2024 AIME I Answer Key

1. 204

2. 025

3. 809

4. 116

5. 104

6. 294

7. 540

8. 197

9. 480

10. 113

11. 371

12. 384 (385?)

13. 110

14. 104

15. 721

Note

Both me and another person on AoPS got 385 for problem 12. It seems that 385 is the correct answer, and someone verified it by Desmos. The Solution page's answer has already been updated to 385. --Furaken

\textbf{Response to Furaken: }

You are correct. The number of intersecting points is 385, not 384. The controversy is whether there is one more solution near (beyond the solution at this point). The correct answer is YES, there is one more solution. This point is very very close to , but not the same as .

First, if you use any plotting tool and zoom in the region near , you can see two distinct solutions.

Second, a more realistic thing is how could we find this second solution in the contest since we were not allowed to use any graphing calculator. On the page Solution page, I provided my solution (both my text solution and video solution) to answer this question. The key idea is as follows. I denote and . If such a second solution exists, then we should get a solution that are strictly positive and very close to 0. Since I restrict to small and , I can get closed forms without any absolution signs in the two given functions. After this step, we still need to solve a system of two non-trivial equations. Again, because and are sufficiently small, we can use approximations that and . This reduces two complicated equations to one linear and one quadratic equation. I can then easily find a non-zero solution and even get the closed form.

Third, based on my above analysis, the closed-form (up to the second order approximation) of the second solution near is .