1.Choa,bge0. CM: a^3b^3left(a^2-ab+b^2right)lefrac{left(a+bright)^8}{256}.2.Choa,b,cge0 và frac{1}{1+a}+frac{1}{1+b}+frac{1}{1+c}ge2. CM:abclefrac{1}{8}.3.Choa,b,c,dge0 và frac{a}{1+a}+frac{2b}{b+1}+frac{3c}{1+c}le1. CM:ab^2c^3 frac{1}{5^6}.4.Với ∀a,b,cge0. CM:a^4b^2c+b^4c^2a+c^4a^2ble a^7+b^7+c^7.5.Choa,b,c0. CM:frac{a^5}{b^3c}+frac{b^5}{c^3a}+frac{c^5}{a^3b}ge a+b+c.6.Choa,b,c0. CM:frac{a^3b}{c}+frac{b^3c}{a}+frac{c^3a}{b}ge ab^2+bc^2+ca^2.7.Choa,b,c0 và a+b+c3. CM:frac{a}{b^2+1}...
Đọc tiếp
\(1.\)\(Cho\)\(a,b\ge0.\)
\(CM: \)\(a^3b^3\left(a^2-ab+b^2\right)\le\frac{\left(a+b\right)^8}{256}.\)
\(2.\)\(Cho\)\(a,b,c\ge0\) và \(\frac{1}{1+a}+\frac{1}{1+b}+\frac{1}{1+c}\ge2.\)
\(CM:\)\(abc\le\frac{1}{8}.\)
\(3.\)\(Cho\)\(a,b,c,d\ge0\) và \(\frac{a}{1+a}+\frac{2b}{b+1}+\frac{3c}{1+c}\le1.\)
\(CM:\)\(ab^2c^3< \frac{1}{5^6}.\)
\(4.\)Với ∀\(a,b,c\ge0.\)
\(CM:\)\(a^4b^2c+b^4c^2a+c^4a^2b\le a^7+b^7+c^7.\)
\(5.\)\(Cho\)\(a,b,c>0.\)
\(CM:\)\(\frac{a^5}{b^3c}+\frac{b^5}{c^3a}+\frac{c^5}{a^3b}\ge a+b+c.\)
\(6.\)\(Cho\)\(a,b,c>0.\)
\(CM:\)\(\frac{a^3b}{c}+\frac{b^3c}{a}+\frac{c^3a}{b}\ge ab^2+bc^2+ca^2.\)
\(7.\)\(Cho\)\(a,b,c>0\) và \(a+b+c=3.\)
\(CM:\)\(\frac{a}{b^2+1}+\frac{b}{c^2+1}+\frac{c}{a^2+1}\ge\frac{3}{2}.\)
\(8.\)\(Cho\)\(a,b,c>0.\)
\(CM:\)\(\frac{a^2}{b+c}+\frac{b^2}{c+a}+\frac{c^2}{a+b}\ge\frac{a+b+c}{2}.\)
\(9.\)\(Cho\)\(a,b,c>0\) và \(a+b+c=1.\)
\(CM:\)\(\frac{ab}{c+1}+\frac{bc}{a+1}+\frac{ca}{b+1}\le\frac{1}{4}.\)
\(10.\)\(Cho\)\(a,b,c>0.\)
\(CM:\)\(\frac{1}{a^2+bc}+\frac{1}{b^2+ac}+\frac{1}{c^2+ab}\le\frac{a+b+c}{2abc}.\)
