Đề bài

a) Chứng minh rằng nếu \(a,b,c \ne 0,a + b + c = 0\) thì \(\frac{1}{{ab}} + \frac{1}{{bc}} + \frac{1}{{ca}} = 0\) b) Chứng minh rằng nếu \(x \ne y,y \ne z,z \ne x\) thì \(\frac{1}{{\left( {x - y} \right)\left( {y - z} \right)}} + \frac{1}{{\left( {y - z} \right)\left( {z - x} \right)}} + \frac{1}{{\left( {z - x} \right)\left( {x - y} \right)}} = 0\)

Lời giải chi tiết

a) Ta có: \(\frac{1}{{ab}} + \frac{1}{{bc}} + \frac{1}{{ca}} = \frac{c}{{abc}} + \frac{a}{{abc}} + \frac{b}{{abc}} = \frac{{a + b + c}}{{abc}}\)Theo đầu bài, \(a + b + c = 0\) nên \(\frac{1}{{ab}} + \frac{1}{{bc}} + \frac{1}{{ca}} = 0\)b) Ta có: \(\frac{1}{{\left( {x - y} \right)\left( {y - z} \right)}} + \frac{1}{{\left( {y - z} \right)\left( {z - x} \right)}} + \frac{1}{{\left( {z - x} \right)\left( {x - y} \right)}}\)\( = \frac{{z - x}}{{\left( {x - y} \right)\left( {y - z} \right)\left( {z - x} \right)}} + \frac{{x - y}}{{\left( {x - y} \right)\left( {y - z} \right)\left( {z - x} \right)}} + \frac{{y - z}}{{\left( {z - x} \right)\left( {x - y} \right)\left( {y - z} \right)}}\)\( = \frac{{z - x + x - y + y - z}}{{\left( {x - y} \right)\left( {y - z} \right)\left( {z - x} \right)}} = \frac{0}{{\left( {x - y} \right)\left( {y - z} \right)\left( {z - x} \right)}} = 0\)