Cho \[{\left( {{a_1}m - {b_1}n} \right)^{2024}} + {\left( {{a_2}m - {b_2}n} \right)^{2024}} + {\left( {{a_3}m - {b_3}n} \right)^{2024}} + ... + {\left( {{a_{2025}}m - {b_{2025}}n} \right)^{2024}} \le 0\].

Chứng minh rằng: \[\frac{{{a_1} + {a_2} + {a_3} + ... + {a_{2025}}}}{{{b_1} + {b_2} + {b_3} + ... + {b_{2025}}}} = \frac{n}{m}\].

c) \[\frac{{x - 214}}{{86}} + \frac{{x - 132}}{{84}} + \frac{{x - 54}}{{82}} = 6\]

\[x\left( {\frac{1}{{86}} + \frac{1}{{84}} + \frac{1}{{82}}} \right) = 6 + \frac{{214}}{{86}} + \frac{{132}}{{84}} + \frac{{54}}{{82}}\]

\[x\left( {\frac{1}{{86}} + \frac{1}{{84}} + \frac{1}{{82}}} \right) = \left( {1 + \frac{{214}}{{86}}} \right) + \left( {2 + \frac{{132}}{{84}}} \right) + \left( {3 + \frac{{54}}{{82}}} \right)\]

\[x\left( {\frac{1}{{86}} + \frac{1}{{84}} + \frac{1}{{82}}} \right) = \frac{{300}}{{86}} + \frac{{300}}{{84}} + \frac{{300}}{{82}}\]

\[x\left( {\frac{1}{{86}} + \frac{1}{{84}} + \frac{1}{{82}}} \right) = 300\left( {\frac{1}{{86}} + \frac{1}{{84}} + \frac{1}{{82}}} \right)\]

\[x = 300\]

Vậy \[x = 300\].

d) \[\left| {x + \frac{1}{{1.2.3}}} \right| + \left| {x + \frac{1}{{2.3.4}}} \right| + \left| {x + \frac{1}{{3.4.5}}} \right| + .... + \left| {x + \frac{1}{{18.19.20}}} \right| = 19x\]

Do \[\left| {x + \frac{1}{{1.2.3}}} \right| + \left| {x + \frac{1}{{2.3.4}}} \right| + \left| {x + \frac{1}{{3.4.5}}} \right| + .... + \left| {x + \frac{1}{{18.19.20}}} \right| \ge 0\] với mọi \[x\].

Do đó, \[19x \ge 0\], suy ra \[x \ge 0\].

Với mọi \[x \ge 0\], ta có:

\[x + \frac{1}{{1.2.3}} + x + \frac{1}{{2.3.4}} + x + \frac{1}{{3.4.5}} + .... + x + \frac{1}{{18.19.20}} = 19x\]

\[x + \frac{1}{{1.2.3}} + x + \frac{1}{{2.3.4}} + x + \frac{1}{{3.4.5}} + .... + x + \frac{1}{{18.19.20}} = 19x\]

\[18x + \left( {\frac{1}{{1.2.3}} + \frac{1}{{2.3.4}} + \frac{1}{{3.4.5}} + .... + \frac{1}{{18.19.20}}} \right) = 19x\]

\[x = \frac{1}{{1.2.3}} + \frac{1}{{2.3.4}} + \frac{1}{{3.4.5}} + .... + \frac{1}{{18.19.20}}\]

\[x = \frac{1}{2}\left( {\frac{1}{{1.2}} - \frac{1}{{2.3}} + \frac{1}{{2.3}} - \frac{1}{{3.4}} + \frac{1}{{3.4}} - \frac{1}{{4.5}}.... + \frac{1}{{18.19}} - \frac{1}{{19.20}}} \right)\]

\[x = \frac{1}{2}\left( {\frac{1}{{1.2}} - \frac{1}{{19.20}}} \right)\]

\[x = \frac{1}{2}\left( {\frac{1}{2} - \frac{1}{{380}}} \right)\]

\[x = \frac{1}{2}\left( {\frac{1}{2} - \frac{1}{{380}}} \right)\]

\[x = \frac{{189}}{{760}}\] (thỏa mãn)

Vậy \[x = \frac{{189}}{{760}}\].