MathDB

6

Part of 2021 Argentina National Olympiad

Problems(3)

Argentina MO 2021 National Level 1 P6

Source:

4/23/2022
Milly chooses a positive integer nn and then Uriel colors each integer between 11 and nn inclusive red or blue. Then Milly chooses four numbers a,b,c,da, b, c, d of the same color (there may be repeated numbers). If a+b+c=da+b+c= d then Milly wins. Determine the smallest nn Milly can choose to ensure victory, no matter how Uriel colors.
combinatoricscontests
Argentina MO 2021 National Level 2 P6

Source:

4/23/2022
Decide if it is possible to choose 330330 points in the plane so that among all the distances that are formed between two of them there are at least 17001700 that are equal.
algebracontests
Argentina MO 2021 National Level 3 P6

Source:

4/24/2022
We say that a positive integer kk is tricubic if there are three positive integers a,b,c,a, b, c, not necessarily different, such that k=a3+b3+c3.k=a^3+b^3+c^3.
a) Prove that there are infinitely many positive integers nn that satisfy the following condition: exactly one of the three numbers n,n+2n, n+2 and n+28n+28 is tricubic. b) Prove that there are infinitely many positive integers nn that satisfy the following condition: exactly two of the three numbers n,n+2n, n+2 and n+28n+28 are tricubic. c) Prove that there are infinitely many positive integers nn that satisfy the following condition: the three numbers n,n+2n, n+2 and n+28n+28 are tricubic.
number theorycontestsTricubic numbers