由 kaseikami 於 星期日 十二月 05, 2004 2:10 am
令 N = 100 * a + 10 * b + c
a - b + c = 11 * k ( k = 0 or 1 )
a^2 + b^2 + c^2 = 100 * a / 11 + 10 * b / 11 + c / 11
=> a^2 + b^2 + c^2 = 9 * a + b + k
(I) k = 0 => a + c = b
=> a^2 + ( a + c )^2 + c^2 = 9 * a + ( a + c )
=> 2 * a^2 - 10 * a + 2 * c^2 - c + 2 * a * c = 0
=> 2 * a^2 + ( 2 * c - 10 ) * a + 2 * c^2 - c = 0
2 = A , 2 * c - 10 = B , 2 * c^2 - c = C
B^2 - 4 * A * C = 4 * c^2 - 40 * c +100 - 16*c^2 + 8 * c
= -12* c^2 - 32 * c + 100 >= 0 且左式需為完全平方數
所以滿足條件的 c = 0 => a = 5 , b = 5 => N = 550
(II) k = 1 => a + c - 11 = b
=> a^2 + ( a + c - 11 )^2 + c^2 = 9 * a + ( a + c - 11 )
=> 2 * a^2 + ( 2 * c - 32 ) * a + 2 * c^2 - 23 * c + 110 = 0
2 = A , 2 * c - 32 = B , 2 * c^2 - 23 * c = C
B^2 - 4 * A * C = -12 * c^2 + 56 *c + 1024 >= 0 且左式需為完全平方數
滿足條件的 c 無解
所以 N = 550 #