Show that for all real numbers $a$ and $b$, $\,\, ab \le (1/2)(a^2+b^2)$
105047
Prove the inequality $|xy|\leq\frac{1}{2}(x^2+y^2)$
51706
If $(k,l) = 1$, show that $(b_2k-b_1l, a_1l-a_2k) = 1$ for $a_1b_2-a_2b_1=1$
82612
Show that $\frac c {1+c} \le \frac a {1+a} + \frac b {1+b}$ , for $c \le a+b$ and $a,b,c \ge 0$
6970
If $abc=1$ and $a,b,c$ are positive real numbers, prove that ${1 \over a+b+1} + {1 \over b+c+1} + {1 \over c+a+1} \le 1$.
158844
If $a$ and $b$ are positive real numbers such that $a+b=1$, prove that $(a+1/a)^2+(b+1/b)^2\ge 25/2$
105164
If $abc=1$, prove that $\frac{2}{(a+1)^2+b^2+1} + \frac{2}{(b+1)^2+c^2+1} + \frac{2}{(c+1)^2+a^2+1} \le 1$.
154822
Properties of the composition law $a @ b= ab + b + a$ on the real numbers
98017
If $a_{n+1}/a_n\to\ell$ show $a_n^{1/n}\to\ell$
57125
Prove the distributive law $a(b+c)=ab+ac$ for real numbers?
89175
Proof of $(a+b)^{n+1}$
105641
Prove that $ab\leq\frac{1}{2\varepsilon}a^2+\frac{\varepsilon}{2}b^2$
149889
Prove that $|\log(1 + x^2) - \log(1 + y^2)| \le |x-y|$
130281
Proving $a^ab^b + a^bb^a \le 1$, given $a + b = 1$
25623
Does $\sqrt{a+b} \le \sqrt a + \sqrt b$ hold for all positive real numbers a and b?
11506
Alternative proof that $(a^2+b^2)/(ab+1)$ is a square when it's an integer
118507
Prove $\frac{ab}{1+c^2}+\frac{bc}{1+a^2}+\frac{ca}{1+b^2}\le\frac{3}{4}$ if $a^2+b^2+c^2=1$
85977
Show that if $\sqrt{a}+\sqrt{b}\neq 0$, then $\mathbb{Q}(\sqrt{a}+\sqrt{b}) = \mathbb{Q}(\sqrt{a},\sqrt{b}), \forall a,b\in\mathbb{Q}$