On Qingqing Grassland, there are 7 sheep numberd $1,2,3,4,5,6,7$ and 2017 wolves numberd $1,2,\cdots,2017$. We have such strange rules: (1) Define $P(n)$: the number of prime numbers that are smaller than $n$. Only when $P(i)\equiv j\pmod7$, wolf $i$ may eat sheep $j$ (he can also choose not to eat the sheep). (2) If wolf $i$ eat sheep $j$, he will immediately turn into sheep $j$. (3) If a wolf can make sure not to be eaten, he really wants to experience life as a sheep. Assume that all wolves are very smart, then how many wolves will remain in the end?
Problem
Source: 2017 China Northern MO, Grade 11, Problem 8
Tags: number theory, prime numbers
25.02.2020 01:07
Small typo: The first rule should be "wolf $i$ may eat sheep $j$ only if $P(i) \equiv j \pmod 7$, but even if this is the case, wolf $i$ does not necessarily have to eat sheep $j$.
25.02.2020 02:34
stroller wrote: Small typo: The first rule should be "wolf $i$ may eat sheep $j$ only if $P(i) \equiv j \pmod 7$, but even if this is the case, wolf $i$ does not necessarily have to eat sheep $j$. Sorry for my carelessness. Thank you.
15.08.2020 13:13
Quote: If a wolf can make sure not to be eaten, he really wants to experience life as a sheep. What does this mean?
15.08.2020 18:28
richardrichard wrote: Quote: If a wolf can make sure not to be eaten, he really wants to experience life as a sheep. What does this mean? I think this means that a wolf won't eat a sheep which was once a wolf
19.11.2024 07:09
Why is there no solution yet