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PUMaC 2013 · 团队赛 · 第 7 题

PUMaC 2013 — Team Round — Problem 7

专题
Discrete Math / 离散数学
难度
L3
来源
PUMaC

题目详情

  1. Find the total number of triples of integers ( x, y, n ) satisfying the equation + = , where 2 x y n n is either 2012 or 2013.
解析
  1. Find the total number of triples of integers ( x, y, n ) satisfying the equation + = , where 2 x y n n is either 2012 or 2013. 2 2 4 SOLUTION: For n , the equation is the same as ( x − n )( y − n ) = n . For each particular n , 4 there are 2 d ( n ) − 1 solution pairs ( x, y ) in N , where d ( m ) is the number of positive divisors 2 2 2 4 of m ∈ N (since ( − n ) · ( − n ) = n would lead to x = y = 0, which is rejected). The answer 4 4 follows from calculating d (2012 ) and d (2013 ). ANSWER: 338