Sum of fractions Inequality: $ {a\over b} + {b \over c} + {c \over a} \ge 3 $
14563
Show $\frac{b}{c} + \frac{c}{a} + \frac{a}{b} \ge 3$ for $a,b,c > 0$.
107494
prove $\frac {ab}{a+b} \geq \sum ^n_{i=1} \frac{a_ib_i}{a_i+b_i}$
117281
How prove this inequality $x^3y+y^3z+z^3x\ge xyz(x+y+z)$
138099
A cyclic inequality $\sum\limits_{cyc}{\sqrt{3x+\frac{1}{y}}}\geqslant 6$
89128
Proof of Nesbitt's Inequality: $\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}\ge \frac{3}{2}$?
25203
with inequality $\frac{y}{xy+2y+1}+\frac{z}{yz+2z+1}+\frac{x}{zx+2x+1}\le\frac{3}{4}$
60073
Proving $4(a^3 + b^3) \ge (a + b)^3$ and $9(a^3 + b^3 + c^3) \ge (a + b + c)^3$
152053
Proof that $\frac{a^2}{b} + \frac{b^2}{c} + \frac{c^2}{a} \geq a+b+c$
26606
Prove the inequality $\,\frac{1}{\sqrt{1}+ \sqrt{3}} +\frac{1}{\sqrt{5}+ \sqrt{7} }+\ldots+\frac{1}{\sqrt{9997}+\sqrt{9999}}\gt 24$
73987
Prove the inequality $({1+\frac{a}b})^n$ + $(1+\frac{b}a)^n$ $\geq$ $2^{n+1}$
4099
Inequality $\sum\limits_{cyc}\frac{a^3}{13a^2+5b^2}\geq\frac{a+b+c}{18}$
90670
Prove $a^ab^bc^c\ge (abc)^{\frac{a+b+c}3}$ for positive numbers.
112676
Inequality $(a+b+c)(ab+bc+ca)(a^3+b^3+c^3)\le (a^2+b^2+c^2)^3$
126557
Proving the inequality $\sum_{\text{cyc}} \frac{a}{a^2+b^3+c^3}\le\frac1{5abc}$
108782
Inequality $\frac{1}{n+1}+\frac{1}{n+2}+...+\frac{1}{3n+1} \ge 1$
124968
How prove this inequality $\sum\limits_{cyc}\frac{x+y}{\sqrt{x^2+xy+y^2+yz}}\ge 2+\sqrt{\frac{xy+yz+xz}{x^2+y^2+z^2}}$
141525
Cyclic Inequality $\frac{a^3}{(a+b)(a+c)}$