Proof help: Prove that $x^2+y^2+z^2 \geq xy+xz+yz$
5694
How to prove $a^2 + b^2 + c^2 \ge ab + bc + ca$?
72081
$\forall x,y>0, x^x+y^y \geq x^y + y^x$
18313
show this inequality with $xy+yz+zx=3$
111847
For $a,b,c>0$, prove that $\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a}\ge a+b+c+\frac{4(a-b)^2}{a+b+c}$
17100
Prove that $\frac{b^2+c^2}{b+c}+\frac{c^2+a^2}{c+a}+\frac{a^2+b^2}{a+b} \ge a+b+c$
66387
Prove $(x_1+.....+x_n)^2 \leq n(x_1^2 + .....+x_n^2)$
14677
Proving the exponential inequality: $x^y+y^x\gt1$
45619
Prove: $|a\sin x+b \cos x|\leq \sqrt{a^2+b^2}$
29342
Prove the inequality: $(y+z)*x^4+(x+z)*y^4+(x+y)*z^4\le((x+y+z)^5)/12$ for positive reals.
152167
Prove that if $x,y,z$ are integers such that $x^2+y^2=z^2$, then $xyz \equiv o \pmod{60}$.
117281
How prove this inequality $x^3y+y^3z+z^3x\ge xyz(x+y+z)$
111295
Prove that $a^7 + b^7 + c^7 \geq a^4b^3 + b^4c^3 + c^4a^3$
136868
Prove that $\sqrt{a^2+3b^2}+\sqrt{b^2+3c^2}+\sqrt{c^2+3a^2}\geq6$ if $(a+b+c)^2(a^2+b^2+c^2)=27$
36507
If $x,y>0$ and $x^2+y^3\ge x^3+y^4$, prove that $x^3+y^3 \le 2$.
118467
Prove that $a^2+ab+b^2\ge 0$
64096
Prove that $||z_1|-|z_2|| \leq |z_1-z_2|$.
13572
$x^y+y^x>1$ for all $(x, y)\in \mathbb{R_+^2}$