MathDB

5

Part of 2023 All-Russian Olympiad

Problems(3)

Heaps of stones

Source: All-Russian MO 2023 Final stage 9.5

4/23/2023
If there are several heaps of stones on the table, it is said that there are <spanclass=′latex−italic′>many</span><span class='latex-italic'>many</span> stones on the table, if we can find 5050 piles and number them with the numbers from 11 to 5050 so that the first pile contains at least one stone, the second - at least two stones,..., the 5050-th has at least 5050 stones. Let the table be initially contain 100100 piles of 100100 stones each. Find the largest n≤10000n \leq 10 000 such that after removing any nn stones, there will still be <spanclass=′latex−italic′>many</span><span class='latex-italic'>many</span> stones left on the table.
combinatorics
Easy NT divisibility

Source: All-Russian MO 2023 Final stage 10.5

4/23/2023
Find the largest natural number nn for which the product of the numbers n,n+1,n+2,…,n+20n, n+1, n+2, \ldots, n+20 is divisible by the square of one of them.
number theory
Two players want to obtain a number divisible by 2023

Source: All-Russian MO 2023 Final stage 11.5

4/23/2023
Initially, 1010 ones are written on a blackboard. Grisha and Gleb are playing game, by taking turns; Grisha goes first. On one move Grisha squares some 55 numbers on the board. On his move, Gleb picks a few (perhaps none) numbers on the board and increases each of them by 11. If in 10,00010,000 moves on the board a number divisible by 20232023 appears, Gleb wins, otherwise Grisha wins. Which of the players has a winning strategy?
number theory