a. Theo tính chất dãy tỉ số bằng nhau ta có:
\(\frac{a}{{2007}} = \frac{b}{{2009}} = \frac{c}{{2011}} = \frac{{a - c}}{{2007 - 2011}} = \frac{{a - b}}{{207 - 2009}} = \frac{{b - c}}{{2009 - 2011}}\)
\(\frac{{a - c}}{4} = \frac{{a - b}}{2} = \frac{{b - c}}{2}\)
\(\frac{{a - c}}{4} = \frac{{a - b}}{2} \Rightarrow {\left( {\frac{{a - c}}{4}} \right)^2} = \frac{{a - b}}{2}.\frac{{b - c}}{2}\)
\( \Rightarrow \frac{{{{(a - c)}^2}}}{4} = (a - b)(b - c)\)
b.
\(\frac{{a + c + d}}{b} + 1 = \frac{{a + b + d}}{c} + 1 = \frac{{a + b + c}}{d} + 1 = \frac{{b + c + d}}{a} + 1\)
\(\frac{{a + c + d + b}}{b} = \frac{{a + b + d + c}}{c} = \frac{{a + b + c + d}}{d} = \frac{{b + c + d + a}}{a}\)
Nếu a + b + c+ d = 0 thì b + c + d = -a; a + b = - (c + d); a + b + c = - d; b + c + d = a.
\(A = \frac{a}{{b + c + d}} + \frac{{a + b}}{{c + d}} + \frac{{a + b + c}}{d} + \frac{{a + b + c + d}}{a}\)
\(=\frac{a}{{ - a}} + \frac{{a + b}}{{ - \left( {a + b} \right)}} + \frac{{ - d}}{d} + \frac{0}{a}=-3\)
Nếu a + b + c + d e 0 thì a = b = c = d
\(A = \frac{1}{3} + \frac{2}{2} + \frac{3}{1} + \frac{4}{1} = \frac{{25}}{3}\)
Vậy A = - 4 nếu a + b + c + d = 0 và A = 25/3 nếu \( a + b + c + d e 0\)
