Question

Cho a,b  >0 sao cho a+b+c=1 CMR \(\frac{ab}{c+1}+\frac{bc}{a+1}+\frac{ac}{b+1}\le\frac{1}{4}\)

Answer 1

c+1 hay c-1 vậy  xem lại đề ik 

Answer 2

Áp dụng tính chất : 1/x+y < = 1/4.(1/x + 1/y) với x,y > 0 thì :

ab/c+1 = ab/c+a+b+c = ab/(c+a)+(c+b) < = ab/4.(1/c+a + 1/c+b) = 1/4.(ab/c+a + ab/c+b)

Tương tự : bc/a+1 < = 1/4.(bc/a+c + bc/a+b) ; ca/b+a < = 1/4.(ca/b+c + ca/b+a)

=> ab/c+1 + bc/a+1 + ca/b+1 < = 1/4.(ab/c+a + ab/c+b + bc/a+c + bc/a+b + ca/b+c + ca/b+a ) 

= 1/4.[(ab/c+a + bc/a+c) + (ab/c+b + ca/b+c) + (bc/a+b + ca/a+b)]

= 1/4.(a+b+c) = 1/4

=> ĐPCM

Tk mk nha

Answer 3

Áp dụng tính chất : 1/x+y < = 1/4.(1/x + 1/y) với x,y > 0 thì :

ab/c+1 = ab/c+a+b+c = ab/(c+a)+(c+b) < = ab/4.(1/c+a + 1/c+b) = 1/4.(ab/c+a + ab/c+b)

Tương tự : bc/a+1 < = 1/4.(bc/a+c + bc/a+b) ; ca/b+a < = 1/4.(ca/b+c + ca/b+a)

=> ab/c+1 + bc/a+1 + ca/b+1 < = 1/4.(ab/c+a + ab/c+b + bc/a+c + bc/a+b + ca/b+c + ca/b+a ) 

= 1/4.[(ab/c+a + bc/a+c) + (ab/c+b + ca/b+c) + (bc/a+b + ca/a+b)]

= 1/4.(a+b+c) = 1/4( ĐPCM)