ta có:
\(VT+4=\left(a^2-2ad+d^2\right)+\left(b^2-2bc+c^2\right)+\left(a^2-2ab+b^2\right)=\left(a-d\right)^2+\left(b-d\right)^2+\left(a-b\right)^2\)theo AM-GM:\(\left(a-d\right)^2+\left(b-c\right)^2\ge2\left(a-d\right)\left(b-c\right)=2\)
và \(\left(a-b\right)^2\ge0\)
do đó \(VT+4\ge2\Leftrightarrow VT\ge2\)
Dấu = xảy ra khi a=b=1;c=d=0 ...
Cho \(a^2+b^2=2\) và \(\left(a-d\right)\left(b-c\right)=1\). Chứng minh \(c^2+d^2-2ad-2bc-2ab\ge-2\)
chứng minh các đẳng thức sau
a)\(\left(a+b+c\right)^2+\left(b+c-a\right)^2\left(c+a-b\right)^2\left(a+b+c\right)^2=4\left(a^2+b^2+c^2\right)\)
b) \(\left(a+b+c+d\right)^2+\left(a+b-c-d\right)^2+\left(a+c-b-d\right)^2+\left(a+d-b-c\right)^2=4\left(a^2+b^2+c^2+d^2\right)\)
1. Cho \(\left(a-b\right)^2+\left(b-c\right)^2+\left(c-a\right)^2=\left(a+b-2c\right)^2+\left(b+c-2a\right)^2+\left(c+a-2b\right)^2.\)
Chứng minh: a=b=c.
2. Chứng minh rằng:
a, A= x4 - 4x3 - 2x2 +12x +9 là số chính phương \(\forall\)x,y,z \(\in Z\).
b, B = 4x(x+y)(x+y+z)(x+z) + y2z2 là số chính phương với \(\forall\)x,y,z\(\in N\).
Cho 0 < a < b < c < d. Chứng minh: \(\left(b+c\right).\left(\dfrac{1}{b}+\dfrac{1}{c}\right)< \dfrac{\left(a+d\right)^2}{ad}\)
CMR:
a/\(a^2+b^2+c^2\ge\text{ab}+bc+c\text{a}\)
b/\(3\left(\text{a}b+bc+c\text{a}\right)\le\left(\text{a}+b+c\right)^2\le3\left(\text{a}^2+b^2+c^2\right)\)
c/\(\text{a}^3+b^3\ge\text{a}b\left(\text{a}+b\right)\)
a/ \(\left(x+y\right)^2-y^2=x\left(x+2y\right)\)
b/ \(\left(x^2+y^2\right)^2-\left(2xy\right)^2=\left(x+y\right)^2\left(x-y\right)^2\)
c/ \(\left(a+b+c\right)^2+\left(a+b-c\right)^2+\left(2a-b\right)^2\)
d/ \(\left(a+b+c\right)^2+\left(a+b-c\right)^2+2\left(a+b\right)\)
Cho \(\left(a-b\right)^2+\left(b-c\right)^2+\left(c-a\right)^2=4.\left(a^2+b^2+c^2-ac-bc-ca\right)\). Chứng minh rằng : a = b = c
Bài 1: X = \(\dfrac{b^2+c^2-a^2}{2bc}\) ; Y = \(\dfrac{a^2-\left(b-c\right)^2}{\left(b+c\right)^2-a^2}\)
Tính P = x +y +xy
Giup mk nha các bạn
Dùng phương pháp xét giá trị riêng:
\(M=a\left(b+c-a\right)^2+b\left(c+a-b\right)^2+c\left(a+b-c\right)^2+c\left(a+b-c\right)^2+\left(a+b-c\right)\left(b+c-a\right)\left(c+a-b\right)\)
