Show that $\frac{(x + y + z)^2}{3} \ge x\sqrt{yz} + y\sqrt{zx} + z\sqrt{xy}$ for all non-negative reals $x, y, z$.

Prove that for any real $ x $ and $ y $ the inequality $x^2 \sqrt {1+2y^2} + y^2 \sqrt {1+2x^2} \geq xy (x+y+\sqrt{2})$ .

Let $a,b,c$ be real numbers such that $0 \leq a \leq b \leq c$ and $a+b+c=ab+bc+ca >0.$ Prove that $\sqrt{bc}(a+1) \geq 2$ and determine the equality cases. (Edit: Proposed by sir Leonard Giugiuc, Romania)

Given an integer $n \geq2$ and real numbers $a,b$ such that $0<a<b$. Let $x_1,x_2,\ldots, x_n\in [a,b]$ be real numbers. Find the maximum value of $$\frac{\frac{x^2_1}{x_2}+\frac{x^2_2}{x_3}+\cdots+\frac{x^2_{n-1}}{x_n}+\frac{x^2_n}{x_1}}{x_1+x_2+\cdots +x_{n-1}+x_n}.$$

Let $a, b, c$ be positive real numbers such that $abc = 1$. Prove that $$(a + b + c)(ab + bc + ca) + 3\ge 4(a + b + c).$$

Prove that the following inequality holds for all positive real numbers $x,y,z$: \[\dfrac{x^3}{y^2+z^2}+\dfrac{y^3}{z^2+x^2}+\dfrac{z^3}{x^2+y^2}\ge \dfrac{x+y+z}{2}.\]

Let $x_1,x_2,\cdots,x_n \in(0,1)$ , $n\geq2$. Prove that$$\frac{\sqrt{1-x_1}}{x_1}+\frac{\sqrt{1-x_2}}{x_2}+\cdots+\frac{\sqrt{1-x_n}}{x_n}<\frac{\sqrt{n-1}}{x_1 x_2 \cdots x_n}.$$

Let $x,y,z$ be positive real numbers . Prove: $\frac{x}{\sqrt{\sqrt[4]{y}+\sqrt[4]{z}}}+\frac{y}{\sqrt{\sqrt[4]{z}+\sqrt[4]{x}}}+\frac{z}{\sqrt{\sqrt[4]{x}+\sqrt[4]{y}}}\geq \frac{\sqrt[4]{(\sqrt{x}+\sqrt{y}+\sqrt{z})^7}}{\sqrt{2\sqrt{27}}}$

(a) A function $f$ is defined by $f(x)=x^x$ for all $x>0$. Find the minimum value of $f$. (b) If $x$ and $y$ are two positive real numbers, show that $x^y+y^x>1$.

A linear binomial $l(z) = Az + B$ with complex coefficients $A$ and $B$ is given. It is known that the maximal value of $|l(z)|$ on the segment $-1 \leq x \leq 1$ $(y = 0)$ of the real line in the complex plane $z = x + iy$ is equal to $M.$ Prove that for every $z$ \[|l(z)| \leq M \rho,\] where $\rho$ is the sum of distances from the point $P=z$ to the points $Q_1: z = 1$ and $Q_3: z = -1.$

Let $a_1,a_2,a_3,a_4$ be positive real numbers satisfying \[\sum_{i<j}a_ia_j=1.\]Prove that \[\sum_{\text{sym}}\frac{a_1a_2}{1+a_3a_4}\geq\frac{6}{7}.\][i]* * *[/i]

1) Two nonnegative real numbers $x, y$ have constant sum $a$. Find the minimum value of $x^m + y^m$, where m is a given positive integer. 2) Let $m, n$ be positive integers and $k$ a positive real number. Consider nonnegative real numbers $x_1, x_2, . . . , x_n$ having constant sum $k$. Prove that the minimum value of the quantity $x^m_1+ ... + x^m_n$ occurs when $x_1 = x_2 = ... = x_n$.

Let $a_1,a_2,...,a_{2020}$ be positive real numbers. Prove that: $$\max{(a^2_1-a_2,a^2_2-a_3,...,a^2_{2020}-a_1)}\ge\max{(a^2_1-a_1,a^2_2-a_2,...,a^2_{2020}-a_{2020})}$$

Prove the following inequality where positive reals $a$, $b$, $c$ satisfies $ab+bc+ca=1$. \[ \frac{a+b}{\sqrt{ab(1-ab)}} + \frac{b+c}{\sqrt{bc(1-bc)}} + \frac{c+a}{\sqrt{ca(1-ca)}} \le \frac{\sqrt{2}}{abc} \]

Let $a$, $b$, $c$, $d$ be positive real numbers such that $abcd=1$. Prove that $1/[(1/2 +a+ab+abc)^{1/2}]+ 1/[(1/2+b+bc+bcd)^{1/2}] + 1/[(1/2+c+cd+cda)^{1/2}] + 1/[1(1/2+d+da+dab)^{1/2}]$ is greater than or equal to $2^{1/2}$.

In a contest, there are $m$ candidates and $n$ judges, where $n\geq 3$ is an odd integer. Each candidate is evaluated by each judge as either pass or fail. Suppose that each pair of judges agrees on at most $k$ candidates. Prove that \[{\frac{k}{m}} \geq {\frac{n-1}{2n}}. \]

Positive real numbers $a_1,a_2,\ldots, a_n$ satisfy the equation $$2a_1+a_2+\ldots+a_{n-1}=a_n+\frac{n^2-3n+2}{2}$$ For every positive integer $n \geq 3$ find the smallest possible value of the sum $$\frac{(a_1+1)^2}{a_2}+\ldots+\frac{(a_{n-1}+1)^2}{a_n}$$ [i]M. Zorka[/i]

Let $a$, $b$, $c$ be real numbers such that $0 < a \le b \le c$. Prove that $(a + 3b)(b + 4c)(c + 2a) \ge 60abc$. When does equality hold?

Let $a$, $b$, $c$ and $d$ be positive real numbers with $a+b+c+d = 4$. Prove that $\frac{a}{b^2 + 1} + \frac{b}{c^2+1} + \frac{c}{d^2+1} + \frac{d}{a^2+1} \geq 2$.

Let $a_0,a_1,\dots$ be a sequence of positive reals such that $$ a_{n+2} \leq \frac{2023a_n}{a_na_{n+1}+2023}$$ for all integers $n\geq 0$. Prove that either $a_{2023}<1$ or $a_{2024}<1$.

$(YUG 1)$ Suppose that positive real numbers $x_1, x_2, x_3$ satisfy $x_1x_2x_3 > 1, x_1 + x_2 + x_3 <\frac{1}{x_1}+\frac{1}{x_2}+\frac{1}{x_3}$ Prove that: $(a)$ None of $x_1, x_2, x_3$ equals $1$. $(b)$ Exactly one of these numbers is less than $1.$

Let $p \geq 5$ be a prime and let \begin{align*} (p-1)^p +1 = \prod _{i=1}^n q_i^{\beta_i} \end{align*} where $q_i$ are primes. Prove, \begin{align*} \sum_{i=1}^n q_i \beta_i >p^2 \end{align*}

Find all triples of positive real numbers $(a, b, c)$ so that the expression $M = \frac{(a + b)(b + c)(a + b + c)}{abc}$ gets its least value.

When a company releases a new social media software, the marketing development of the company researches and analyses the characteristics of the customer group apart from paying attention to the active customer depending on the change of the time. We use $n(t, x)$ to express the customer density (which will be abbreviated as density). Here $t$ is the time and $x$ is the time of the customer spent on the social media software. In the instant time $t$, for $0<x_1<x_2$, the number of customers of spending time between $x_1$ and $x_2$ is $\int_{x_1}^{x_2}n(t,x)dx$. We assume the density $n(t,x)$ depends on the time and the following factors: Assumption 1. When the customer keeps using that social media software, their time spent on social media increases linearly. Assumption 2. During the time that the customer uses the social media software, they may stop using it. We assumption the speed of stopping using it $d(x)>0$ only depends on $x$. Assumption 3. There are two sources of new customer. (i) The promotion from the company: A function of time that expresses the increase of number of people in a time unit, expressed by $c(t)$. (ii) The promotion from previous customer: Previous customer actively promotes this social media software to their colleagues and friends actively. The speed of promoting sucessfully depends on $x$, denoted as $b(x)$. Assume if in an instant time, denoted as $t=0$, the density function is known and $n(0,x)=n_0(x)$. We can derive. The change of time $n(t,x)$ can satisfy the equation: $\begin{cases} \frac{\partial}{\partial t}n(t,x)+\frac{\partial}{\partial x}n(t,x)+d(x)n(t,x)=0, t\ge 0, x\ge 0 \\ N(t):=n(t,x=0)=c(t)+\int_{0}^{\infty}b(y)n(t,y)dy \end{cases}\,$ where $N(t)$ iis the speed of the increase of new customers. We assume $b, d \in L^\infty_-(0, \infty)$. $b(x)$ and $d(x)$ is bounded in essence. The following, we first make a simplified assumption: $c(t)\equiv 0$, i.e. the increase of new customer depends only on the promotion of previous customer. (a) According to assumption 1 and 2, formally derive the PDE that $n(t, x)$ satisfies in the two simtaneous equation above. You are required to show the assumption of model and the relationship between the Maths expression. Furthermore, according to assumption 3, explain the definition and meaning of $N(t)$ in the simtaneous equation above. (b) We want to research the relationship of the speed of the increase of the new customers $N(t)$ and the speed of promoting sucessfully $b(x)$. Derive an equation that $N(t)$ satisfies in terms of $N(t), n_0(x), b(x), d(x)$ only and does not include $n(t, x)$. Prove that $N(t)$ satifies the estimation $|N(t)|\le ||b||_\infty e^{||b||_\infty t}\int_{0}^{\infty}|n_0(x)|dx$, where $||\cdot||_\infty$ is the norm of $L^\infty$. (c) Finally, we want to research, after sufficiently long time, what trend of number density function $n(t, x) $\frac{d} has. As the total number of customers may keep increasing so it is not comfortable for us to research the number density function $n(t, x)$. We should try to find a density function which is renormalized. Hence, we first assume there is one only solution $(\lambda_0,\varphi(x))$ of the following eigenvalue problem: $\begin{cases} \varphi'(x)+(\lambda_0+d(x))\varphi(x)=0, x\ge 0 \\ \varphi(x)>0,\varphi(0)=\int_{0}^{\infty}b(x)\varphi(x)dx=1 \end{cases}\, $ and its dual problem has only solution $\psi(x)$: $\begin{cases} -\varphi'(x)+(\lambda_0+d(x))\psi(x)=\psi(0)b(x), x\ge 0 \\ \psi(x)>0,\int_{0}^{\infty}\psi(x)\varphi(x)dx=1 \end{cases}\,$ Prove that for any convex function $H:\mathbb{R}^+\to \mathbb{R}^+$ which satisfies $H(0)=0$. We have $\frac{d}{dx}\int_{0}^{\infty}\psi(x)\varphi(x)H(\frac{\tilde{n}(t,x)}{\varphi(x)})dx\le 0, \forall t\ge 0$. Furthermore, prove that $\int_{0}^{\infty}\psi(x)n(t,x)dx=e^{\lambda_0t}\int_{0}^{\infty}\psi(x)n_0(x)dx$ To simplify the proof, the contribution of boundary terms in $\infty$ is negligible.

Find all triples of non-negative real numbers $(a,b,c)$ which satisfy the following set of equations $$a^2+ab=c$$ $$b^2+bc=a$$ $$c^2+ca=b$$