Violympic toán 9
Các câu hỏi tương tự
Cho a, b, c, d > 0 với abcd=1. Chứng minh rằng: \(a^2+b^2+c^2+d^2+a\left(b+c\right)+b\left(c+d\right)+d\left(c+a\right)\ge10\)
cho \(7\left(\dfrac{1}{a^2}+\dfrac{1}{b^2}+\dfrac{1}{c^2}\right)=6\left(\dfrac{1}{ab}+\dfrac{1}{bc}+\dfrac{1}{ac}\right)+2017\)
tìm max \(\dfrac{1}{\sqrt{3\left(2a^2+b^2\right)}}+\dfrac{1}{\sqrt{3\left(2b^2+c^2\right)}}+\dfrac{1}{\sqrt{3\left(2c^2+a^2\right)}}\)
Cho a, b, c là độ dài 3 cạnh tam giác. CMR:
1, \(\dfrac{1}{\left(a+b-c\right)^n}+\dfrac{1}{\left(a-b+c\right)^n}+\dfrac{1}{\left(b+c-a\right)^n}\ge\dfrac{1}{a^n}+\dfrac{1}{b^n}+\dfrac{1}{c^n}\)
2, \(\dfrac{1}{a^n}+\dfrac{1}{b^n}+\dfrac{1}{c^n}\ge4^n\left[\dfrac{1}{\left(2a+b+c\right)^n}+\dfrac{1}{\left(a+2b+c\right)^n}+\dfrac{1}{\left(a+b+2c\right)^n}\right]\)
cho các số thực không âm a , b , c ( a khác b ) thỏa mãn (a+c)(b+c)=1
Tìm min A \(\dfrac{1}{\left(a-b\right)^2}\)+\(\dfrac{1}{\left(a+c\right)^2}\)+\(\dfrac{1}{\left(b+c\right)^2}\)
Cho a,b,c \(\ge1.CMR:a\left(b+c\right)+b\left(c+a\right)+c\left(a+b\right)+2\left(\dfrac{1}{1+a^2}+\dfrac{1}{1+b^2}+\dfrac{1}{1+c^2}\right)\ge9\)
Cho \(a,b,c\ge1\). CMR:
\(a\left(b+c\right)+b\left(c+a\right)+c\left(a+b\right)+2\left(\dfrac{1}{a^2+1}+\dfrac{1}{b^2+1}+\dfrac{1}{c^2+1}\right)\ge9\)
Cho a,b,c >0 và a+b+c = abc. Tìm Max
\(S=\dfrac{a}{\sqrt{cb\left(1+a^2\right)}}+\dfrac{b}{\sqrt{ac\left(1+b^2\right)}}+\dfrac{c}{\sqrt{ab\left(1+c^2\right)}}\)
Cho a, b , c dương thỏa mãn a + b + c = abc. Tìm Max
\(S=\dfrac{a}{\sqrt{bc\left(1+a^2\right)}}+\dfrac{b}{\sqrt{ca\left(1+b^2\right)}}+\dfrac{c}{\sqrt{ab\left(1+c^2\right)}}\)
cho 3 số thực dương a,b,c thay đổi thỏa mãn \(\dfrac{1}{a^2}+\dfrac{1}{b^2}+\dfrac{1}{c^2}=3\)
tìm max của \(P=\dfrac{1}{\left(2a+b+c\right)^2}+\dfrac{1}{\left(2b+c+a\right)^2}+\dfrac{1}{\left(2c+a+b\right)^2}\)