IMO 2024 Problem 5 - most *TROLL* problem in IMO history?

#mathematics #olympiad #math
International Mathematical Olympiad (IMO) 2024 Day 2
Solutions and discussion of problem 5

65th International Mathematical Olympiad Bath UK
Problem 5 - Combinatorics

[polyquadratus]

There is an easier solution:
Spend 1 attempt finding the monster on the first row. If it is not at the end, then you can guarantee to sneak under it in at most 2 more attempts.

If it is at the end, go to the opposite end of the second row, and explore up to 2 squares before the monster. If you find another monster, then you can sneak under the first row monster, otherwise do something similar for the third row etc, creating a kind of staircase pattern. If no monsters are found, then at the end, the staircase will reach the bottom row.

[dedekindcuts]

Some consider this a troll problem because the answer is wayyy smaller than what some contestants may expect. What do you think of this problem? Regardless of your views, I think we can all agree that it's a pretty fun problem!

[Ma]

I initially thought we could do binary search and so that way we get log(2023) base 2 = 11 but realised it was wrong. Nice solution!

[laszloliptak]

I think this is a very reasonable problem, and it is very natural to think that the answer will be quite small, which turns out to be the case. It would only be a troll problem if the solution turns out very different from your expectations. Regarding the 2010 problem 5, there one might say that the end result seems so outrageously large, they would not ask it that way if there would be a "normal" upper bound on the number of coins one can get in the last box. So the answer must be yes, and one needs to find a way to get it.

Of course, everyone may box him- or herself into a wrong answer, and it is very difficult to get out of that, especially during the competition.

[quite_unknown_]

If this appears as P4, everyone solves it.

[sunillGD]

As someone who participated in some of the hong kong TST this year, I can say that I can definitely see some of our trainers trolling us, but I definitely didnt see this problem being selected for P5 xdd

[pietergeerkens]

Fun problem. I enjoyed solving it, with the single staircase. Thank you.

However, here's my two cents: I placed consistently in the top 10% nationally in both high school contests and then Putnam. But "top 10%, barely" results also mean I am nowhere near Olympiad level either now or when I was younger. If I can solve this problem in about 15 minutes, as I did, the question is not whether this problem is too hard for Olympiad; but rather whether it is too easy. Especially for a Problem 5, I would expect the difficulty to be such as to require me to spend at least 60 minutes to solve.

[juanmelo]

In the motivation part you read my thinking process as a book 😭 (until 5:42)

[PetersLabAviation]

Nice solution! The official solution involves walking in a staircase pattern instead, but seems like it is a more ingenious construction compared to the one shown in the video, does anyone know what's the motivation behind it?

Also, is there a solution where you can guarantee 3 attempts without exploring the second row much? Seems like knowing where the uppermost monster is a must.

[ekxo]

my strategy:
wherever you start (unless it's the first or last column, then move to another one) go down until you find a monster
1. if you don't you managed in one try.
2. if you do then next attempt rego down the same column and arrived at the row before the monster go to the right and down,
3. if you don't find a monster, go to point 5.
4. if you do find a monster, you are guaranteed that there's no monster on the cell two on the left of where you are then on the third attempt you can move to the left and then down instead of right and shouldn't find a monster
5. go just under the monster that you first found by going one right (or left if you skipped point 4.) and you are guaranteed not to find any other monster in that column, so you will finish

[rendoesmath]

This was pretty similar to the last year's P5!

[Asmonix]

i don't understand the win in 3
how this could work for the case like this:
1, 2, 3, 4, 5, 6, 7, 8
1⬇🟢🟢🟢🟢🟢⬇
2❌🟢🟢🟢🟢🟢⬇
3⬜❌🟢🟢🟢🟢🟢
4⬜⬜⬜⬜⬜⬜❌
5⬜⬜⬜⬜⬜❌⬜
…

green is "known safe to explore" fields after 2 failed tries of checking the sides: (1, 2) and (8, 3).
We still can be catched by the monster from the row 4, wherever we try to explore next.

[selwynliu]

I am curious if it's really the placement of the problem or the overestimation of difficulty that caused it to be poorly solved, or are there other factors?

[juancarlosortiz]

This is a beautiful problem, in my opinion! ❤

Spent an hour or two on it, but finally cracked it

[saumyashah]

I love ur vids and effort-keep up the work❤

[dedekindcuts]

Do stay till the end of the video to hear some cool stories about this problem. I hope you enjoyed the little bit of story-telling!

[mrpringles]

Alright i know i am little annoying but can you explain the hungry goat problem btw love your channel ioqm aspirant here.

[delaxer]

idk, this is very easy problem, but at the same time its P5!?
ive solved p1, p2, p4, p5 and 1 point for p3 -> gold medal
what i can say about problem? its beautiful.😊

[saraswatasadhu]

I have an alternate idea for this. Can you please check...

[BigAsciiHappyStar]

Turbo the Snail should have been replaced with a troll 😁