AoPS - Problem

Source: 2018 Taiwan APMO preliminary

Tags: combinatorics, number theory

hsiangshen

10.11.2020 18:02

$240$ students are participating a big performance show. They stand in a row and face to their coach. The coach askes them to count numbers from left to right, starting from $1$. (Of course their counts be like $1,2,3,...$)The coach askes them to remember their number and do the following action: First, if your number is divisible by $3$ then turn around. Then, if your number is divisible by $5$ then turn around. Finally, if your number is divisible by $7$ then turn around. (a) How many students are face to coach now? (b) What is the number of the $66^{\text{th}}$ student counting from left who is face to coach?

I generated a list of numbers by hand and it confirms my answer above. Also, I am getting part (b) to be $105+38=143$. 1,2,_,4,_,_,_,8,_,_,11,_,13,_,_,16,17,_,19,_,_,22,23,_,_,26,_,_,29,_,31,32,_,34,_,_,37,38,_,_,41,_,43,44,_,46,47,_,_,_,_,52,53,_,_,_,_,58,59,_,61,62,_,64,_,_,67,68,_,_,71,_,73,74,_,76,_,_,79,_,_,82,83,_,_,86,_,88,89,_,_,92,_,94,_,_,97,_,_,_,101,_,103,104,_ 105+(1,2,_,4,_,_,_,8,_,_,11,_,13,_,_,16,17,_,19,_,_,22,23,_,_,26,_,_,29,_,31,32,_,34,_,_,37,38,_,_,41,_,43,44,_,46,47,_,_,_,_,52,53,_,_,_,_,58,59,_,61,62,_,64,_,_,67,68,_,_,71,_,73,74,_,76,_,_,79,_,_,82,83,_,_,86,_,88,89,_,_,92,_,94,_,_,97,_,_,_,101,_,103,104,_) 2*105+(1,2,_,4,_,_,_,8,_,_,11,_,13,_,_,16,17,_,19,_,_,22,23,_,_,26,_,_,29,_)