Prove that If $a$ and $b$ are real numbers with $a  b^2$
113168
If $a > b$, is $a^2 > b^2$?
13478
Prove that if $a^p-b^p$ is divisible by $p$, then it is also divisible by $p^2$
154822
Properties of the composition law $a @ b= ab + b + a$ on the real numbers
152039
Prove that $a^{2^n}=1 \mod 2^{n+2}$
11127
Let $a$, $b$, and $c$ be positive real numbers.
13914
If $a^2+b^2=c^2$ then $a$ or $b$ is even.
137718
Prove that the complex expression is real
153313
Prove that for all integers $ a $ and $ b $, if $ a $ divides $ b $, then $ a^{2} $ divides $ b^{2} $.
50614
If $4\mid a+bc$ and $6\mid b+ac$ prove that $2\mid a^2-b^2$
45090
Given any positive real numbers $a,b,c$, we have $(a^{2}+2)(b^{2}+2)(c^{2}+2)\geq 9(ab+bc+ca)$
5410
Prove that $a^2 + b^2 \geq 8$ if $ x^4 + ax^3 + 2x^2 + bx + 1 = 0 $ has at least one real root.
80148
Prove $\gcd(a+b,a^2+b^2)$ is $1$ or $2$ if $\gcd(a,b) = 1$
121700
If $a$ and $b$ are positive real numbers, then $a + b \geq 2 \sqrt{ab}$.
142000
Prove that there is an irrational number and a rational number between any two distinct real numbers
96630
Let a and b be real numbers such that 0  \sqrt{ab} > \frac{2ab}{a + b}$
76568
$a,b,c$ are real numbers $>0$. If $a+b+c=1$, show that $(a+\frac{1}{a})^2+(b+\frac{1}{b})^2+(c+\frac{1}{c})^2\ge\frac{100}{3}$
80009
Prove that $| |a|-|b| | \le |a-b|$