Find all functions $f : \mathbb{R} \to \mathbb{R}$, such that $$f\left(xy+f(y)\right)f(x)=x^2f(y)+f(xy)$$ for all $x,y \in \mathbb{R}$ [i]Proposed by chengbilly[/i]

For any positive integers $n>m$ prove the following inequality: $$[m,n]+[m+1,n+1]\geq 2m\sqrt{n}$$ As usual, [x,y] denotes the least common multiply of $x,y$ [I]Proposed by A. Golovanov[/i]

Let $\alpha \in (1, +\infty)$ be a real number, and let $P(x) \in \mathbb{R}[x]$ be a monic polynomial with degree $24$, such that (i) $P(0) = 1$. (ii) $P(x)$ has exactly $24$ positive real roots that are all less than or equal to $\alpha$. Show that $|P(1)| \le \left( \frac{19}{5}\right)^5 (\alpha-1)^{24}$.

Given an integer $n > 2$, consider all sequences $x_1,x_2,...,x_n$ of nonnegative real numbers such that $$x_1+ 2x_2 + ... + nx_n = 1.$$ Find the maximum value and the minimum value of $x_1^2+x_2^2+...+x_n^2$ and determine all the sequences $x_1,x_2,...,x_n$ for which these values are obtained.

Two lines, $l_1$ and $l_2$, are tangent to the parabola $x^2-4(x+y)+y^2=2xy+8$ such that they intersect at a point whose coordinates sum to $-32$. The minimum possible sum of the slopes of $l_1$ and $l_2$ can be written as $\frac{m}{n}$ for relatively prime integers $m$ and $n$. Find $m+n$. [I] Proposed by [b]AOPS12142015[/b][/I]

Let $G$ be a finite simple graph and let $k$ be the largest number of vertices of any clique in $G$. Suppose that we label each vertex of $G$ with a non-negative real number, so that the sum of all such labels is $1$. Define the [i]value of an edge[/i] to be the product of the labels of the two vertices at its ends. Define the [i]value of a labelling[/i] to be the sum of values of the edges. Prove that the maximum possible value of a labelling of $G$ is $\frac{k-1}{2k}$. (A [i]finite simple graph[/i] is a graph with finitely many vertices, in which each edge connects two distinct vertices and no two edges connect the same two vertices. A [i]clique[/i] in a graph is a set of vertices in which any two are connected by an edge.)

Let $a, b, c$ be three positive real numbers. a) Prove that there exists a unique positive real number $d$ that satisfies $$\frac{1}{a + d}+ \frac{1}{b + d}+\frac{1}{c + d}=\frac{2}{d} .$$ b) With $x, y, z$ being positive real numbers such that $ax + by + cz = xyz$, prove the inequality $$x + y + z \ge \frac{2}{d}\sqrt{(a + d)(b + d)(c + d)}.$$

Let $n\geqslant 2$ be a positive integer and $a_1,a_2, \ldots ,a_n$ be real numbers such that \[a_1+a_2+\dots+a_n=0.\] Define the set $A$ by \[A=\left\{(i, j)\,|\,1 \leqslant i<j \leqslant n,\left|a_{i}-a_{j}\right| \geqslant 1\right\}\] Prove that, if $A$ is not empty, then \[\sum_{(i, j) \in A} a_{i} a_{j}<0.\]

For each positive integer $n$ find the largest subset $M(n)$ of real numbers possessing the property: \[n+\sum_{i=1}^{n}x_{i}^{n+1}\geq n \prod_{i=1}^{n}x_{i}+\sum_{i=1}^{n}x_{i}\quad \text{for all}\; x_{1},x_{2},\cdots,x_{n}\in M(n)\] When does the inequality become an equality ?

Given \(n \in \mathbb{N}\), let \(\sigma (n)\) denote the sum of the divisors of \(n\) and \(\phi (n)\) denote the number of integers \(n \geq m\) for which \(\gcd(m,n) = 1\). Show that for all \(n \in \mathbb{N}\), \[\large \frac{1}{\sigma (n)} + \frac{1}{\phi (n)} \geq \frac{2}{n}\] and determine when equality holds.

Show that for any positive real numbers $a,b,c$ the following inequality is true: $$4(a^3+b^3+c^3+3)\geq 3(a+1)(b+1)(c+1)$$ When does equality hold?

Two masses are connected with spring constant $k$. The masses have magnitudes $m$ and $M$. The center-of-mass of the system is fixed. If $ k = \text {100 N/m} $ and $m=\dfrac{1}{2}M=\text{1 kg}$, let the ground state energy of the system be $E$. If $E$ can be expressed in the form $ a \times 10^p $ eV (electron-volts), find the ordered pair $(a,p)$, where $ 0 < a < 10 $, and it is rounded to the nearest positive integer and $p$ is an integer. For example, $ 4.2 \times 10^7 $ should be expressed as $(4,7)$. [i](Trung Phan, 10 points)[/i]

Let $a,b,c,d$ be real numbers such that $a^2+b^2+c^2+d^2=1$. Determine the minimum value of $(a-b)(b-c)(c-d)(d-a)$ and determine all values of $(a,b,c,d)$ such that the minimum value is achived.

Prove that for any positive integer $k$, \[(k^2)!\cdot\displaystyle\prod_{j=0}^{k-1}\frac{j!}{(j+k)!}\]is an integer.

a,b,c>0 and $abc\ge 1$.Prove that: $\dfrac{1}{a^3+2b^3+6}+\dfrac{1}{b^3+2c^3+6}+\dfrac{1}{c^3+2a^3+6} \le \dfrac{1}{3}$