Articles of inequality

Prove $(a_1+b_1)^{1/n}\cdots(a_n+b_n)^{1/n}\ge \left(a_1\cdots a_n\right)^{1/n}+\left(b_1\cdots b_n\right)^{1/n}$

consider positive numbers $a_1,a_2,a_3,\ldots,a_n$ and $b_1,b_2,\ldots,b_n$. does the following in-equality holds and if it does then how to prove it $\left[(a_1+b_1)(a_2+b_2)\cdots(a_n+b_n)\right]^{1/n}\ge \left(a_1a_2\cdots a_n\right)^{1/n}+\left(b_1b_2\cdots b_n\right)^{1/n}$

If $0<a<b$, prove that $a<\sqrt{ab}<\frac{a+b}{2}<b$

This question already has an answer here: Proving the AM-GM inequality for 2 numbers $\sqrt{xy}\le\frac{x+y}2$ 5 answers

Proving the inequality $|a-b| \leq |a-c| + |c-b|$ for real $a,b,c$

Let $a,b,c$ real numbers. Prove the inequality $|a-b| \leq |a-c| + |c-b|$. Prove that equality holds if and only if $a \leq c \leq b$ or $b \leq c \leq a$. I’ve tried starting with just $a \leq c$ and using field properties to reconstruct the inequality, however I haven’t been able to make it […]

Does $y(y+1) \leq (x+1)^2$ imply $y(y-1) \leq x^2$?

Can anyone see how to prove the following? If $x$ and $y$ are real numbers with $y\geq 0$ and $y(y+1) \leq (x+1)^2$ then $y(y-1) \leq x^2$. It seems it is true at least according to Mathematica.

What is the value of $\sum\limits_{i=1}^\infty\frac{1}{p_{p_i}}$ where $p_{i}$ is the $i$th prime?

What is the value of $\sum\limits_{i=1}^\infty\dfrac{1}{p_{p_i}}$ where $p_n$ is the nth prime (and so $p_{p_n}$ is the $k$th prime, where $k$ is the $n$th prime) ? Thus $\frac{1}{3}+\frac{1}{5}+\frac{1}{11}+…=$ ?? How do I efficiently decide if $\sum_{i=1}^\infty\dfrac{1}{p_{p_i}}\gt1$ is true without using a computer ? I assume this infinite sum is an irrational number. Has this already […]

Ramsey Number Inequality: $R(\underbrace{3,3,…,3,3}_{k+1}) \le (k+1)(R(\underbrace{3,3,…3}_k)-1)+2$

I want to prove that: $$R(\underbrace{3,3,…,3,3}_{k+1}) \le (k+1)(R(\underbrace{3,3,…3}_k)-1)+2$$ where R is a Ramsey number. In the LHS, there are $k+1$ $3$’s, and in the RHS, there are $k$ $3’s$. I really have no clue how to start this proof. Any help is appreciated!

If $a^3+b^3+c^3=3$ so $\frac{a^3}{a+b}+\frac{b^3}{b+c}+\frac{c^3}{c+a}\geq\frac{3}{2}$

Let $a$, $b$ and $c$ be positive numbers such that $a^3+b^3+c^3=3$. Prove that: $$\frac{a^3}{a+b}+\frac{b^3}{b+c}+\frac{c^3}{c+a}\geq\frac{3}{2}.$$ This inequality we can prove by BW with computer. I am looking for an human proof, which we can use during competition. For example, we need to prove that $$\sum_{cyc}\left(\frac{a^3}{a+b}-\frac{a^2}{2}\right)\geq\frac{3}{2}-\frac{a^2+b^2+c^2}{2}$$ or $$\sum_{cyc}\frac{a^2(a-b)}{a+b}\geq\sqrt[3]{3(a^3+b^3+c^3)^2}-a^2-b^2-c^2$$ or $$\sum_{cyc}\frac{a^2(a-b)}{a+b}\geq\frac{3(a^3+b^3+c^3)^2-(a^2+b^2+c^2)^3}{\sqrt[3]{9(a^3+b^3+c^3)^4}+(a^2+b^2+c^2)\sqrt[3]{3(a^3+b^3+c^3)^2}+(a^2+b^2+c^2)^2}$$ and since $\sqrt[3]{3(a^3+b^3+c^3)^2}\geq a^2+b^2+c^2$ or $3(a^3+b^3+c^3)^2\geq(a^2+b^2+c^2)^3$, it […]

Equality of triangle inequality in complex numbers

$z$ and $w$ be nonzero complex numbers. How do I show that $|z+w|=|z|+|w|$ if and only if $z=sw$ for some real positive number $s$. I approached this by letting $z=a+ib$, and $w=c+id$, and kinda play around with it. I also tried to square both sides when proving forward direction, but I could not get it […]

If $x_n\leq y_n$ then $\lim x_n\leq \lim y_n$

This question already has an answer here: Suppose that $(s_n)$ converges to $s$, $(t_n)$ converges to $t$, and $s_n \leq t_n \: \forall \: n$. Prove that $s \leq t$. 3 answers

$\forall x,y>0, x^x+y^y \geq x^y + y^x$

Prove that $\forall x,y>0, x^x+y^y \geq x^y + y^x$ A friend of mine told me none of the teachers in my school have succeeded in proving this seemingly simple inequality (it was asked at an oral exam last year). I tried it myself, but I made no significant progress.