Prove that \[ \log_ab\plus{}\log_bc\plus{}\log_ca\le \log_ba\plus{}\log_cb\plus{}\log_ac\] for all $ 1<a\le b\le c$.

Let $a$, $b$, $c$ be real numbers. Prove that \begin{align*}&\quad\,\,\sqrt{2(a^2+b^2)}+\sqrt{2(b^2+c^2)}+\sqrt{2(c^2+a^2)}\\&\ge \sqrt{3[(a+b)^2+(b+c)^2+(c+a)^2]}.\end{align*}

If $a,b,c$ are positive numbers with $a^2 +b^2 +c^2 = 1$, prove that $a+b+c+\frac{1}{abc} \ge 4\sqrt3$

Positive reals $a$, $b$, and $c$ obey $\frac{a^2+b^2+c^2}{ab+bc+ca} = \frac{ab+bc+ca+1}{2}$. Prove that \[ \sqrt{a^2+b^2+c^2} \le 1 + \frac{\lvert a-b \rvert + \lvert b-c \rvert + \lvert c-a \rvert}{2}. \][i]Proposed by Evan Chen[/i]

What is the largest real number $C$ that satisfies the inequality $x^2 \geq C \lfloor x \rfloor (x-\lfloor x \rfloor)$ for every real $x$? $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ 9 \qquad\textbf{(E)}\ 25 $

Let $ P(x)$ be a quadratic polynomial with real coefficients satisfying \[x^2 \minus{} 2x \plus{} 2 \le P(x) \le 2x^2 \minus{} 4x \plus{} 3\] for all real numbers $ x$, and suppose $ P(11) \equal{} 181$. Find $ P(16)$.

Let $a,b,c \in \mathbb {R}^{+}$ and $abc=1$ prove that: $\frac {a+b}{(a+b+1)^2}+\frac {b+c}{(b+c+1)^2}+\frac {c+a}{(c+a+1)^2} \geq \frac {2}{a+b+c}$

In a small town, there are $n \times n$ houses indexed by $(i, j)$ for $1 \leq i, j \leq n$ with $(1, 1)$ being the house at the top left corner, where $i$ and $j$ are the row and column indices, respectively. At time 0, a fire breaks out at the house indexed by $(1, c)$, where $c \leq \frac{n}{2}$. During each subsequent time interval $[t, t+1]$, the fire fighters defend a house which is not yet on fire while the fire spreads to all undefended [i]neighbors[/i] of each house which was on fire at time t. Once a house is defended, it remains so all the time. The process ends when the fire can no longer spread. At most how many houses can be saved by the fire fighters? A house indexed by $(i, j)$ is a [i]neighbor[/i] of a house indexed by $(k, l)$ if $|i - k| + |j - l|=1$.

Let $a,b,c $ be the lengths of the three sides of a triangle and $a,b$ be the two roots of the equation $ax^2-bx+c=0 $$ (a<b) . $ Find the value range of $ a+b-c .$

Some computers of a computer room have a following network. Each computers are connected by three cable to three computers. Two arbitrary computers can exchange data directly or indirectly (through other computers). Now let's remove $K$ computers so that there are two computers, which can not exchange data, or there is one computer left. Let $k$ be the minimum value of $K$. Let's remove $L$ cable from original network so that there are two computers, which can not exchange data. Let $l$ be the minimum value of $L$. Show that $k=l$.

Prove that for all positive real numbers $a$ and $ b$: $$\frac{(a + b)^3}{4} \ge a^2b + ab^2$$

Let $x_1, x_2, \ldots ,x_n(n\ge 2)$ be real numbers greater than $1$. Suppose that $|x_i-x_{i+1}|<1$ for $i=1, 2,\ldots ,n-1$. Prove that \[\frac{x_1}{x_2}+\frac{x_2}{x_3}+\ldots +\frac{x_{n-1}}{x_n}+\frac{x_n}{x_1}<2n-1\]

Let $n\ge 2$ be a positive integer. There are $n$ real coefficient polynomials $P_1(x),P_2(x),\cdots ,P_n(x)$ which is not all the same, and their leading coefficients are positive. Prove that $$\deg(P_1^n+P_2^n+\cdots +P_n^n-nP_1P_2\cdots P_n)\ge (n-2)\max_{1\le i\le n}(\deg P_i)$$ and find when the equality holds.

Let $f$ be a linear function such that $f(0) = -5$ and $f(f(0)) = -15$. Find the values of $ k \in R$ for which the solutions of the inequality $f(x) \cdot f(k - x) > 0$, lie in an interval of[u][/u] length $2$.

A non-negative integer $n$ is said to be [i]squaredigital[/i] if it equals the square of the sum of its digits. Find all non-negative integers which are squaredigital.

Find the largest real number $ C_1 $ and the smallest real number $ C_2 $, such that, for all reals $ a,b,c,d,e $, we have \[ C_1 < \frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+d}+\frac{d}{d+e}+\frac{e}{e+a} < C_2 \]

Prove that if \( a \), \( b \), and \( c \) are positive real numbers, then the following inequality holds: \[ \frac{a + 3c}{a + b} + \frac{c + 3a}{b + c} + \frac{4b}{c + a} \geq 6. \]

Let $n>k>1$ be positive integers. Determine all positive real numbers $a_1, a_2, \ldots, a_n$ which satisfy $$ \sum_{i=1}^n \sqrt{\frac{k a_i^k}{(k-1) a_i^k+1}}=\sum_{i=1}^n a_i=n . $$

a) Prove that \[ \dfrac{1}{2} \plus{} \dfrac{1}{3} \plus{} ... \plus{} \dfrac{1}{2^{2n}} > n, \] for all positive integers $ n$. b) Prove that for every positive integer $ n$ we have $ \min\left\{ k \in \mathbb{Z}, k\geq 2 \mid \dfrac{1}{2} \plus{} \dfrac{1}{3} \plus{} \cdots \plus{} \dfrac{1}{k}>n \right\} > 2^n$.

[b]Problem 3[/b] : Let $x_1,x_2,\cdots,x_{2022}$ be non-negative real numbers such that $$x_k + x_{k+1}+x_{k+2} \leq 2$$ for all $k = 1,2,\cdots,2020$. Prove that $$\sum_{k=1}^{2020}x_kx_{k+2}\leq 1010$$

The real numbers $x,y,z, m, n$ are positive, such that $m + n \ge 2$. Prove that $x\sqrt{yz(x + my)(x + nz)} + y\sqrt{xz(y + mx)(y + nz)} + z\sqrt{xy(z + mx)(x + ny) }\le \frac{3(m + n)}{8} (x + y)(y + z)(z + x)$

Let $ n$ and $ N$ be natural number. Prove that for any $ \alpha$, $ 0\le\alpha\le N$, and any real $ x$, it holds that \[{ |\sum_{k=0}^n}\frac{\sin((\alpha+k)x)}{N+k}|\le\min\{(n+1)|x|, \frac{1}{N|\sin\frac{x}{2}|}\}\]

Show that the radian measure of an acute angle is less than the arithmetic mean of its sine and its tangent.

Let $y^{\alpha}=\sum_{i=1}^n x_i^{\alpha}$ where $\alpha \neq 0, y > 0, x_i > 0$ are real numbers, and let $\lambda \neq \alpha$ be a real number. Prove that $y^{\lambda} > \sum_{i=1}^n x_i^{\lambda}$ if $\alpha (\lambda - \alpha) > 0,$ and $y^{\lambda} < \sum_{i=1}^n x_i^{\lambda}$ if $\alpha (\lambda - \alpha) < 0.$

In a triangle $ABC$ with incircle $\omega$ and incenter $I$ , the segments $AI$ , $BI$ , $CI$ cut $\omega$ at $D$ , $E$ , $F$ , respectively. Rays $AI$ , $BI$ , $CI$ meet the sides $BC$ , $CA$ , $AB$ at $L$ , $M$ , $N$ respectively. Prove that: \[AL+BM+CN \leq 3(AD+BE+CF)\] When does equality occur?