Question

1) cho a,b,c dương thỏa abc<1

C/M : \(\dfrac{1}{1+a+ab}+\dfrac{1}{1+b+bc}+\dfrac{1}{1+c+ca}>1\)

2) cho a,b,c không âm thỏa a+b+c=1

CMR \(a^2+b^2+c^2\ge4\left(ab+bc+ca\right)-1\)

3)cho x,y,z,t thỏa \(x^2+y^2+z^2+t^2\le1\)

CMR :\(\sqrt{\left(x+z\right)^2+\left(y-t\right)^2}+\sqrt{\left(x-z\right)^2+\left(y+t\right)^2}\le2\)

Answer 1

Câu 3/ \(\sqrt{\left(x+z\right)^2+\left(y-t\right)^2}+\sqrt{\left(x-z\right)^2+\left(y+t\right)^2}\)

\(\le\sqrt{1+2xz-2yt}+\sqrt{1-2xz+2yt}\)

\(\le\dfrac{1+1+2xz-2yt}{2}+\dfrac{1+1-2xz+2yt}{2}=1+1=2\)

Answer 2

Đăng nhiều thế???

Answer 3

Câu 1/ \(\dfrac{1}{1+a+ab}+\dfrac{1}{1+b+bc}+\dfrac{1}{1+c+ca}\)

\(>\dfrac{1}{1+a+ab}+\dfrac{1}{1+b+\dfrac{1}{a}}+\dfrac{1}{1+\dfrac{1}{ab}+\dfrac{1}{b}}\)

\(=\dfrac{1}{1+a+ab}+\dfrac{a}{a+ab+1}+\dfrac{ab}{ab+1+a}=\dfrac{1+a+ab}{1+a+ab}=1\)

Vậy \(\dfrac{1}{1+a+ab}+\dfrac{1}{1+b+bc}+\dfrac{1}{1+c+ca}>1\)

Answer 4

Câu 2/ Ta có: \(1=\left(a+b+c\right)^2\ge3\left(ab+bc+ca\right)\)

Quay lại bài toán ta có:

\(a^2+b^2+c^2+1\ge ab+bc+ca+3\left(ab+bc+ca\right)=4\left(ab+bc+ca\right)\)

\(\Leftrightarrow a^2+b^2+c^2\ge4\left(ab+bc+ca\right)-1\)

Dấu = xảy ra khi \(a=b=c=\dfrac{1}{3}\)