Question

Cho a+b+c=1. Chứng minh rằng

ab + bc + ca < 1/2

Answer 1

<=>2ab+2bc+2ca<=1=1^2=(a+b+c)^2

<=>a^2+b^2+c^2+2ab+2bc+2ca>=2ab+2bc+2ca

<=>a^2+b^2+c^2>=0

a,b,c khong dong thoi =0

=> dang thuc khong xay ra

=> ab+bc+ca<1/2=>dpcm

Answer 2

(a+b+c)=1

a^2+b^2+c^2+2ab+2bc+2ca=1

a^^2+b^2+c^2>=0

=>2ab+2bc+2ca<=1

Đẳng thức khi (a+b+c=1 &0=> vô nghiệm

=> 2ab+2bc+2ca<1

=>ab+2bc+2ca<1/2

=>đpcm