IMO 2020 Day 1 solutions and discussion of statistics

#mathematics #olympiad #math
International Mathematical Olympiad (IMO) 2020 Day 1
Solutions and discussion of problems 1, 2 and 3

61st International Mathematical Olympiad Saint-Petersburg Russia
00:00 Introduction
00:24 Problem 1 - Geometry
03:06 Problem 2 - Algebra
07:30 Problem 3 - Combinatorics
15:12 Statistics, discussion, and thoughts

[dedekindcuts]

What are your thoughts on this year's IMO problems?

[williamchurcher]

Easiest solution to Q3:

Call the piles A and B with weights wA and wB. Let x be wA - wB. We need to show that we can choose A and B so they contain 2 stones of each colour such that x = 0. Now 2n^2+n \leq wA \leq 6n^2 +n and similarly for wB. So -4n^2 \leq x \ leq 4n^2. So x is in a set of size at most 8n^2+1.

How many ways can we choose A and B? Well for each colour we may split them between piles in 4C2 = 6 ways, so we have 6^n ways of choosing A and B.

Case n=1 is obviously true and 6^n \geq 8n^2+1 for n \geq 2 so the result is true by pigeonhole.

[wasitahmid]

I have personally done another solution of Q1..I Think it is more elegant than that...

[thayanithirk]

Can I know that what are you doing currently?as a career, and what did you studied