If $a+b=c+d$, and $a^3+b^3=c^3+d^3$, then prove that: $a^{2011}+b^{2011} = c^{2011}+d^{2011}$
68396
$a+b=c+d$ , $a^3+b^3=c^3+d^3$ , prove that $a^{2009}+b^{2009}=c^{2009}+d^{2009}$
6621
Prove the inequality $a^2bc+b^2cd+c^2da+d^2ab \leq 4$ with $a+b+c+d=4$
25258
Prove that if $ a,b,c > 0 $, then $ [(1 + a) (1 + b) (1 + c)]^{7} > 7^{7} (a^{4} b^{4} c^{4}) $.
127891
If $ a^{2} + b^{2} = c^{2}$ and $c$ is even, prove that $a$ and $b$ are both even.
41383
Prove that $(x+y+z)^2(yz+zx+xy)^2 \leq 3(y^2+yz+z^2)(z^2+zx+x^2)(x^2+xy+y^2)$
132098
Prove $(a^2+b^2)(c^2+d^2)\ge (ac+bd)^2$ for all $a,b,c,d\in\mathbb{R}$.
44705
Prove that $\frac{1}{a^3(b+c)}+\frac{1}{b^3(a+c)}+\frac{1}{c^3(a+b)}\ge \frac32$
86378
Prove that $3(a^3+b^3+c^3) > (a+b+c)(a^2+b^2+c^2)$
60122
Prove $(1+a+b)(1+b+c)(1+c+a)\ge 9(ab+bc+ca)$
136868
Prove that $\sqrt{a^2+3b^2}+\sqrt{b^2+3c^2}+\sqrt{c^2+3a^2}\geq6$ if $(a+b+c)^2(a^2+b^2+c^2)=27$
46814
$ a+b+c+d=6 , a^2+b^2+c^2+d^2=12$ $\implies$ $ 36 \leq 4(a^3+b^3+c^3+d^3)-(a^4+b^4+c^4+d^4) \leq48 $
16382
Find $a$ such that $(x+a)(x+1991)+1=(x+b)(x+c)$ with $a,b,c\in\Bbb Z$
95852
Prove that $(a+b)(b+c)(c+a) \ge8$
71299
Prove that $\frac{a}{b} < \frac{a+c}{b+d} <\frac{ c}{d}$
29935
If $a, b, c, d>0$ and $abcd=1$ prove that an inequality holds true
118507
Prove $\frac{ab}{1+c^2}+\frac{bc}{1+a^2}+\frac{ca}{1+b^2}\le\frac{3}{4}$ if $a^2+b^2+c^2=1$
133428
How prove this inequality $\frac{a}{b+3}+\frac{b}{c+3}+\frac{c}{d+3}+\frac{d}{a+3}\le 1$