cho a,b,c la ba so thuc duong thoa man dieu kien a+b+c=1
chung minh rang P=\(\sqrt{\frac{ab}{c+ab}}+\sqrt{\frac{bc}{a+bc}}+\sqrt{\frac{ca}{b+ca}}\le\frac{3}{2}\)

1. Cho a,b,c,d la cac so nguyen thoa man \(a^2=b^2+c^2+d^2\)

chung minh rang a.b.c.d + 2015 viet duoc duoi dang hieu cua 2 so chinh phuong.

2. Cho a,b la cac so duong thoa man dieu kien a+b=1. tim gia tri nho nhat cua bieu thuc

\(P=\frac{2+a}{\sqrt{2-a}}+\frac{2+b}{\sqrt{2-b}}\)

cho 3 so thuc x,y,z khac khong va thoa man hai dieu kien \(ax^3=by^3=cz^3\) va \(\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}=1\)

chung minh rang : \(\sqrt[3]{ax^2+by^2+cz^2}=\sqrt[3]{a}+\sqrt[3]{b}+\sqrt[3]{c}\)

tim tat ca cac so duong a,b,c thoa man dieu kien \(\left\{{}\begin{matrix}\sqrt{a}+\sqrt{b}+\sqrt{c}=6\\\frac{1}{\sqrt{a}}+\frac{4}{\sqrt{b}}+\frac{9}{\sqrt{c}}=6\end{matrix}\right.\)

1)Rut gon bieu thuc:P=\(\sqrt{4+\sqrt{10+2\sqrt{5}}}+\sqrt{4-\sqrt{10+2\sqrt{5}}}\)

2) Cho 3 so duong a,b,c thoa man dieu kien:a+b+c=\(\sqrt{ab}+\sqrt{bc}+\sqrt{ca}\)

Chung minh rang:a=b=c

cho a,b,c la do dai 3 canh cua mot tam giac thoa man dieu kien \(\sqrt{a+b-c}+\sqrt{b+c-a}+\sqrt{c+a-b}=\sqrt{a}+\sqrt{b}+\sqrt{c}\)

chung minh a,b,c la 3 canh cua mot tam giac deu

cho cac so a,b,c duong thoa man ab+bc+ca=1  chung minh : \(p=\frac{2a}{\sqrt{1+a^2}}+\frac{b}{\sqrt{1+b^2}}+\frac{c}{\sqrt{1+c^2}}\)

cho ca so a,b,c duong thoa man ab+bc+ca =1  chung minh   \(P=\frac{2a}{\sqrt{1+a^2}}+\frac{b}{\sqrt{1+b^2}}+\frac{c}{\sqrt{1+c^2}}\le\frac{1}{4}\)

Cho a,b,c > \(\dfrac{-1}{4}\). Chứng minh rằng

\(\sqrt{4a+1}+\sqrt{4b+1}+\sqrt{4c+1}\le\sqrt{21}\)