Found problems: 29
2019 Teodor Topan, 4
Ana choses two real numbers $ y>0,x $ and Bogdan repeatedly tries to guess these in the following manner: at step $ j $ he choses a real number $ b_j, $ asks her if $ b_j=x+jy, $ and she tells him the truth.
[b]a)[/b] If $ x=0, $ can Bogdan find Ana's numbers in a finite number of steps?
[b]b)[/b] If $ x\neq 0, $ can Bogdan find Ana's numbers in a finite number of steps?
2017 Israel National Olympiad, 7
A table with $m$ rows and $n$ columns is given. In each cell of the table an integer is written. Heisuke and Oscar play the following game: at the beginning of each turn, Heisuke may choose to swap any two columns. Then he chooses some rows and writes down a new row at the bottom of the table, with each cell consisting the sum of the corresponding cells in the chosen rows. Oscar then deletes one row chosen by Heisuke (so that at the end of each turn there are exactly $m$ rows). Then the next turn begins and so on. Prove that Heisuke can assure that, after some finite amount of turns, no number in the table is smaller than the number to the number on his right.
Example: If we begin with $(1,1,1),(6,5,4),(9,8,7)$, Heisuke may choose to swap the first and third column to get $(1,1,1),(4,5,6),(7,8,9)$. Then he chooses the first and second rows to obtain $(1,1,1),(4,5,6),(7,8,9),(5,6,7)$. Then Oscar has to delete either the first or the second row, let's say the second. We get $(1,1,1),(7,8,9),(5,6,7)$ and Heisuke wins.
2023 Ukraine National Mathematical Olympiad, 10.2
On a rectangular board $100 \times 300$, two people take turns coloring the cells that have not yet been colored. The first one colors cells in yellow, and the second one in blue. Coloring is completed when every cell of the board is colored. A [i]connected sequence[/i] of cells is a sequence of cells in which every two consecutive cells share a common side (and all cells in the sequence are different). Consider all possible connected sequences of yellow cells. The result of the first player is the number of cells in the connected sequence of yellow cells of maximum length. The first player's goal is to maximize the result, and the second player's goal is to make the first player's result as small as possible. Prove that if each player tries to achieve his goal, the result of the first player will be no more than $200$.
[i]Proposed by Mykhailo Shtandenko and Fedir Yudin[/i]
2021 Israel TST, 1
An ordered quadruple of numbers is called [i]ten-esque[/i] if it is composed of 4 nonnegative integers whose sum is equal to $10$. Ana chooses a ten-esque quadruple $(a_1, a_2, a_3, a_4)$ and Banana tries to guess it. At each stage Banana offers a ten-esque quadtruple $(x_1,x_2,x_3,x_4)$ and Ana tells her the value of
\[|a_1-x_1|+|a_2-x_2|+|a_3-x_3|+|a_4-x_4|\]
How many guesses are needed for Banana to figure out the quadruple Ana chose?