Prove that for all $a, b > 0, a^b + b^a > 1$
13572
$x^y+y^x>1$ for all $(x, y)\in \mathbb{R_+^2}$
8366
Proof of $a^n+b^n$ divisible by $a+b$ when $n$ is odd
80009
Prove that $| |a|-|b| | \le |a-b|$
56315
Prove $\frac{|a+b|}{1+|a+b|}<\frac{|a|}{1+|a|}+\frac{|b|}{1+|b|}$.
21781
Prove for any $a>1$ there exist a natural number, $N$, such that $a^n>n$ for all $n \ge N$
46831
Show that $ a，b, \sqrt{a}+ \sqrt{b} \in\mathbb Q \implies \sqrt{a},\sqrt{b} \in\mathbb Q $
98855
Prove $aba^{-1}b^{-1}\in{N}$ for all $a,b$
33903
Prove that $(1+a_1) \cdot (1+a_2) \cdot \dots \cdot (1+a_n) \geq 2^n$
88687
Proof of the inequality $||a+b|^p - |a|^p| \leq \epsilon |a|^p + C_{\epsilon} |b|^p$
81818
Prove or disprove : $\det(A^k + B^k) \geq 0$
25981
Prove that $\sum_{k=0}^n a_k x^k = 0$ has at least $1$ real root if $\sum_{k=0}^n \frac{a_k}{k+1} = 0$
150565
Prove $(a+b)^\frac{1}{n} \le a^\frac{1}{n} + b^\frac{1}{n}$
119436
Proving $2^n\leq 2^{n+1}-2^{n-1}-1$ for all $n\geq 1$ by induction
130281
Proving $a^ab^b + a^bb^a \le 1$, given $a + b = 1$
111267
Prove that: $2^a+3^b<3a+4b$
27281
How to prove $\,x^a-1 \mid x^b-1 \iff a\mid b$
10236
Show that $a^n \mid b^n$ implies $a \mid b$