C/m nếu a,b là các số nguyên dương thì
\(\left(\left[a,b\right]c\right)=\left[\left(a,c\right),\left(b,c\right)\right]\)
\(\left[\left(a,b\right),c\right]=\left(\left[a,c\right],\left[b,c\right]\right)\)
\(\left[a,b,c\right]=\frac{abc.\left(a,b,c\right)}{\left(a,b\right),\left(b,c\right),\left(c,a\right)}\)
\(\left(a,b,c\right)=\frac{abc.\left[a,b,c\right]}{\left[a,b\right],\left[b,c\right],\left[c,a\right]}\)
C/m rằng:
\(\frac{\left(x-a\right)\left(x-c\right)}{\left(c-a\right)\left(c-b\right)}+\frac{\left(x-a\right)\left(x-c\right)}{\left(b-a\right)\left(b-c\right)}+\frac{\left(x-b\right)\left(x-c\right)}{\left(a-b\right)\left(a-c\right)}=1\)với a, b, c khác nhau
Tính A= \(\frac{\left(x-a\right)\left(x-b\right)}{\left(c-a\right)\left(c-b\right)}+\frac{\left(x-a\right)\left(x-c\right)}{\left(b-a\right)\left(b-c\right)}+\frac{\left(x-b\right)\left(x-c\right)}{\left(a-b\right)\left(a-c\right)}\)
Tính :
\(\frac{\left(x-a\right)\left(x-b\right)}{\left(c-a\right)\left(c-b\right)}+\frac{\left(x-a\right)\left(x-c\right)}{\left(b-a\right)\left(b-c\right)}+\frac{\left(x-b\right)\left(x-c\right)}{\left(a-b\right)\left(a-c\right)}\)
\(\frac{\left(x-a\right)\left(x-b\right)}{\left(c-a\right)\left(c-b\right)}+\frac{\left(x-a\right)\left(x-c\right)}{\left(b-a\right)\left(b-c\right)}+\)\(\frac{\left(x-b\right)\left(x-c\right)}{\left(a-b\right)\left(a-c\right)}\)
Tính tổng \(S=\frac{\left(x-a\right)\left(x-b\right)}{\left(c-a\right)\left(c-b\right)}+\frac{\left(x-a\right)\left(x-c\right)}{\left(b-a\right)\left(b-c\right)}+\frac{\left(x-b\right)\left(x-c\right)}{\left(a-b\right)\left(a-c\right)}\) với a,b,c đôi một khác nhau
CMR : Nếu a,b,c khác nhau thì :
\(\frac{b-c}{\left(a-b\right)\left(a-c\right)}+\frac{c-a}{\left(b-c\right)\left(b-a\right)}+\frac{a-b}{\left(c-a\right)\left(c-b\right)}=\frac{2}{a-b}+\frac{2}{b-c}+\frac{2}{c-a}\)
CMR
\(\frac{1}{2}\left[\frac{b-c}{\left(a-b\right)\left(a-c\right)}+\frac{c-a}{\left(b-c\right)\left(b-a\right)}+\frac{a-b}{\left(c-a\right)\left(c-b\right)}\right]=\frac{1}{a-b}+\frac{1}{b-c}+\frac{1}{c-a}\)
Cho a,b,c là các số thực bất kì. CMR:
\(\left|a\right|+\left|b\right|+\left|c\right|+\left|a+b+c\right|\ge\left|a+b\right|+\left|b+c\right|+\left|c+a\right|\)
Một bạn nào đó (không biết rõ tên) đã giải như sau:
Ta có: \(\left|a\right|+\left|b\right|+\left|c\right|+\left|a+b+c\right|\ge\left|a+b\right|+\left|a+c\right|+\left|b\right|+\left|c\right|\)
\(\ge\left|a+b\right|+\left|b+c\right|+\left|c+a\right|\)
Em hãy cho nhận xét về lời giải trên! Nếu sai hãy sửa lại cho đúng!
Cho 3 số a,b,c đôi một phân biệt. CMR:
\(\frac{b-c}{\left(a-b\right)\left(a-c\right)}+\frac{c-a}{\left(b-c\right)\left(b-a\right)}+\frac{a-b}{\left(c-a\right)\left(c-b\right)}=2\left(\frac{1}{a-b}+\frac{1}{b-c}+\frac{1}{c-a}\right)\\ \)
