一个奇妙的分式方程.doc

一个奇妙的分式方程Math Horizons 杂志 2010 年 4 月刊上发表了一个有点搞笑的题目，很有些愚人节玩笑的味道。观察下面这个分式方程：它可以化简为 x3 - 42x + 36 = 0 ，如果分式方程存在整数解，这个解一定是 36 的约数。把 36 的约数一个一个代进去便可得到，这个分式方程的唯一整数解为：现在，你能快速求解出下面这个方程的整数解吗？只需要注意到，新的分式方程是由原方程旋转 180 度得到的，因此它的解应该为：有意思的是，这个“推导”虽然是荒谬的，但 x = 9 真的就是第二个方程的唯一整数解！来源：/proofs/FalseSymmetry.shtmlFaulty SymmetryThe Math Horizons magazine (v 17, n 4, April 2010, pp. 31-33) published a solution to Problem 239 that was posed by former problem section editor Andy Liu, who found inspiration for the problem from the book Inversions by Scott Kim. Doubled Over asked readers to find an integer root of each of the following two equations:(a)9 (8 - x)(9 - 8) x+8 - x9 - 8+11 - xx - 1=x(b)x =1 - xx - 11+8 - 6x - 8+x (8 - 6)(x - 8) 6Equation (a) is easily converted to a polynomial equation:x - 42x + 36 = 0. According to the rational root theorem, all integer roots of the equation must divide the free term, 36 in this case. The latter has the following factors1, 2, 3, 4, 6, 9, 12, 18, 36. It takes a little time to check that only x = 6 satisfies the equation.Reaching this point the problem editor made a remark:The same type of analysis could be applied to equation (b). But James Hochschild and Zhiheng Li both noticed a very useful symmetry: equation (b) is equation (a) rotated by 180 degrees! Thus, we may simply rotate the solution to (a) by 180 degrees to arrive at 9 = x, the solution for (b). This argument is a very hilarious mathematical joke. Given that it appeared in the Aprils issue of the magazine, I am inclined to think that this is what it meant to be. Lest it be perceived otherwise, I offer a couple of simplified equations that follow the ingenious Andy Lius exercise:(a)x - 111 - x+8 - x9 - 8+6=x(b)x =9+8 - 6x - 8+x - 111 - xAs can be easily seen, x = 9 solves (a) while x = 6 does not fit into (b). The wink at the symmetry for solving (b) had to be an Aprils Fools joke, for, otherwise, it would serve an example of a faulty use of symmetry. Trying to come