Proving that $a^2+b^2\neq 3c^2$ for any positive nonzero integers $a$, $b$, and $c$
52960
Show that the curve $x^2+y^2-3=0$ has no rational points
151153
Prove $\frac{a(a^2+2)}{3}$ is an integer for all integer $a\geqslant 1$
57670
Show that $|z_1 + z_2|^2 < (1+C)|z_1|^2 + \left(1 + \frac{1}{C}\right) |z_2|^2$
10604
Prove $3^n > n^2$ by induction
20964
Proof that $n^2 < 2^n$
153241
Four different positive integers a, b, c, and d are such that $a^2 + b^2 = c^2 + d^2$
23638
$a^2b^2+a^2c^2+b^2c^2\leq 3$
37536
Inequality: $ab^2+bc^2+ca^2 \le 4$, when $a+b+c=3$.
69673
Prove that $\left (\frac{a^2 + b^2 +c^2}{a+b+c} \right) ^ {(a+b+c)} > a^a b^b c^c$
2019
Prove $\sqrt{a} + \sqrt{b} + \sqrt{c} \ge ab + bc + ca$
115460
Proving $a\mid b^2,\,b^2\mid a^3,\,a^3\mid b^4,\ldots\implies a=b$ - why is my approach incorrect
104498
Show that $x^3y^3(x^3+y^3) \leq 2$
156513
Prove that inequality $\sum _{cyc}\frac{a^2+b^2}{a+b}\le \frac{3\left(a^2+b^2+c^2\right)}{a+b+c}$
67666
show that $a^2+b^2+c^2+2abc+3\ge (1+a)(1+b)(1+c)$
137626
If $a,b,c$ are positive integers,and $\cfrac{a}{b}+\cfrac{b}{c}+\cfrac{c}{a}$ and $\cfrac{b}{a}+\cfrac{c}{b}+\cfrac{a}{c}$ are integers then $a=b=c$
124868
Prove that $\frac{bc}{b+c}+\frac{ca}{c+a}+\frac{ab}{a+b}<\frac{a+b+c}{2}$
6621
Prove the inequality $a^2bc+b^2cd+c^2da+d^2ab \leq 4$ with $a+b+c+d=4$