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a+b+c=1 <=> a+b=1-c
+) Nếu 1-c=0 => a+b=0 <=> a=-b
=> A = a2015+b2015+c2015
A = (-b)2015+b2015+c2015
A = c2015 => A = 1 (Vì 1-c=0) (1)
Ta có: a3+b3+c3=1
a3+b3=1-c3
(a+b)(a2-ab+b20=(1-c)(1+c+c2)
=> (1-c)(a2-ab+b2)=(1-c)(1+c+c2)
=> a2-ab+b2=1+c+c2
(a+b)2-3ab=(1-c)2+3c
=> -3ab=3c <=> -ab=c
Thay -ab = c vào a+b+c=1, ta có:
a+b+(-ab)=1 <=> a+b-ab-1=0 <=> a(1-b)-(1-b)=0 <=> (a-1)(1-b)=0
=> a-1=0 hoặc 1-b = 0 <=> a=1 hoặc b=1
+) Nếu a=1 => b+c=0 <=> b=-c
=> A=a2015+b2015+c2015
=> A=a2015+b2015-b2015
=> A=a2015 => A=1 (2)
+) Nếu b=1 => a+c=0 <=>a=-c
=> A=a2015+b2015+c2015
=> A=a2015+b2015+-a2015
=> A=b2015 => A=1 (3)
Từ (1)(2)(3) => A = 1
Vậy A = 1 với a+b+c=1 và a3+b3+c3=1
b) B = x2-3x+2016
B=x2-3x+2,25+2013,75
B=(x-1,5)2+2013,75
Vì (x-1,5)2 ≥ 0 => (x-1,5)2+2013,75 ≥ 2013,75
=> B ≥ 2013,75
=> GTNN của B bằng 2013,75
Dấu '=' xảy ra khi (x-1,5)2=0 <=> x-1,5=0 <=> x=1,5
Vậy GTNN của B bằng 2013,75 tại x = 1,5
a+b+c=0 => a^2+b^2+c^2+2ab+2bc+2ca = 0 => a^2+b^2+c^2=0
=> a^2+b^2+c^2 = ab+bc+ca
=> 2a^2+2b^2+2c^2 = 2ab+2bc+2ca
=> (a-b)^2 + (b-c)^2 + (c-a)^2 = 0
=> a=b=c, mà a+b+c=0 => a=b=c=0
thay vào
M=(0-2016)2016+(0-2016)2016-(0-2016)2016=(-2016)2016=20162016
Chúc bạn hoc tốt ùng hộ nha
\(a^2+b^2+c^2=1\Rightarrow a^2,b^2,c^2\le1\)\(\Rightarrow a,b,c\le1\)
Ta lại có: \(a^2+b^2+c^2=a^3+b^3+c^3\)
\(\Leftrightarrow a^3-a^2+b^3-b^2+c^3-c^2=0\)
\(\Leftrightarrow a^2\left(a-1\right)+b^2\left(b-1\right)+c^2\left(c-1\right)=0\)
Mà \(a^2\left(a-1\right)+b^2\left(b-1\right)+c^2\left(c-1\right)\le0\forall a,b,c\)(vì \(a^2,b^2,c^2\le0\) và \(a,b,c\le1\))
Suy ra ta phải có: \(a^2\left(a-1\right)=b^2\left(b-1\right)=c^2\left(c-1\right)=0\)
Kết hợp gt suy ra 3 số a,b,c phải là 1 số bằng 1 và 2 số còn lại bằng 0
Vì a,b,c vai trò như nhau nên giả sử \(a=1\Rightarrow b=c=0\)
Khi đó \(A=0^{2014}+1^{2015}+1^{2016}=1+1=2\)
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Ta có
(m+n+p)^q >= m^q+n^q+p^q
=>a+b+c=1
=>(a+b+c)^2016=1 >= a2016 + b2016 + c2016
Mà a2016 + b2016 + c2016 >=0
=> a2016 + b2016 + c2016=1