1, a, Cho a khác-b; a khác -c; b khác -c. CMR: \(\frac{b^2-c^2}{\left(a+b\right)\left(a+c\right)}+\frac{c^2-a^2}{\left(b+c\right)\left(b+a\right)}+\frac{a^2-b^2}{\left(c+a\right)\left(c+b\right)}=\frac{b-c}{b+c}+\frac{c-a}{c+a}+\frac{a-b}{a+b}\) b, CHo hai sô x,y thỏa mãn \(\left(x+\sqrt{2017+x^2}\right)\left(y+\sqrt{2017+y^2}\right)=2017\)Tính giá trị của biểu thức:\(P=x^{2017}+y^{2017}+2017\)
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a) Ta có \(\frac{b^2-c^2}{\left(a+b\right).\left(a+c\right)}=\frac{\left(b+c\right)\left(b-c\right)}{\left(a+b\right).\left(a+c\right)}=\frac{\left(b+c\right)\left(a+b-a-c\right)}{\left(a+b\right).\left(a+c\right)}\)
\(=\frac{\left(b+c\right)\left(a+b\right)-\left(b+c\right).\left(a+c\right)}{\left(a+b\right).\left(a+c\right)}=\frac{b+c}{a+c}-\frac{b+c}{a+b}\)
Tương tự \(\frac{c^2-a^2}{\left(b+c\right)\left(b+a\right)}=\frac{c+a}{b+a}-\frac{c+a}{b+c}\)
\(\frac{a^2-b^2}{\left(c+a\right).\left(c+b\right)}=\frac{a+b}{c+b}-\frac{a+b}{c+a}\)
Do đó \(\frac{b^2-c^2}{\left(a+b\right)\left(a+c\right)}+\frac{c^2-a^2}{\left(b+c\right)\left(b+a\right)}+\frac{a^2-b^2}{\left(c+a\right).\left(c+b\right)}\)
\(=\frac{b+c}{a+c}-\frac{b+c}{a+b}+\frac{c+a}{b+a}-\frac{c+a}{b+c}+\frac{a+b}{c+b}-\frac{a+b}{c+a}\)
\(=\frac{b+c-a-b}{a+c}+\frac{a+b-c-a}{b+c}+\frac{c+a-b-c}{a+b}\)
\(=\frac{c-a}{a+c}+\frac{b-c}{b+c}+\frac{a-b}{a+b}\)
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