HMMT 二月 2017 · COMB 赛 · 第 3 题
HMMT February 2017 — COMB Round — Problem 3
题目详情
- There are 2017 jars in a row on a table, initially empty. Each day, a nice man picks ten consecutive jars and deposits one coin in each of the ten jars. Later, Kelvin the Frog comes back to see that N of the jars all contain the same positive integer number of coins (i.e. there is an integer d > 0 such that N of the jars have exactly d coins). What is the maximum possible value of N ?
解析
- There are 2017 jars in a row on a table, initially empty. Each day, a nice man picks ten consecutive jars and deposits one coin in each of the ten jars. Later, Kelvin the Frog comes back to see that N of the jars all contain the same positive integer number of coins (i.e. there is an integer d > 0 such that N of the jars have exactly d coins). What is the maximum possible value of N ? Proposed by: Kevin Sun Answer: 2014 Label the jars 1 , 2 , . . . , 2017 . I claim that the answer is 2014 . To show this, we need both a construction and an upper bound. For the construction, for 1 ≤ i ≤ 201, put a coin in the jars 10 i + 1 , 10 i + 2 , . . . , 10 i + 10 . After this, each of the jars 1 , 2 , . . . , 2010 has exactly one coin. Now, put a coin in each of the jars 2008 , 2009 , . . . , 2017 . Now, the jars 1 , 2 , . . . , 2007 , 2011 , 2012 , . . . , 2017 all have exactly one coin. This gives a construction for N = 2014 (where d = 1). Now, we show that this is optimal. Let c , c , . . . , c denote the number of coins in each of the jars. 1 2 2017 For 1 ≤ j ≤ 10, define s = c + c + c + . . . . j j j +10 j +20 Note that throughout the process, s = s = · · · = s . It is also easy to check that the sums s , s , . . . , s 1 2 j 1 2 7 each involve 202 jars, while the sums s , s , s each involve 201 jars. 8 9 10 Call a jar good if it has exactly d coins. If there are at least 2015 good jars, then one can check that it is forced that at least one of s , s , . . . , s only involves good jars, and similarly, at least one of s , s , s 1 2 7 8 9 10 only involves good jars. But this would mean that 202 d = 201 d as all s are equal, contradiction. i