HMMT 二月 2000 · 代数 · 第 9 题

英文解析

9. W e an view these onditions as a geometry diagram as seen b elo w. So, we kno w thatpe 3 3 3 3
   2 2 = (sin e e = a b = d = f and we kno w that e + f = 15 (sin e this isf 4 4 4 4
   p p
   2 2 2 2
   a + b + d ). Also, note that a + bd ad b = ( a b )( d ) = ef . So, solving
   2 2
   2 2 2 2
   for e and f , we nd that e + f = 225, so 16 e + 16 f = 3600, so (4 e ) + (4 f ) = 3600,
   2 2
   2 2 2 2 2
   so (3 f ) + (4 f ) = 3600, so f (3 + 4 ) = 3600, so 25 f = 3600, so f = 144 and f = 12.
   Th us, e = 12 = 9. Therefore, ef = 9 12 = 108 .3
   e 4
   dbafc
   4 2