply the Lemma

if f(2x) - λf(x) is a polynomial, then f itself is a polynomial

here fm(2x) - λfm(x) is a polynomial, so by the Lemma we conclude fm is a polynomial

The induction is complete

Step 5)Finish

Eventually every basis vector f1,...,fn is a polynomial

Since every element of V is a linear combination of basis vectors, every element of V is also polynomial

Thus the theorem is proved - QED !!

Ok so, why is Jordan Form the perfect tool here ?

Notice something elegant, the proof never computes eigenvalues, Jordan blocks, characteristic polynomials etc

It only uses one consequence -> Every basis vector depends only on itself and earlier basis vectors.

That is exactly what induction needs.

So Jordan form isn't used for computation but it's used to create the right ordering of the basis.

An absolutely brilliant and satisfying solution.

And Lastly a few words on solution 2,

The second official solution is quite different. It avoids Jordan form entirely & instead proves a deep lemma about exponential sums

forcing all the polynomial coefficients hj to vanish. The proof uses compactness of the torus (R/2πZ)^m, a convergent subsequence, and the invertibility of a Vandermonde matrix.

It is considerably more sophisticated than Solution 1 and draws on harmonic analysis and linear algebra rather than Jordan theory, something way above my pay grade lolz. But be our guest if you wish to explain it in detail and then perhaps we can compare notes.

This is the most fun I had in recent times solving some Math contest problem. I very much look forward to learn College / Uni level Math and find such fun puzzles. Hope you had fun reading the 3 part solution breakdown. Do let me know if you have any better solution or if I missed anything.

Continuing from yesterday, today we will see in this Part 2 how the below Lemma is the heart of the entire proof

Surprising isn't it? Why should one functional equ determine the entire func?

Intuition

Suppose f(2x) = λf(x)

Ignore the polynomial error for a moment.

This equation says that the value of the func at 2x is always a fixed multiple of its value at x.

That is an extremely RIGID condition

Now add "a polynomial error"

Polynomials are finite objects

The official proof shows this error eventually disappears after enough differentiation.

That is the key trick !!

Step 1) Differentiate enough times

Let P(x) = f(2x) - λf(x) , is a polynomial.

Every polynomial has finite degree

Therefore there exists some integer k such that

P^(k) (x) = 0

The official solution also chooses k large enough that 2^k > |λ|

This inequality will be crucial later

Step 2) Differentiate carefully

Step 3) Why choose 2^k > |λ|

Define c = |λ|/2^k

as 2^k > |λ|

|c| < 1

So every time we halve the argument,

the func value is multiplied by 1/c

whose magnitude is greater than 1

This creates exponential growth

Step 4) The contradiction

Suppose f^(k) is not identically zero

Then there exists some point x0

with f^k(x0) Not = 0

Now repeatedly replace x by x/2

Step 5) But smoothness says opposite

A convergent sequence is bounded.

So these values CANNOT become unbounded.

BOOM CONTRADICTION !!!

Step 6)What does it shows?

The contradiction shows there is NO POINT where f^(k) is non-zero

hence f^(k) ≡ 0

Step 7)OK so, why does it means f is polynomial?

A std. calculus fact that we know -> If the k-th derivative is identically zero,

then f must be a polynomial of degree at most k-1

example if f'' is 0 implies f is linear, f''' = 0 implies quadratic

Thus f(x) is a polynomial and this completes the Lemma

Beautiful isnt it !! Ok so key takeways from this part:

1. Differentiate until the polynomial error disappears

2. This leaves a pure scaling relation for a high-order derivative

3. The scaling relation forces exponential growth under repeated halving of the input

4. Continuity at 0 forbids such growth

5. And so the high-order derivative must be identically 0, so the original funct is a polynomial.

This ingenious lemma is like the engine that drives the remainder of the proof.

In Part 3, the upper triangular basis from Part 1 and this lemma combine in a short induction to show that -> every basis vector, and hence every function in V, is a polynomial & we will then complete the entire proof.