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Omni-MATH / Alice Czarina is bored and is playing a game with a pile of rocks. The pile…

Problem

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problem

Alice Czarina is bored and is playing a game with a pile of rocks. The pile initially contains 2015 rocks. At each round, if the pile has NN rocks, she removes kk of them, where 1≤k≤N1 \leq k \leq N, with each possible kk having equal probability. Alice Czarina continues until there are no more rocks in the pile. Let pp be the probability that the number of rocks left in the pile after each round is a multiple of 5. If pp is of the form 5a⋅31b⋅cd5^{a} \cdot 31^{b} \cdot \frac{c}{d}, where a,ba, b are integers and c,dc, d are positive integers relatively prime to 5⋅315 \cdot 31, find a+ba+b.
Plain-text mathematical notation (without MathML)
Alice Czarina is bored and is playing a game with a pile of rocks. The pile initially contains 2015 rocks. At each round, if the pile has N rocks, she removes k of them, where 1≤k≤N, with each possible k having equal probability. Alice Czarina continues until there are no more rocks in the pile. Let p be the probability that the number of rocks left in the pile after each round is a multiple of 5. If p is of the form 5^(a)⋅31^(b)⋅(c)/(d), where a,b are integers and c,d are positive integers relatively prime to 5⋅31, find a+b.
Original LaTeX notation
Alice Czarina is bored and is playing a game with a pile of rocks. The pile initially contains 2015 rocks. At each round, if the pile has $N$ rocks, she removes $k$ of them, where $1 \leq k \leq N$, with each possible $k$ having equal probability. Alice Czarina continues until there are no more rocks in the pile. Let $p$ be the probability that the number of rocks left in the pile after each round is a multiple of 5. If $p$ is of the form $5^{a} \cdot 31^{b} \cdot \frac{c}{d}$, where $a, b$ are integers and $c, d$ are positive integers relatively prime to $5 \cdot 31$, find $a+b$.

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Source and history

task_id
9c2472fd-db53-5303-853d-3e9342c1e586
task_key
test--9c2472fd-db53-5303-853d-3e9342c1e586
task_revision_id
1
config / split
/ test

Official source

initial import