PUMaC 2014 · 团队赛 · 第 9 题

PUMaC 2014 — Team Round — Problem 9

[ 7 ] Find the largest p such that p + p + p + ... + p  100, n n n 1 n 2 1 th where p denotes the n prime number. n

解析

[ 7 ] Find the largest p such that p + p + p + ... + p ≤ 100, where p denotes n n n − 1 n − 2 1 n th the n prime number. Solution: 2 For a rough estimate of the answer, we would like to have (100 − p ) − p to be no n n − 1 less than 100 − p . Letting p = p and we get that 100 − p has to be bigger than n n n − 1 n 44, so p ≤ 90. The biggest prime that satisfies the condition is 89. We see that p + n n √ √ √ √ √ √ √ 2 p + p + ... + p < 89 + 89 + 89 + 89 + ... = S . Since S = ( S − 89) , n − 1 n − 2 1 179 + sqrt (357) 179 + 20 we see that S = < < 100. The next prime after 89 is 97 which 2 2 obviously fails.