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- TL;DR Summary
- Prove you are smarter than the bot!

Chatgpt is actually pretty good at generating math problems. It's awful at solving them. I guarantee every question posted here cannot be solved by chatgpt, but maybe can be solved by a human? My plan is to spend a couple minutes getting a question I think it's cool and then posting it here - I don't know if I'll actually do it each day.

Since chatgpt is bad at knowing whether things are true or not, and I'm not going to try to solve all of these before posting, most will be of the form prove or find a counterexample

#5.) .I take a piece of string of length 1, and do the following k times: cut the string into a ratio of 2 to 1 (so a string of length 1 is cut into a string of length 2/3 and a string of length 1/3) I then throw out one of the pieces, leaving myself with a single piece that I can repeat this process on, until I have made k cuts.

What is the expected length of the final piece of the string in terms of k, if

a.) Each time I pick one of the two pieces to throw out randomly with equal probability

b.) Each time I "grab" a random spot on the string and throw that piece out - i.e. I have a 2/3 probability of throwing out the longer piece

6.) If you start an infinite random walk on the integers where each step is 50/50 to be one unit left or one unit right, show the expected number of times you return to the origin is infinite.

Bonus question: what about a walk in 2 or more dimensions? Slightly handywavy arguments are acceptable.

#11) given three points on a plane, not collinear, prove there exists a unique circle passing through them.

#12) Prove or disprove: Every graph of 30 vertices has two vertices with the same degree (same number of edges)

Is 30 important here? It actually asked me a question about friends that might have been a mix between a graph theory question and the birthday problem but this is what I reduced it to.

#13) Prove or disprove: Every infinite dimensional Banach space has a linearly independent sequence that is not a basis.

Solved problems (no solutions in the spoiler so feel free to take a crack at them)

#1.) Either prove every compact connected metric space is path connected, or find a counterexample.

fresh42 found a counterexample

#2.) Prove every finite group ##G## has a normal subgroup ##H## such that ##G/H## is simple, or find an example of ##G## for which this is not true. Bonus: if it's true, does it require that ##G## is finite?

solved by Infrared for the finite case

and the bonus infinite counterexample!

#3) if ##f:\mathbb{C}\to \mathbb{C}## is holomorphic, and ##f(z)\in \mathbb{R}## whenever ##|z|=1##, either prove ##f## must be constant or find a counterexample

solved by Infrared

#4.) If ##a,b,c## are positive integers such that ##a+b+c=72##, find the maximal possible value of ##(a-b)^2+(b-c)^2+(c-a)^2##.

fresh_42 with a very elegant transformation

#7). Prove or give a counterexample: Every irreducible complex representation of a compact group is unitary

Infrared observes it doesn't have to be unitary under a random choice of inner product

but you can pick an inner product that makes this work

#8) Prove or give a counterexample:

Every ideal in a commutative ring is the intersection of the prime ideals containing it.

fresh_42 with a simple counterexample

#9) Prove every lebesgue measurable set of positive measures contains measurable subsets of arbitrarily small positive measure, or provide a counterexample.

Infrared shows there exists a subset of every smaller measure

#10) Prove or disprove: Every uniformly continuous function that is differentiable on a closed interval is necessarily Lipschitz continuous.

Counterexample by infrared

Since chatgpt is bad at knowing whether things are true or not, and I'm not going to try to solve all of these before posting, most will be of the form prove or find a counterexample

#5.) .I take a piece of string of length 1, and do the following k times: cut the string into a ratio of 2 to 1 (so a string of length 1 is cut into a string of length 2/3 and a string of length 1/3) I then throw out one of the pieces, leaving myself with a single piece that I can repeat this process on, until I have made k cuts.

What is the expected length of the final piece of the string in terms of k, if

a.) Each time I pick one of the two pieces to throw out randomly with equal probability

b.) Each time I "grab" a random spot on the string and throw that piece out - i.e. I have a 2/3 probability of throwing out the longer piece

6.) If you start an infinite random walk on the integers where each step is 50/50 to be one unit left or one unit right, show the expected number of times you return to the origin is infinite.

Bonus question: what about a walk in 2 or more dimensions? Slightly handywavy arguments are acceptable.

#11) given three points on a plane, not collinear, prove there exists a unique circle passing through them.

#12) Prove or disprove: Every graph of 30 vertices has two vertices with the same degree (same number of edges)

Is 30 important here? It actually asked me a question about friends that might have been a mix between a graph theory question and the birthday problem but this is what I reduced it to.

#13) Prove or disprove: Every infinite dimensional Banach space has a linearly independent sequence that is not a basis.

Solved problems (no solutions in the spoiler so feel free to take a crack at them)

fresh42 found a counterexample

#2.) Prove every finite group ##G## has a normal subgroup ##H## such that ##G/H## is simple, or find an example of ##G## for which this is not true. Bonus: if it's true, does it require that ##G## is finite?

solved by Infrared for the finite case

and the bonus infinite counterexample!

#3) if ##f:\mathbb{C}\to \mathbb{C}## is holomorphic, and ##f(z)\in \mathbb{R}## whenever ##|z|=1##, either prove ##f## must be constant or find a counterexample

solved by Infrared

#4.) If ##a,b,c## are positive integers such that ##a+b+c=72##, find the maximal possible value of ##(a-b)^2+(b-c)^2+(c-a)^2##.

fresh_42 with a very elegant transformation

#7). Prove or give a counterexample: Every irreducible complex representation of a compact group is unitary

Infrared observes it doesn't have to be unitary under a random choice of inner product

but you can pick an inner product that makes this work

#8) Prove or give a counterexample:

Every ideal in a commutative ring is the intersection of the prime ideals containing it.

fresh_42 with a simple counterexample

#9) Prove every lebesgue measurable set of positive measures contains measurable subsets of arbitrarily small positive measure, or provide a counterexample.

Infrared shows there exists a subset of every smaller measure

#10) Prove or disprove: Every uniformly continuous function that is differentiable on a closed interval is necessarily Lipschitz continuous.

Counterexample by infrared

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