\(a^{2012}+b^{2012}+c^{2012}\ge3\sqrt[3]{\left(abc\right)^{2012}}=3\)

\(\Rightarrow\dfrac{1}{a^{2012}+b^{2012}+c^{2012}}\le\dfrac{1}{3}\)

\(\Rightarrow-\dfrac{1}{a^{2012}+b^{2012}+c^{2012}}\ge-\dfrac{1}{3}\)

Lại có:

\(a^{2013}+a^{2013}+...+a^{2013}\left(\text{2012 số hạng}\right)+1\ge2013\sqrt[2013]{\left(a^{2013}\right)^{2012}}=2013.a^{2012}\)

\(\Rightarrow2012.a^{2013}+1\ge2013.a^{2012}\)

Tương tự: \(2012.b^{2013}+1\ge2013.b^{2012}\) ; \(2012.c^{2013}+1\ge2013.c^{2012}\)

Cộng vế với vế:

\(\Rightarrow a^{2013}+b^{2013}+c^{2013}\ge\dfrac{2013\left(a^{2012}+b^{2012}+c^{2012}\right)-3}{2012}\)

\(\Rightarrow A\ge\dfrac{2013\left(a^{2012}+b^{2012}+c^{2012}\right)-3}{2012\left(a^{2012}+b^{2012}+c^{2012}\right)}=\dfrac{2013}{2012}-\dfrac{3}{2012}.\dfrac{1}{a^{2012}+b^{2012}+c^{2012}}\ge\dfrac{2013}{2012}-\dfrac{3}{2012}.\dfrac{1}{3}=1\)

\(A_{min}=1\) khi \(a=b=c=1\)

Ta có: \(a^2+b^2+c^2=1\)

\(\Rightarrow\left\{{}\begin{matrix}\left|a\right|\le1\\\left|b\right|\le1\\\left|c\right|\le1\end{matrix}\right.\)

Ta lại có:

\(a^3+b^3+c^3=a^2+b^2+c^2\)

\(\Leftrightarrow a^2\left(1-a\right)+b^2\left(1-b\right)+c^2\left(1-c\right)=0\)

Vì \(\left\{{}\begin{matrix}1-a\ge0\\1-b\ge0\\1-c\ge0\end{matrix}\right.\)

\(\Rightarrow a^2\left(1-a\right)+b^2\left(1-b\right)+c^2\left(1-c\right)\ge0\)

Dấu = xảy ra khi: \(\left(a,b,c\right)=\left(1,0,0;0,1,0;0,0,1\right)\)

\(\Rightarrow S=1\)

Đề \(\Rightarrow a^{2014}+b^{2014}-2\left(a^{2013}+b^{2013}\right)+a^{2012}+b^{2012}=0\)

\(\Leftrightarrow a^{2012}\left(a^2-2a+1\right)+b^{2012}\left(b^2-2b+1\right)=0\)

\(\Leftrightarrow a^{2012}\left(a-1\right)^2+b^{2012}\left(b-1\right)^2=0\)

\(\Leftrightarrow\left(a=0\text{ hoặc }a=1\right)\text{ và }\left(b=0\text{ hoặc }b=1\right)\)

\(+a=0\text{ hoặc }a=1\text{ thì }a^{2014}=a^{2010}\)

\(+b=0\text{ hoặc }b=1\text{ thì }b^{2014}=b^{2010}\)

Suy ra \(a^{2014}+b^{2014}=a^{2010}+b^{2010}\)

ta có \(a^{2012}+b^{2012}=a^{2013}+b^{2013}\)

\(\Rightarrow a^{2012}-a^{2013}+b^{2012}_{ }-b^{2013}=0\)

\(\Rightarrow a^{2012}\left(1-a\right)+b^{2012}\left(1-b\right)=0\)\(\left(1\right)\)

tương tự \(a^{2013}+b^{2013}=a^{2014}+b^{2014}\)

\(\Leftrightarrow a^{2013}\left(1-a\right)+b^{2013}\left(1-b\right)=0\)\(\left(2\right)\)

trừ (1) cho (2)

ta có \(\left(a^{2012}-a^{2013}\right)\left(1-a\right)\)\(+\left(b^{2012}-b^{2013}\right)\left(1-b\right)=0\)

\(\Leftrightarrow a^{2012}\left(1-a\right)^2+b^{2012}\left(1-b\right)^2=0\)

mà\(a^{2012}\left(1-a\right)^2\ge0;b^{2012}\left(1-b\right)^2\ge0\)

\(\Rightarrow a=1;b=1\)

\(\Rightarrow M=20\times1+11\times1+2013=2044\)

lay cai dau tru cai thu 2

xong lay cai thu 2 tru cai thu 3

xong lay ket qua dau tim dc tru ket qua sau la tim dc a=b=1

roi thay vao tinh M la xong

Ta có: \(a^{2012}+b^{2012}=a^{2013}+b^{2012}=a^{2014}+b^{2014}\)

\(\Rightarrow a^{2012}+b^{2012}-2\left(a^{2013}+b^{2013}\right)+a^{2014}+b^{2014}=0\)

\(\Rightarrow a^{2012}+b^{2012}-2\left(a^{2013}+b^{2013}\right)+a^{2014}+b^{2014}=0\)

\(\Leftrightarrow\left(a^{1006}-a^{1007}\right)^2+\left(b^{1006}-b^{1007}\right)=0\)

Từ đó ta có 2 TH

\(\hept{\begin{cases}a^{1006}-a^{1007}=0\\b^{1006}-b^{1007}=0\end{cases}\hept{\begin{cases}a=0;a=1\\b=0;b=1\end{cases}}}\)

Vậy P=20.0+11.0+2013=2013

       P=20.1+11.0+2013=2033

       P=20.0+11.1+2013=2024

Chả biết

a+b+c=0=> a²-b²-c²=2bc,b²-c²-a²=2ac,c²+a²-b²=-2ac,c²-a²-b²=2ab

=>\(P=\frac{a}{c}.\frac{2bc}{2ac}.\frac{-2ac}{2ab}=-1\)

a+b+c=0 <=> a+b=-c; b+c=-a;c+a=-b

\(\frac{a^2-b^2-c^2}{b^2-c^2-a^2}=\frac{\left(a-c\right)\left(a+c\right)-b^2}{\left(b-a\right)\left(b+a\right)-c^2}=\frac{\left(a-c\right)\left(-b\right)-b^2}{\left(b-a\right)\left(-c\right)-c^2}=\frac{b\left(c-a-b\right)}{c\left(a-b-c\right)}\)

\(=\frac{b\left[c-\left(a+b\right)\right]}{c\left[a-\left(b+c\right)\right]}=\frac{b\left[c-\left(-c\right)\right]}{c\left[a-\left(-a\right)\right]}=\frac{b.2c}{c.2a}=\frac{b}{a}\)

***

\(\frac{c^2+a^2-b^2}{c^2-a^2-b^2}=\frac{\left(c-b\right)\left(c+b\right)+a^2}{\left(c-b\right)\left(c+b\right)-a^2}=\frac{\left(c-b\right)\left(-a\right)+a^2}{\left(c-b\right)\left(-a\right)-a^2}=\frac{a\left(a+b-c\right)}{a\left(b-c-a\right)}\)

\(=\frac{a+b-c}{b-\left(c+a\right)}=\frac{-c-c}{b-\left(-b\right)}=\frac{-2c}{2b}=\frac{-c}{b}\)

\(P=\frac{a}{c}.\frac{a^2-b^2-c^2}{b^2-c^2-a^2}.\frac{c^2+a^2-b^2}{c^2-a^2-b^2}=\frac{a}{c}.\frac{b}{a}.\frac{-c}{b}=-1\)