Show that $3=\sqrt{1+2\sqrt{1+3\sqrt{1+4...}}}$
13617
Evaluating the nested radical $ \sqrt{1 + 2 \sqrt{1 + 3 \sqrt{1 + \cdots}}} $.
86378
Prove that $3(a^3+b^3+c^3) > (a+b+c)(a^2+b^2+c^2)$
73277
Prove $[\sqrt{x}+\sqrt{x+1}]=[\sqrt{4x+2}], x \in \Bbb N$
19940
How to show that $A^3+B^3+C^3 - 3ABC = (A+B+C)(A+B\omega+C\omega^2)(A+B\omega^2+C\omega)$ indirectly?
11612
How to show that $\sum_{k=1}^n k(n+1-k)=\binom{n+2}3$?
19457
Show that: $\inf(A+B) = \inf(A)+ \inf(B)$
120954
Prove that: $\sqrt[3]{a_1^3+ a_2^3 +\cdots+a_n^3} \le \sqrt{a_1^2 + a_2^2 +\cdots+a_n^2}$
138894
Prove that if $3\mid(a^2+b^2)$,then $3\mid a$ and $ 3\mid b$
108429
Show that if $x_n \to x$ then $\sqrt{x_n} \to \sqrt{x}$
7201
Prove that $x_1^2+x_2^2+x_3^2=1$ yields $ \sum_{i=1}^{3}\frac{x_i}{1+x_i^2} \le \frac{3\sqrt{3}}{4} $
104498
Show that $x^3y^3(x^3+y^3) \leq 2$
109133
$a+b+c = 3$, prove that :$a\sqrt{a+3}+b\sqrt{b+3}+c\sqrt{c+3} \geq 6$
28347
Prove that $\sqrt 2 + \sqrt 3$ is irrational
126018
Show that $ e^{A+B}=e^A e^B$
157884
For any prime $p>3$ show that 3 divides $2p^2+1$
109953
How to prove that $\sqrt 3$ is an irrational number?
40348
Proving by mathematical induction: $1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+...+\frac{1}{\sqrt{n}}>2(\sqrt{n+1}-1)$