Olympiad problems

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Found problems: 2

1984 Polish MO Finals, 4

A coin is tossed $n$ times, and the outcome is written in the form ($a_1,a_2,...,a_n$), where $a_i = 1$ or $2$ depending on whether the result of the $i$-th toss is the head or the tail, respectively. Set $b_j = a_1 +a_2 +...+a_j$ for $j = 1,2,...,n$, and let $p(n)$ be the probability that the sequence $b_1,b_2,...,b_n$ contains the number $n$. Express $p(n)$ in terms of $p(n-1)$ and $p(n-2)$.

2024 Irish Math Olympiad, P7

Tags: Coin , irmo

A game of coins is played as follows: You start with $1$ head and $1$ tail on a table. At each turn, you can perform any one of the following moves: [list=a] [*]You can turn over all the coins on the table. [*]You can triple the number of heads and tails at the table. [*]If there are at least $4$ tails on the table, you can turn over $4$ tails. [*]If there are at least $5$ tails on the table, you can turn over $3$ of the tails and discard $2$ of the tails. [/list] Knowing that at the end of the game you have $2024$ heads, what are all possible numbers of tails at the end of that game?