Question

[Nick Đặt Cho Vui]

Cho a,b,c thuộc N* ; x+y+z = 5

S1= b/a.x + c/a.z 

S2= a/b.x + c/b.y

S3= a/c.z + b/c.y

Chứng minh rằng S = S1 +S2 + S3 lớn hơn hoặc bằng 10

Answer 1

[Huỳnh Quang Sang]

\(S_1+S_2+S_3=\left[\frac{b}{a}x+\frac{c}{a}z\right]+\left[\frac{a}{b}x+\frac{c}{b}y\right]+\left[\frac{a}{c}z+\frac{b}{c}y\right]\)

\(=\left[\frac{b}{a}x+\frac{a}{b}x\right]+\left[\frac{c}{b}y+\frac{b}{c}y\right]+\left[\frac{c}{a}z+\frac{a}{c}z\right]\)

\(=\left[\frac{b}{a}+\frac{a}{b}\right]x+\left[\frac{c}{b}+\frac{b}{c}\right]y+\left[\frac{c}{a}+\frac{a}{c}\right]z\)

\(S_1+S_2+S_3\ge2x+2y+2z=2\left[x+y+z\right]=2\cdot5=10\)

Vậy : \(S_1+S_2+S_3\ge10\)