Let the lengths of the sides of triangle $ABC$ be denoted by $a,b,$ and $c,$ using the standard notations. Let $G$ denote the centroid of triangle $ABC.$ Prove that for an arbitrary point $P$ in the plane of the triangle the following inequality is true: \[a\cdot PA^3+b\cdot PB^3+c\cdot PC^3\geq 3abc\cdot PG.\][i]Proposed by János Schultz, Szeged[/i]

How many integers $n$ are there those satisfy the following inequality $n^4 - n^3 - 3n^2 - 3n - 17 < 0$? A. $4$ B. $6$ C. $8$ D. $10$ E. $12$

For any three positive real numbers $a$, $b$, $c$, prove the inequality \[\frac{\left(b+c\right)^{2}}{a^{2}+bc}+\frac{\left(c+a\right)^{2}}{b^{2}+ca}+\frac{\left(a+b\right)^{2}}{c^{2}+ab}\geq 6.\] Darij

For a positive integer $n$ and real numbers $a_1, a_2, \dots ,a_n$ we'll define $b_1, b_2, \dots ,b_{n+1}$ such that $b_k=a_k+\max({a_{k+1},a_{k+2}})$ for all $1\leq k \leq n$ and $b_{n+1}=b_1$. (Also $a_{n+1}=a_1$ and $a_{n+2}=a_2$) Find the least possible value of $\lambda$ such that for all $n, a_1, \dots, a_n$ the inequality $$\lambda \Biggl[ \sum_{i=1}^n(a_i-a_{i+1})^{2024} \Biggr] \geq \sum_{i=1}^n(b_i-b_{i+1})^{2024}$$ holds.

Define a positive number sequence sequence $\{a_n\}$ by \[a_{1}=1,(n^2+1)a^2_{n-1}=(n-1)^2a^2_{n}.\]Prove that\[\frac{1}{a^2_1}+\frac{1}{a^2_2}+\cdots +\frac{1}{a^2_n}\le 1+\sqrt{1-\frac{1}{a^2_n}} .\]

Let $n$ be a positive integer, and $x$ be a positive real number. Prove that $$\sum_{k=1}^{n} \left( x \left[\frac{k}{x}\right] - (x+1)\left[\frac{k}{x+1}\right]\right) \leq n,$$ where $[x]$ denotes the largest integer not exceeding $x$.

Let $ \{a_n\}_{n\geq 1}$ be a sequence of real numbers such that $ |a_{n\plus{}1}\minus{}a_n|\leq 1$, for all positive integers $ n$. Let $ \{b_n\}_{n\geq 1}$ be the sequence defined by \[ b_n \equal{} \frac { a_1\plus{} a_2 \plus{} \cdots \plus{}a_n} {n}.\] Prove that $ |b_{n\plus{}1}\minus{}b_n | \leq \frac 12$, for all positive integers $ n$.

Let $m$ and $n$ be positive integers. Show that $\frac{(m+n)!}{(m+n)^{m+n}} < \frac{m!}{m^m}\cdot\frac{n!}{n^n}$

Show that for real numbers $x_1, x_2, ... , x_n$ we have: \[ \sum\limits_{i=1}^n \sum\limits_{j=1}^n \dfrac{x_ix_j}{i+j} \geq 0 \] When do we have equality?

Prove that for any positive integer $n\ge 2$ the following inequality holds: $$\frac{1}{n+1}+\frac{1}{n+2}+...+\frac{1}{2n}>\frac{1}{2}$$

Let $a, b, c \ge 0$ so that $ab + bc + ca = 3$. Prove that: $$\frac{a}{a^2+7}+\frac{b}{b^2+7}+\frac{c}{c^2+7}\le \frac38$$

The incircle and the excircle of triangle $ABC$ touch the side $AC$ at points $P$ and $Q$ respectively. The lines $BP$ and $BQ$ meet the circumcircle of triangle $ABC$ for the second time at points $P'$ and $Q'$ respectively. Prove that $$PP' > QQ'$$

Let $P$, $Q$, and $R$ be the points where the incircle of a triangle $ABC$ touches the sides $AB$, $BC$, and $CA$, respectively. Prove the inequality $\frac{BC} {PQ} + \frac{CA} {QR} + \frac{AB} {RP} \geq 6$.

Let $x, y,z$ satisfy the following inequalities $\begin{cases} | x + 2y - 3z| \le 6 \\ | x - 2y + 3z| \le 6 \\ | x - 2y - 3z| \le 6 \\ | x + 2y + 3z| \le 6 \end{cases}$ Determine the greatest value of $M = |x| + |y| + |z|$.

*. Positive numbers $x_1, x_2, ..., x_{100}$ satisfy the system $$\begin{cases} x^2_1+ x^2_2+ ... + x^2_{100} > 10 000 \\ x_1 + x_2 + ...+ x_{100} < 300 \end{cases}$$ Prove that among these numbers there are three whose sum is greater than $100$.

Let $ a $ and $ b $ be different real numbers. Prove that for any real numbers $ c_1, c_2, \ldots,c_n $ there exists a sequence of $ n $-elements $ (x_i) $, each term of which is equal to one of the numbers $ a $ or $ b $ such that $$ |x_1c_1 + x_2c_2 + \ldots + x_nc_n| \geq \frac{|b-a|}{2}(|c_1|+|c_2|+\ldots+|c_n|).$$

Find all monotonically increasing or monotonically decreasing functions $f: \mathbb{R}_+\to\mathbb{R}_+$ which satisfy the equation $f\left(xy\right)\cdot f\left(\frac{f\left(y\right)}{x}\right)=1$ for any two numbers $x$ and $y$ from $\mathbb{R}_+$. Hereby, $\mathbb{R}_+$ is the set of all positive real numbers. [i]Note.[/i] A function $f: \mathbb{R}_+\to\mathbb{R}_+$ is called [i]monotonically increasing[/i] if for any two positive numbers $x$ and $y$ such that $x\geq y$, we have $f\left(x\right)\geq f\left(y\right)$. A function $f: \mathbb{R}_+\to\mathbb{R}_+$ is called [i]monotonically decreasing[/i] if for any two positive numbers $x$ and $y$ such that $x\geq y$, we have $f\left(x\right)\leq f\left(y\right)$.

Show that there exists a positive number t such that for all positive numbers $a,b,c,d$ with $abcd = 1$, $$\frac{1}{1+a}+\frac{1}{1+b}+\frac{1}{1+c}+\frac{1}{1+d}> t.$$ and find the largest $t$ with this property.

[color=darkred]Prove that if $n\ge 2$ is a natural number and $x_1,x_2,\ldots,x_n$ are positive real numbers, then: \[4\left(\frac {x_1^3-x_2^3}{x_1+x_2}+\frac {x_2^3-x_3^3}{x_2+x_3}+\ldots+\frac {x_{n-1}^3-x_n^3}{x_{n-1}+x_n}+\frac {x_n^3-x_1^3}{x_n+x_1}\right)\le \\ \\ \le(x_1-x_2)^2+(x_2-x_3)^2+\ldots+(x_{n-1}-x_n)^2+(x_n-x_1)^2\, .\][/color]

Let $f(x) = x^n + a_{n-2} x^{n-2} + a_{n-3}x^{n-3} + \dots + a_1x + a_0$ be a polynomial with real coefficients $(n \ge 2)$. Suppose all roots of $f$ are real. Prove that the absolute value of each root is at most $\sqrt{\frac{2(1-n)}n a_{n-2}}$.

Prove or disprove that $\forall a,b,c,d \in \mathbb{R}^+$ we have the following inequality: \[3 \leq \frac{4a+b}{a+4b} + \frac{4b+c}{b+4c} + \frac{4c+a}{c+4a} < \frac{33}{4}\]

Let $ x_{1} ,x_{2} ,\ldots ,x_{9}$ be real numbers on $ \left[ \minus{} 1,1\right]$. If $ \sum _{i \equal{} 1}^{9}x_{i}^{3} \equal{} 0$, then what is the largest possible value of $ \sum _{i \equal{} 1}^{9}x_{i}$? $\textbf{(A)}\ 1 \qquad\textbf{(B)}\ \frac {3}{2} \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ \frac {9}{2} \qquad\textbf{(E)}\ \text{None}$

For each side of a given pentagon, divide its length by the total length of all other sides. Prove that the sum of all the fractions obtained is less than 2.

We define \[f(x,y,z)=|xy|\sqrt{x^2+y^2}+|yz|\sqrt{y^2+z^2}+|zx|\sqrt{z^2+x^2}.\] Find the best constants $c_1,c_2\in\mathbb{R}$ such that \[c_1(x^2+y^2+z^2)^{3/2}\leq f(x,y,z)\leq c_1(x^2+y^2+z^2)^{3/2}\] hold for all reals $x,y,z$ satisfying $x+y+z=0$. [i]Proposed by Untro368.[/i]

Let $a,b,c$ be the sides of a triangle, with $a+b+c=1$, and let $n\ge 2$ be an integer. Show that \[ \sqrt[n]{a^n+b^n}+\sqrt[n]{b^n+c^n}+\sqrt[n]{c^n+a^n}<1+\frac{\sqrt[n]{2}}{2}. \]