HMMT 十一月 2015 · 冲刺赛 · 第 5 题

英文解析

5. [ 6 ] James writes down three integers. Alex picks some two of those integers, takes the average of them,
   and adds the result to the third integer. If the possible final results Alex could get are 42, 13, and 37,
   what are the three integers James originally chose?
   Proposed by: Alexander Katz
   Answer: − 20 , 28 , 38
   Let x, y, z be the integers. We havex + y + z = 42
   y + z 2 + x = 13
   x + z 2 + y = 37
   x + y 2
   Adding these three equations yields 2( x + y + z ) = 92, so + z = 23 + = 42 so z = 38. Similarly,z
   2 2
   x = − 20 and y = 28.