MathDB

Problem 4

Part of 2023 Kyiv City MO Round 1

Problems(4)

No right triangle on a grid

Source: Kyiv City MO 2023 7.4

5/14/2023
For n≥2n \ge 2 consider n×nn \times n board and mark all n2n^2 centres of all unit squares. What is the maximal possible number of marked points that we can take such that there don't exist three taken points which form right triangle? Proposed by Mykhailo Shtandenko
combinatorics
Similar numbers

Source: Kyiv City MO 2023 Round 1, Problem 8.4

12/16/2023
Let's call a pair of positive integers a1a2…ak‾\overline{a_1a_2\ldots a_k} and b1b2…bk‾\overline{b_1b_2\ldots b_k} kk-similar if all digits a1,a2,…,ak,b1,b2,…,bka_1, a_2, \ldots, a_k , b_1 , b_2, \ldots, b_k are distinct, and there exist distinct positive integers m,nm, n, for which the following equality holds:
a1m+a2m+…+akm=b1n+b2n+…+bkna_1^m + a_2^m + \ldots + a_k^m = b_1^n + b_2^n + \ldots + b_k^n
For which largest kk do there exist kk-similar numbers?
Proposed by Oleksiy Masalitin
number theoryalgebraDigits
Cutting grid into corners squared

Source: Kyiv City MO 2023 Round 1, Problem 10.4

12/16/2023
Positive integers m,nm, n are such that mnmn is divisible by 99 but not divisible by 2727. Rectangle m×nm \times n is cut into corners, each consisting of three cells. There are four types of such corners, depending on their orientation; you can see them on the figure below. Could it happen that the number of corners of each type was the exact square of some positive integer?
Proposed by Oleksiy Masalitin
https://i.ibb.co/Y8QSHyf/Kyiv-MO-2023-10-4.png
gridcombinatorics
AoPS suggested tag 3D geometry for this number theory

Source: Kyiv City MO 2023 Round 1, Problem 11.4

12/16/2023
Find all pairs (m,n)(m, n) of positive integers, for which number 2n−13m2^n - 13^m is a cube of a positive integer.
Proposed by Oleksiy Masalitin
number theory