IMO 2024 - Problem 1: A cool Algebra with integer parts and bounding :)

[technolan]

I've been selected to represent my country (Morocco) in the 2024 IMO after acing at the TST. I've put so much time and effort in, and spent the last three years preparing for this event. Just imagine one second being so much excited to get to that event, but for a very dumb reason, you couldn't attend the IMO, the most prestigious mathematical competition in the world! Do you know why? because my country didn't assume its responsibility to get us our visa to travel to England and didn't apply for them until the LAST DAY! They were expecting to get them in the SAME DAY but of course they didn't. It's the first time it happens in 42 years. Me and my team were left devastated, and our mentors were so disappointed. it's a shame for my country to waste such great talents. Shame on you Morocco. 😔😔😢😢

[AryanSingh-iuvt]

Continue with the pen and paper format! It's much clearer than than the white board!
Keep Up the good work!

[tollspiller]

I think the problem was fairly nice but its very susceptible for very bashy solutions - and when you sit in a contest and see that a certain approach works you'd rather write that one fown and move on instead of trying to search for a nicer one.
The bash I quickly came up with:
Write a=k+c, for c in (0, 1)
then fo cases on c where we bound it between 1/m and 1/(m+1) and then you can do cases kn parity on k and m, so in total 4 cases, and get contradictions for all but k odd and m = 1, then you can write c=1-b and do again 2 cases on b (bound it again by reciprocals) and finally get contradictions

[Maths_.]

What do you want to say about the difficulty of this year's problems?

[annashakaia]

shef i really need your advice my best vice now i want to improve my geometry im 15 years old and able to do ew imo p4 or p1 and is it good at my age?

[filipkrakus]

hey, just solved the problem, about to watch the video ;). But I'm curious - where can I find the problemu? They're not available at the imo website yet

[quite_unknown_]

Solved this one in 10 minutes and Q2 in 30 min. Couldnt get Q3 in 2 hours though. A solution to Q3 would be much appreciated.

[huseyinyigitemekci-fmvf]

Who could predict this AAA day 1. After solving p1, will come back here. Can we use here Hermite's identity