It is the year 2005 now. According to a legend there is a monster that awakes every now and then to swallow everyone who is solving this problem, and then falls back asleep for as many years as the sum of the digits of that year. The monster first hit AoPS in the year +234. Prove you're safe this year, as well as for the coming 10 years.
Problem
Source: flanders junior olympiad '05
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K81o7
25.04.2005 15:50
We can assume that the monster doesn't ever wake up for other reasons, right? We know that every year during which the monster wakes up is a multiple of 3...so we're safe this year. If the monster wakes up in 1992, then some of us will die, but if it does in 1995, we're safe. In other words, if and only if it wakes up on one of the years in the span from 1986-1992 (we're safe in 1983 and 1995) then we're doomed. I'll figure out more later...
thematrix
25.04.2005 20:54
the monster wakes up only if the year is a multiple of 9 ($234= 9*26$);and 2005 is not.In the coming 10 years,the monster can only wake up in the year 2007;the sum of the digits of any number $\leq 2007$ is at maximum 28;for this reason,if the monster will wake in 2007,it woke in one of the years between $1979$($2007-28$) and $1998$,that are multiples of nine.But $1980+1+9+8=1998$;$1989+1+9+8+9=2016$;$1998+1+9+9+8=2025$,so it won't wake up in $2007$