Math and Logic Puzzles: Redux

[DeathRowKitty]

Prove that in any sequence of n^2 + 1 real numbers there is an increasing subsequence of length n+1 or a decreasing subsequence of length n+1.

Spoiler:

Let A = {a_i}, i=0,1,...,n²+1 be the sequence. Let f(a_k) denote the length of the longest increasing subsequence of A ending with a_k. If there is some k such that f(a_k) = n+1, we've found an increasing subsequence of length n+1. If no such k exists, then, by pigeonhole, there must be some c such that f(a_i)=c for at least n+1 distinct values of i and those n+1 such a_i form a decreasing subsequence of length n+1.

Easier and a different type of problem:

Suppose you have 3 spherical marbles with centers A, B, C resting on a flat table, each touching the other two marbles. Let A', B', C' be the points at which these marbles touch the surface. If each marble has an integer radius and A'B'C' is a right triangle with integer sides, find the minimum possible semiperimeter of ABC.

[DeathRowKitty]

yeps!

[DeathRowKitty]

Can you clarify exactly what replicates? I think different interpretations give different answers here

[DeathRowKitty]

I'm kinda confused tbh because from the clarification it seemed as though this was a version in which the answer was rather trivially that you can't always kill the hydra rather than the usual hydras in which the answer was what implosion said and the intended solution

uses ordinals

. And that seemed weird because I wasn't sure who the intended audience could possibly be. People who already knew enough to know how and why the hydra would normally get slain but who would get tripped up by a trivial version in which none of that applied?? But then if invisibility was on the right track I assume the answer is the usual expected thing, in which case I think I still don't understand the regeneration rule for this one

pedit: started making this post before the previous one was made so it doesn't take that post into account

[DeathRowKitty]

I'm kinda confused tbh because from the clarification it seemed as though this was a version in which the answer was rather trivially that you can't always kill the hydra rather than the usual hydras in which the answer was what implosion said and the intended solution

uses ordinals

. And that seemed weird because I wasn't sure who the intended audience could possibly be. People who already knew enough to know how and why the hydra would normally get slain but who would get tripped up by a trivial version in which none of that applied?? But then if invisibility was on the right track I assume the answer is the usual expected thing, in which case I think I still don't understand the regeneration rule for this one

pedit: started making this post before the previous one was made so it doesn't take that post into account

Frankly it's plausible that we're thinking of the same replication rule and I'm just dogshit at explaining it

Now that I've seen your diagram and looked back at what you said previously, it seems that actually I just can't read perfectly clear explanations. Oops! Thanks for the clarification