A Problem from the Toughest Math Contest

#maths #science #mathematics #interesting #math #geometry #putnam #visualization #animation #easy #fun #proof #proofwithoutwords
This problem is from the 1998 Putnam Competition, a college level math contest that is famous for its incredibly difficult problems. In this video, we solve problem A2 from this edition of the competition with visual aides in a way that turns a seemingly complex problem into a simple one.
The solution method for this problem was based on a "Proof without Words" titled "A Putnam Area Problem". This proof comes from Volume 2 of the book Proofs Without Words: Exercises in Visual Thinking by Roger B. Nelson.
This video was created using Manim, a popular mathematics animation library for Python.
Works Cited:
Nelsen, Roger B. "Geometry and Algebra." Proofs Without Words II: More Exercises in Visual Thinking, American Mathematical Soc., 2020, p. 30.

[@poutineausyropderable7108]

Gotta increase sound quality. Animations are nice though.

[@murrrkkk]

The sound gives off a crazy vibe tho

[@ianweckhorst3200]

@@murrrkkkif you want more of that vibe, try listening to the nurse character from nyan neko sugar girls, but beware, the animation is NOT nice and is probably made on ms paint

[@snowfloofcathug]

An easier way to demonstrate why the rectangles and triangles are related is to slide them over to make parallelograms of the rectangles first / make the triangle first then slide the tip
Echoing what people have said about audio quality. As for the rest of the video I really enjoyed the presentation and I really look forward to see what’ll come of it! Always happy to come across quality maths channels :)

[@Furnischer]

this was glorious the whole way through

[@KrasBadan]

Haven't watched the video, here's my solution:
Let's take radius of the circle as a unit.
Let's say the arc starts at angle a1 and ends at angle a2, a1

[@swarajpatil5829]

I would just probably integrate
Bruteforce way

[@pietergeerkens6324]

Nice problem; and very elegant solution.
I approached it algebraically. With θ as you've assigned, let σ and τ be the 1st quadrant arcs above and below s respectively. Then
A = ½ ( σ - sin(σ)cos(σ) ) - ½ ( τ - sin(τ)cos(τ) )
= ½ [ (σ - τ) - ( sin(τ)cos(τ) - sin(σ)cos(σ) ) ]
= ½ θ - ½ ( sin(τ)cos(τ) - sin(σ)cos(σ) ).
Likewise
B = ½ θ - ½ ( sin(σ)cos(σ) - sin(τ)cos(τ) ).
Summing then gives
A + B = θ = length(s)
as required.
Not as elegant - but I believe also reachable by a young audience.
I'll keep an eye out for your content - but won't subscribe until you get the audio fixed up.

[@ArunKumar-np5ri]

That was good ... too good explaination 😌

[@maths.visualization]

Can you share video code??

[@sclearDevelopment]

amazing video! thanks a lot! keep up the good work!

[@tmahmood4]

One thing is still disturbing me, that is, wether this equation is dimensionally equivalent. Because one side is area and the other side is (arc) length.

[@legamerfandesience2987]

7:32 The constant is the r² that is simplified here

[@BKNeifert]

Good stuff. I call this base 1 math. Because of exponentiation, it wouldn't reduce to Theta if the radius were any larger. (But it would be 2x) But this same principle is how calculus and a circle's area works. The curve determines its area. It's also why Heron's Formula doesn't work that well outside of Base 1. But a+b+c=a*b*c is always a triangle. No matter how you calculate it, but the most elegant nuances are found in base 1.

[@InfinityExt]

Your mic is amazing 🤩

[@Shubhyduby]

Amazing video, thank you!

[@fragadmsc]

great video!

[@copywright5635]

Honestly great video, I remember struggling on this problem before giving up after a few hours.
About the mic quality. I think your setup is likely fine, just needs two things. One, move away from the mic by probably ~6 inches. Second, dont direct your mouth directly at the mic (this is fine with some, but probably not with yours), the main 'gust of air' coming from your mouth should pass just above the mic. This will prevent the sort of super intense sound you get with "p" or "k" sounds.
Good luck and cheers!

[@torispadewtd5494]

🔥🔥🔥💯🥂🥂

[@randomcreations2197]

Cool

[@NoaSolivagus]

amazing

[@ChandrasegaranNarasimhan]

thanks, I liked it.

[@krishpandey854]

A more straight forward approach would be to simply find each of the areas using integration and adding them. take coordinates of the points as (rcosalpha,rsinalpha) and (rcosbeta,rsinbeta) and the curve is x^2 +y^2 =r^2.
Integrating and adding gives
r^2 (beta-alpha)
=r*arc length

[@thenahir14]

Nice

[@jeromeh.348]

cool vid, pls fix your mic plsplsplspls