Solving the hardest question of a British Mathematical Olympiad

Thanks to Nahian for the suggestion! This is a difficult factorial problem.

British Mathematical Olympiad 2002-2003 problem 5
A Mathematical Olympiad Primer Paperback – 1 Aug. 2011
Math Forum post
Math StackExchange post

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[MindYourDecisions]

I am always excited and nervous when I post a video like this! One the one hand, I am thrilled to share a challenging mathematical proof. On the flip side, proofs require perfection and it is very challenging to make a video with no errors. You guys have great attention to detail, so if you see any mistakes, let me know! For major mistakes I will repost a corrected video; for minor mistakes/typos I will leave a note in a comment. Hope you enjoyed this problem!

[sounak]

"Pause the video if you would like to give this problem a try"
Thanks for your concern, I'll just skip 10 seconds instead.

[samuelking]

I understand the steps. What I don’t understand is how *anyone* was able to figure them out in the first place.

[maanasagarwal]

Many brain cells were lost trying to solve this problem.

[JSSTyger]

After 10 minutes of fuming, I proved that my desk is really, really strong.

[Chadicle]

This. THIS is what I’m talking about! Keep this content up Presh!

[GravitationalRealism]

Great solution! One small enhancement I noticed - for the section starting at 4:41 to prove a = b, it is quicker to divide the equation by b! (rather than a!) and notice a! / b! is an integer iff a = b (and everything else are integers).

[bhaswatibhattacharya]

Imagine someone writing "easy" in the comments for this one.

[tarunrathitra]

It's not easy to even understand this proof
I can't even start to imagine solving this on a timed test
These Olympians are probably superhumans

[alexeyrb]

Nice problem!
Some alternative ways that I found when solved it first:
Rewrite the equation as (a! - 1)(b! - 1) = c! + 1, check cases like 0 and 1 and then look at (a! - 1) = (c! + 1) / (b! - 1). This allows to establish that a >= 3 and c > b.
For case a = b: solve a quadratic equation in terms of a!. It gives a! = 1 + sqrt(1 + c!). This then gives a = 3, c = 4, but also an upper bound on c, since if c can't be cubic or higher expression in terms of a (eliminating some low number cases first). But it's sloppy, your proof on c <= a + 2 looks better.

[mohdhassan]

Now I know why exclamation mark is used for factorial!

[dyosorkan]

As a french viewer, your videos make me practice both english and maths... And how to think outside the box. Genius

[vontabi]

The last time I was this early, the Gougu theorem was still called the Pythagoras theorem

[matthewcarlyon]

Wow, great video Presh. I especially loved the section of proof at 5:00. Brilliant maths.

[WOLLEY]

I'm a GCSE student and these kinds of videos are always fun to watch. I love maths but never understood how people even know where to start with these questions. I lost track of what was going on like a minute into the video.

[logannasty]

Constructing these problems, that’s the truly magnificent feat! I imagine it takes weeks or months to come up with such a beauty. Then this is only one of many problems in an Olympiad. So much intellect invested into competitive maths...

[VinaySingh-jmiw]

Great proof, but how would one know they needed to prove that b<c, a=b, and c<=a+2 to begin with?

[pushkarshukla]

Wonderful!!!, it's a treat to see you go through the proof. Keep it up !!

[JuanMataCFC]

actually amazing proof, i think u covered every possibility. i don't think i've ever seen contradiction used better than this before!

[danyd]

You are just amazing. You must have spent days on this video... Thanks for this awesome explanation. Wondering how people solved the exercise in minutes during the olympiade lol