That number is the one my mom and nephew came up with. I’m not sure if they had access to an answer key or not.
I think recognizing that it’s a geometry question, then that structure of the question matches what that particular theorem states and then applying the theorem.
A bit sloppy.
Need to define h1 and h2.
Not sure what K is here.
“Let K = the sum of these two overlapping triangles minus the known overlap”?
Then X = ABD + BCE + ADC - K + 2024
But, X = ABD + ADC, per the picture.
So, you have confused me.
I was thinking “How close is the theorem to the question?”. Some creative person might be able to use a different theorem that isn’t so close, but they can do the extra steps to make it work.
It seems that AI will always be able to take the shortest route because it has the biggest “memory” of learned facts.
I think it depends on part on how well the AI can identify the right theorem for the question as described. Maybe think about it a bit like an escape room if you’ve ever done one. A lot of times, people can be on the right path to a solution and then get hung up on the wrong part of the problem and end up not being able to solve the task. I think the most recent example I can think of was looking in the right spot for the key, but not finding it. Rather than looking further for the key in that location, they tried multiple other solutions. Turns out the key was in the space they’d looked, but had been knocked off the shelf.
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This is the bot’s answer. Looking at it’s “thinking process” it seemed to immediately recognize the application of Menelaus Theorem, but got really stuck on where to go with it.
Problem:
Let the area of a triangle XYZ be denoted by [XYZ].
We are given that D is a point on BC such that BD:DC = 3:2.
We are given that E is a point on AC such that AE = EC.
F is the point of intersection of AD and BE.
The area of the quadrilateral CDFE is 2024.
Answer:
We apply Menelaus’s Theorem to ΔBCE and transversal AFD.
The points A, F, D are collinear. The line intersects the sides BC, BE and the extension of CE at D, F, and A respectively.
According to Menelaus’s Theorem, we have:
(BD/DC) * (CA/AE) * (EF/FB) = 1
We are given BD/DC = 3/2.
Since E is the midpoint of AC, we have CA = 2AE.
Substituting these values into the equation, we get:
(3/2) * (2AE/AE) * (EF/FB) = 1
3 * (EF/FB) = 1
EF/FB = 1/3
So, BF:FE = 3:1.Now, consider the triangles ΔBCF and ΔFCE. They share a common vertex C, and their bases BF and FE lie on the same straight line BE. Therefore, the ratio of their areas is equal to the ratio of their bases:
[BCF]/[FCE] = BF/FE = 3/1
[BCF] = 3 * [FCE]
The area of ΔBCE is the sum of the areas of ΔBCF and ΔFCE:
[BCE] = [BCF] + [FCE] = 3 * [FCE] + [FCE] = 4 * [FCE]
So, [FCE] = (1/4) * [BCE].The area of the quadrilateral CDFE is given as 2024.
[CDFE] = [FCE] + [CDF] = 2024
We have [FCE] = (1/4) * [BCE].
So, [CDFE] = (1/4) * [BCE] + [CDF] = 2024.Now consider ΔABD and ΔACD. They share a common vertex A, and their bases BD and DC lie on the same straight line BC. Therefore, the ratio of their areas is equal to the ratio of their bases:
[ABD]/[ACD] = BD/DC = 3/2
[ABD] = (3/2) * [ACD]
Also, [ABC] = [ABD] + [ACD] = (3/2) * [ACD] + [ACD] = (5/2) * [ACD].
So, [ACD] = (2/5) * [ABC].Now consider ΔFBD and ΔFCD. They share a common vertex F, and their bases BD and DC lie on the same straight line BC. Therefore, the ratio of their areas is equal to the ratio of their bases:
[FBD]/[FCD] = BD/DC = 3/2
[FBD] = (3/2) * [FCD]We have [BCE] = [BCF] + [FCE] = ([FBD] + [FCD]) + [FCE]
[BCE] = (3/2) * [FCD] + [FCD] + [FCE] = (5/2) * [FCD] + [FCE]
We also have [BCE] = 4 * [FCE].
So, 4 * [FCE] = (5/2) * [FCD] + [FCE]
3 * [FCE] = (5/2) * [FCD]
[FCD] = (6/5) * [FCE]Now we can express the area of the quadrilateral CDFE in terms of [FCE]:
[CDFE] = [FCE] + [FCD] = [FCE] + (6/5) * [FCE] = (11/5) * [FCE]
We are given that [CDFE] = 2024.
(11/5) * [FCE] = 2024
[FCE] = 2024 * (5/11) = 184 * 5 = 920.Now we can find the area of ΔBCE:
[BCE] = 4 * [FCE] = 4 * 920 = 3680.Finally, we find the area of ΔABC.
Consider ΔABE and ΔCBE. They share a common base BE. The ratio of their areas is the ratio of the lengths of the altitudes from A and C to the line BE.
Since E is the midpoint of AC, the altitude from A to BE is equal to the altitude from C to BE.
Therefore, [ABE] = [CBE] = 3680.
[ABC] = [ABE] + [CBE] = 3680 + 3680 = 7360.Let’s double check the result.
[FCE] = 920
[FCD] = (6/5) * 920 = 1104
[CDFE] = 920 + 1104 = 2024. Correct.
[BCF] = 3 * [FCE] = 3 * 920 = 2760
[BCE] = 2760 + 920 = 3680. Correct.
[FBD] = (3/2) * [FCD] = (3/2) * 1104 = 1656
[BCF] = [FBD] + [FCD] = 1656 + 1104 = 2760. Correct.
[ABE] = [ABF] + [FBE]
[ABC] = [ABF] + [FBE] + [BCE] = [ABF] + [FBE] + 3680
[ABC] = [ACD] + [ABD] = (2/5)[ABC] + (3/5)[ABC]
[ACD] = [ACE] + [DCE] = [ACE] + [DCF] + [FCE]
[ACE] = (1/2)[ABC]
[ACD] = (2/5)[ABC]
[ACE] = [ABE] = 3680
(1/2)[ABC] = 3680 => [ABC] = 7360.
Oh-- and while I’m ranting about AI, one more point:
It matters a lot which Bot you use, how you prompt it, and its temperature setting.
If you actually use an LLM for work, take a minute to look up the best one. Currently there are a lot of good free bots though, like Gemini 2.5 pro.
X
I’ve been told Claude AI is or was best for programming…
This checks.
Short version:
meneleus shows that BF = 3 * FE
Draw segment FC (for F_A – not DE)
Let Area EFC = N
Let Area FDC = 2 * M
N+2 * M = 2024
Area EFC = Area AFE = N
Area FBD = 3/2 * Area FDC = 3/2 * 2 * M = 3M
Area ABZ = 3 * Area AFE = 3 * N
Clockwise:
Area ABC = Area AFE + Area EFC + Area FDC + Area FBD + Area ABD
= N + N + 2 * M + 3 * M + 3 * N
= 5 * N + 5 * M
The second relationship of N and M:
Area FBC = 3 * Area EFC
So, 5 * M = 3 * N
5 * M = 3 * (2024 - 2 * M)
5 * M = 6072 - 6 * M
11 * M = 6072
M = 552
N = 5/3 * 552 = 920
5 * N + 5 * M = 2760 + 4600 = 7360
That kind of tracks with what I was thinking.
When a person takes a math test, that test is judging the human’s capacity to take a set of concepts and rules, and apply those concepts and rules.
These ai models are predicting the next word. It is a fundamentally different capacity that a person has.
All this means that an ai model achieving a high score on a test does not mean that the ai can do the same kind of work as a person making that same score because the underlying capacities being summarized by that score are fundamentally different.
Yes, this. Though to be fair, any top math contestant will also memorize at least thousands of proofs.
The model also has the unfair advantage of being able to “think” very quickly. Essentially trying hundreds of different approaches (or the same one over and over again) in seconds. Kind of like a chess engine.
I think the challenge for math competitions is for an AI is to verify the chain of logic. It needs to apply every theorem correctly to the problem. Especially as the problems get longer and harder, which can go for pages and pages.
Yes I’d guess that current AI wouldn’t be able to help much with new or niche math because the theorems and lemmas aren’t all laid out nicely like that… Otoh, asking AI to become some kind of math genius is already a weird aside.
I think it’s any real life problem in which the concepts have to be applied to a peculiar situation.
Tests tend not to be like that because it is hard to write tests, and people tend to repeat the same toy problems. (Id consider something like proving the irrationality of sqrt(2) to be a toy problem at this point since it’s already solved, even though it was presumably really hard to set up the conceptual equipment to solve it the first time.) So it seems to me that tests are exactly the kind of thing these models would be best at.
Programming, too, would seem well suited. They are recipes, and recipes tend to be shared and reused.
I will admit i find it very hard to have a good intuition about when a word pdf will and will not do a good job approximating human thinking, or even exceeding it in some cases.
Left as an exercise for the reader. they should be obvious IMO given that we’re giving a formula for the area; and in the end, they’re not used to get an answer.
But is that the only way to determine the area? I think it’s clear that this particular formula while straight forward is not useful apart from seeing that ABD is 3/5 of the total area (per your observation).
Yes, K is the area of the polygon BFAC. I agree that my write up could’ve been more clear on this point. 
My main concern with the use of Menelaus’ Theorem for the picture shown is that this Theorem requires that the transversal uses only “non-vertex” points of the given triangle. That is, the transversal AFD fails the condition to use Menelaus’ Theorem as A is a vertex point of the triangle in question.
4.3: Theorems of Ceva and Menelaus - Mathematics LibreTexts
Menelaus’s theorem - Wikipedia
In the wiki link, it’s very clearly stated as
If someone can present a reference where a vertex point can be used (with corresponding proof), I’d be more accepting of its use; however, I would like to know what is off in my logic if my answer is in fact wrong.
I think the AI generated answer fails to recognize this “non-vertex” requirement in the application of Menelaus’ Theorem. It correctly recognizes that AFD is a transversal, but fails the requirements of the Theorem.
Agreed. Part of the issue is identifying what is really “original”. You can ask an AI for an original video game-- and it might come up with original gameplay, art, music, story. But each of these original things is a recipe made from unoriginal components. In particular the gameplay will be made up of modular code that has all been written before, more or less, with different names for objects and functions and variables.
What AI is surprisingly good at is finding the right components, and adapting them so that they fit into a coherent whole, which compiles, runs, and might even look and sound okay.
If you go by the notion “there is nothing new under the sun” then maybe big data will take us really far.
In this case, the Transversal is FD through the triangle EBC, which crosses EC outside the triangle at A. It gives us the ratio of BF to FE, which is needed to get the Area of Triangle ABF relative to the other triangles.
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“Global Thermal Nuclear War”?
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What is especially surprising is that the coherence is an illusion, in this sense: in a real story or communication, coherence comes from a particular intent, usually rooted in experience, and with a particular target audience in mind.
the models has none of those capacities. it is probabilistic, and works token by token.
it’s perhaps like a movie. the frames don’t really move. and i think maybe our eyes are even good enough to see the different frames? our brains fill in the illusion of motion. so it also fills in the illusion of coherence from these model outputs.