“This probably works”

Advice(?) mainly for consistent Q2/5 solvers.

The way I viewed Math Olympiad and contest problems changed a lot from 2022 to 2023. I think this change of strategy really worked for me, so I want to pen it down before I forget.

Imagine yourself with 2 progress bars, one for knowledge and one for intuition. The aim of every handout or problem you do should be to improve at least one of these bars. (Yes, this is a big simplification, but for our purposes it will suffice.)

The knowledge bar reflects how much theory you know. This is easy (if tedious) to improve – simply use the multitude of handouts available to learn things like Vieta jumping, Hall’s Marriage Theorem, Pascal’s Theorem, etc. Exposing yourself to all the different techniques should be the main focus if you’re struggling to solve Q2/5s.

But there comes a point where you don’t need to know more to solve more.¹ In 2022, I kept reading harder and harder handouts, from MMP to Combinatorial Nullstellensatz to the entire theory of Quadratic Residues, and the issue with that is that not only am I struggling to understand, I rarely have the opportunity to apply anyways.

Case in point – in IMO’22, the hardest theory you need is… LTE. In 2023? Isogonal conjugates.

Which means that a change of plans is needed. Stop doing handouts, and start spamming problems to improve your intuition instead.²^,³

This sounds painfully obvious. Everyone is taught from young that practice makes perfect. But it is very hard to resist the urge to learn this new technique just in case it is the ticket to another 7 marks. I know this from experience, and many people I know do this too.

Spam.

Seriously, spam. In 2023, I relied almost solely on the 2023 contest collections. I tried almost every problem from almost every contest from January to May. That sounds impossible, but here’s the catch – I hardly fully solved any of them⁴. Rather, I explored the structure of the statement for some time, figured out how the solution probably looks like⁵, and then checked if I was right. If I knew I was left with a manageable bash, I wouldn’t complete it.

I know, it sounds ridiculous and perhaps feels like I’m taking shortcuts. But if you’re already at IMO silver level, the only thing that you can learn from most problems is how to see the correct main idea. This is possible even for geometry. And the easiest way to train your speed is to do Q1/4s.

I used to question if simply doing recent contests was a good idea, because there are bound to be trash problems. In hindsight, I think it was the smartest thing I could’ve done. There is value in learning how to find intuition for the ugliest and spammiest problems, because you may just be facing one in the next contest. There is value in doing every question you see because that is literally your job.

Confidence

Knowing your intuition is good can be a huge confidence boost. I felt this the most during IMO training. My observations and conjectures started getting a lot more accurate, and I stopped constantly doubting myself and switching approaches. I would frequently tell myself and others “this probably works” (or even “this obviously works”), and more often than not it would.

This, I think, is the single, most important, thing you need to solve a Q3/6 – the conviction that you can do it. And I don’t mean just reciting “I can solve this” over and over in your head; this belief can’t be forced or instilled, you have to know it is true.

IMO 2023

I will list out the times I told myself “this probably works” during IMO’23, just to give you a concrete example.

Q1. The condition $d_i \mid d_{i+1}+d_{i+2}$ gets very sus when one has (or doesn’t have) a prime divisor the others don’t (or do), so considering just $p^k$ and $\frac{n}{p^k}$ probably works.

Q3. If i can show $f$ is increasing, the equation makes all terms very constrained, and they can take very few values. We can probably uniquely determine the polynomial by infinite pigeonhole. Showing increasing probably works.

Q4. $a_{2023}\ge 3034$ , obviously we want $a_{n+2}\ge a_n+3$ .

Q5. If the answer is $\lfloor \text{log}_2(n) \rfloor+1$ like the construction shows, then inducting on powers of 2 has to work.

(That did not work, which explains why I spent so long on the wrong idea. Obviously I am far from perfect.)

Q6. Coaxial but the circles have no relation. Surely we just find two radical centers.

Q6. We need points that are somehow symmetric about all three vertices. $A_1A_2,B_1B_2,C_1C_2$ , $AA_2,BB_2,CC_2$ or $A_1A,B_1B,C_1C$ probably works.

(I got the last 2 in 45 minutes, and I fully credit the 2 marks to my newly developed habit of trying to guess the main approach instead of a “breadth-first search” exploration from the conditions given.)

Conclusion

I hope I have convinced you to focus on your solution-guessing abilities (yes, that’s what intuition really is). The next time you try a question, try to guess the entire solution outline, or at least the main step, as fast as you can.

And stop learning (D)DIT until you can master that.

Appendix: Q1/4 sources

Here is a list of less-famous contests that have a consistent supply of decent quality Q1-1.5s:

Balkan MO Q1,2
Baltic Way⁶
Canada MO Q1-4
European Mathematical Cup all except S4
IberoAmerican Q1,2,4,5
India MO Q1,2,4,5
Iran MO R2 and R3 Q1,2,4,5
Japan MO Finals Q1-4
Middle European MO
All-Russian Olympiad and its Regional Round.
All Taiwan contests, although the difficulty varies a lot so I can’t say much.

I’m putting this mark at Q2/5, both because it’s convenient and because I think it’s somewhere there. ↩︎
On a similar note, what is the point of training? Also knowledge. I suspect that there is a stage after which organised training is no longer necessary or useful: as long as one is disciplined enough to practise consistently, developing intuition by yourself helps you more. ↩︎
Also, I’m not saying to not learn any more theory. You can, but master the intuition half first. One has a much higher expected rate of return. ↩︎
Especially Q1/4 level questions. I never wrote a single solution for those. Do the hard questions properly though. ↩︎
For example, something like “operate backwards from the final range and show that in each iteration the range gets bigger so we can get to 0/1”. I barely did the actual calculations. ↩︎
I really, really recommend trying to “theory solve” (i.e. find the main idea) of the whole set in a 4.5hr sitting (aim to solve about half). This is because many of the questions are one-trick, i.e. you find the main step and you’re done. The same is true for MEMO (both Individual and Team). ↩︎