设x1,x2,x3,……,x2007为实数,且满足x1x2x3……x2007=x1-x2x3x4x5……x2007=x1x2-x3x4x5……x2007=……=x1x2x3……x2006-x2007=1,则x2000的值是

Question

设x1,x2,x3,……,x2007为实数,且满足x1x2x3……x2007=x1-x2x3x4x5……x2007=x1x2-x3x4x5……x2007=……=x1x2x3……x2006-x2007=1,则x2000的值是_________

Answer 1

考虑以下方程组
ab=1
ab= a-b
(a+1)b = a
b = a/(a+1)
a^2/(a+1) = 1
a^2 - a - 1 = 0
a = (1 +- 5^0.5)/2
b
= (1 +- 5^0.5)/(3 +- 5^0.5)
= (-2 +- 2*5^0.5)/(9-5)
= (-1 +- 5^0.5)/2
---------------------------------------------
因
x1x2...x2007 =1
x1 - x2x3...x2007 = 1
故
x1 = (1 +- 5^0.5)/2
x2x3...x2007 = (-1 +- 5^0.5)/2
同理
x1x2 = (1 +- 5^0.5)/2 (不一定与上组解同号）
x3x4...x2007 = (-1 +- 5^0.5)/2
x1x2x3 = (1 +- 5^0.5)/2 (不一定与上组解同号）
x4x5...x2007 = (-1 +- 5^0.5)/2
...
x1x2...x2006 = (1 +- 5^0.5)/2 (不一定与上组解同号）
x2007 = (-1 +- 5^0.5)/2
x2 = x1x2/x1 = (1 +- 5^0.5)/(1 +- 5^0.5) = 1
或 x2 = x1x2/x1 = (1 -+ 5^0.5)/(1 +- 5^0.5) = (-3 +- 5^0.5)/2
因此
x1 = (1 +- 5^0.5)/2
x2 = 1,(-3 +- 5^0.5)/2 （与x1中的根号5同号）
同理 x3 = 1,(-3 +或- 5^0.5)/2 （与上方最相邻的不为1的解中的根号5反号）
x4 = 1,(-3 +或- 5^0.5)/2（与上方最相邻的不为1的解中的根号5反号）
...
x2006 = 1,(-3 +或- 5^0.5)/2（与上方最相邻的不为1的解中的根号5反号）
x2007 = (1 +或- 5^0.5)/2 （若x2...x2006中有偶数个不为1,则x1与x2007的根号5同号,否则为反号）
共2^2006组解
x2000有三个可能值：1,(-3 + 5^0.5)/2,(-3 - 5^0.5)/2
例：
x1 = (1 + 5^0.5)/2
x2 = (-3 + 5^0.5)/2
x3=x4=...=x2006=1
x2007 = (-1 - 5^0.5)/2
为一组解
x1 = (1 + 5^0.5)/2
x2=x3=...=x1999=1
x2000 = (-3 + 5^0.5)/2
x2001=...=x2006=1
x2007 = (-1 - 5^0.5)/2
也是一组解
x1 = (1 + 5^0.5)/2
x2 = (-3 + 5^0.5)/2
x3=x4=...=x1999=1
x2000= (-3 - 5^0.5)/2
x2001=...=x2006=1
x2007 = (-1 + 5^0.5)/2
也是一组解