Po-Shen Loh (@PoShenLoh)
Oh my goodness. GPT-o1 got a perfect score on my @CarnegieMellon undergraduate #math exam, taking less than a minute to solve each problem. I freshly design non-standard problems for all of my exams, and they are open-book, open-notes. (Problems included below, with links to GPT-o1's answers.)
While eating Pie in the afternoon, I showed the exam to one of our math Ph.D. students (a former International Mathematical Olympiad Gold Medalist from Belarus), and he said "Hmm. Non-Trivial. Good." Our undergraduate students are also very good. This exam was not easy for them, as the score distribution shows.
Today is the 2-year anniversary of the public release of GPT-4. Two years ago, it caught my eye because it exhibited sparks of insight, similar to what I would see when I talked to clever kids who learned quickly. That gave me the instinct and urgency to start warning people. Today's observation of GPT-o1 being able to ace my hard college exam, makes me feel like we're close to the tipping point of being able to do moderately-non-routine technical jobs. I was impressed by every student in my class who got a perfect score. The fastest such person took 30 minutes.
And GPT-o1 only costs $60 per million words output, which means that each problem cost about 5 cents to solve. A total of around 25 cents, for work that most people can't complete in 1 hour.
Problem 1: Consider the recurrence a_n = a_{n-1} + a_{n-2}, with the first initial condition being a_0 = 1. Find all real number values for the second initial condition a_1 such that lim_{n \rightarrow \infty} a_n = 0.
https://chatgpt.com/share/67d4d4bf-1110-8012-96ea-4183acf806bf
Problem 2: Find coefficients such that the sequence a_n = n \sqrt{2} + 2^n \pi satisfies the following recurrence, for some initial conditions. You don't need to find the initial conditions.
a_k = ___ a_{k-1} + ___ a_{k-2} + ___ a_{k-3}
https://chatgpt.com/share/67d4d4fe-6008-8012-950f-704abd78c2a1
Problem 3: Fill in the blanks. The middle entry of the result of:
[[0, 0, 1], [1, 0, 0], [2, 3, 4]]^n [[5], [6], [7]]
is the term a_n of this recurrence:
a_k = ___ a_{k-1} + ___ a_{k-2} + ___ a_{k-3}
a_0 = ___
a_1 = ___
a_2 = ___
https://chatgpt.com/share/67d4d61d-72f4-8012-8e26-0861ed568dce
Problem 4: Consider the recurrence with initial condition a_0 = 1, where for each n \in {1, 2, 3, ...}:
a_n = \sum_{k=0}^{n-1} a_k
Find the generating function f(z) = a_0 + a_1 z + a_2 z^2 + ..., which looks something like f(z) = (1-2z)/(1-z)
https://chatgpt.com/share/67d4d63c-182c-8012-968f-e2fbde54f346
Problem 5: Prove that the coefficient of x^{2025} in (x + x^2)^0 + (x + x^2)^1 + ... + (x + x^2)^{2025} is a Fibonacci number
https://chatgpt.com/share/67d4d657-4bcc-8012-aaaf-1f7f9fc8e855
My main work nowadays is to build and scale up a community of people (through education) to face the challenges of the AI age together. I thought I had more years. Now we have to move faster.
