Problem 2

Part of 2024 Ukraine National Mathematical Olympiad

Problems(4)

Pyramid with largest absolute value

Source: Ukrainian Mathematical Olympiad 2024. Day 1, Problem 8.2

There is a table with

n > 2

cells in the first row,

n-1

cells in the second row is a cell,

n-2

in the third row,

\ldots

1

n

-th row. The cells are arranged as shown below.
https://i.ibb.co/0Z1CR0c/UMO24-8-2.png
In each cell of the top row Petryk writes a number from

1

n

, so that each number is written exactly once. For each other cell, if the cells directly above it contains numbers

a, b

, it contains number

|a-b|

. What is the largest number that can be written in a single cell of the bottom row?
Proposed by Bogdan Rublov

combinatoricsconstructionabsolute value

Rectangle Cover

Source: Ukrainian Mathematical Olympiad 2024. Day 1, Problem 9.2

For some positive integer

n

, consider the board

n\times n

. On this board you can put any rectangles with sides along the sides of the grid. What is the smallest number of such rectangles that must be placed so that all the cells of the board are covered by distinct numbers of rectangles (possibly

0

)? The rectangles are allowed to have the same sizes.
Proposed by Anton Trygub

combinatoricsgridcovering

Smallest representable

Source: Ukrainian Mathematical Olympiad 2024. Day 1, Problem 10.2

You are given a positive integer

n

. Find the smallest positive integer

k

, for which there exist integers

a_1, a_2, \ldots, a_k

, for which the following equality holds:

2^{a_1} + 2^{a_2} + \ldots + 2^{a_k} = 2^n - n + k

Proposed by Mykhailo Shtandenko

number theoryalgebrapower of 2

Source: Ukrainian Mathematical Olympiad 2024. Day 1, Problem 11.2

You are given positive integers

m, n>1

. Vasyl and Petryk play the following game: they take turns marking on the coordinate plane yet unmarked points of the form

(x, y)

x, y

are positive integers with

1 \leq x \leq m, 1 \leq y \leq n

. The player loses if after his move there are two marked points, the distance between which is not a positive integer. Who will win this game if Vasyl moves first and each player wants to win?
Proposed by Mykyta Kharin

number theorygeometrycombinatoricsgame