Find all nonempty sets $S$ of integers such that $3m-2n \in S$ for all (not necessarily distinct) $m,n \in S$.

Let $T$ be a tree with $n$ vertices; that is, a connected simple graph on $n$ vertices that contains no cycle. For every pair $u$, $v$ of vertices, let $d(u,v)$ denote the distance between $u$ and $v$, that is, the number of edges in the shortest path in $T$ that connects $u$ with $v$. Consider the sums \[W(T)=\sum_{\substack{\{u,v\}\subseteq V(T)\\ u\neq v}}d(u,v) \quad \text{and} \quad H(T)=\sum_{\substack{\{u,v\}\subseteq V(T)\\ u\neq v}}\frac{1}{d(u,v)}\] Prove that \[W(T)\cdot H(T)\geq \frac{(n-1)^3(n+2)}{4}.\]

A quadrilateral $ABCD$ is inscribed, as shown, in a square of area one unit. Prove that $$2\le |AB|^2+|BC|^2+|CD|^2+|DA|^2\le 4$$ [asy] size(6cm); draw((0,0)--(10,0)); draw((10,0)--(10,10)); draw((0,10)--(10,10)); draw((0,0)--(0,10)); dot((0,8.5)); dot((3.5,10)); dot((10,3.5)); dot((3.5,0)); label("$D$",(0,8.5),W); label("$A$",(3.5,10),NE); label("$B$",(10,3.5),E); label("$C$",(3.5,0),S); draw((0,8.5)--(3.5,10)); draw((3.5,10)--(10,3.5)); draw((10,3.5)--(3.5,0)); draw((3.5,0)--(0,8.5)); [/asy]

Let $n$ be a positive integer, and let $b_{1}$, $b_{2}$, ..., $b_{n}$ be $n$ positive reals. Set $a_{1}=\frac{b_{1}}{b_{1}+b_{2}+...+b_{n}}$ and $a_{k}=\frac{b_{1}+b_{2}+...+b_{k}}{b_{1}+b_{2}+...+b_{k-1}}$ for every $k>1$. Prove the inequality $a_{1}+a_{2}+...+a_{n}\leq\frac{1}{a_{1}}+\frac{1}{a_{2}}+...+\frac{1}{a_{n}}$.

Find the greatest real number $K$ such that for all positive real number $u,v,w$ with $u^{2}>4vw$ we have $(u^{2}-4vw)^{2}>K(2v^{2}-uw)(2w^{2}-uv)$

Let a triangle $\triangle ABC$ and the real numbers $x$, $y$, $z>0$. Prove that $$x^n\cos\frac{A}{2}+y^n\cos\frac{B}{2}+z^n\cos\frac{C}{2}\geq (yz)^{\frac{n}{2}}\sin A +(zx)^{\frac{n}{2}}\sin B +(xy)^{\frac{n}{2}}\sin C$$

Let $n \ge 3$ be a natural number and let $P$ be a polygon with $n$ sides. Let $a_1,a_2,\cdots, a_n$ be the lengths of sides of $P$ and let $p$ be its perimeter. Prove that \[\frac{a_1}{p-a_1}+\frac{a_2}{p-a_2}+\cdots + \frac{a_n}{p-a_n} < 2 \]

Prove that for arbitrary positive numbers the following inequality holds \[\frac{1}{a} + \frac{1}{b} + \frac{1}{c} \leq \frac{a^8 + b^8 + c^8}{a^3b^3c^3}.\]

Let $a_1,a_2,a_3,\ldots,a_n$ be a sequence of non-negative integers, where $n$ is a positive integer. Let \[ A_n={a_1+a_2+\cdots+a_n\over n}\ . \] Prove that \[ a_1!a_2!\ldots a_n!\ge\left(\lfloor A_n\rfloor !\right)^n \] where $\lfloor A_n\rfloor$ is the greatest integer less than or equal to $A_n$, and $a!=1\times 2\times\cdots\times a$ for $a\ge 1$(and $0!=1$). When does equality hold?

Three travel companies provide transportation between $n$ cities, such that each connection between a pair of cities is covered by one company only. Prove that, for $n \geq 11$, there must exist a round-trip through some four cities, using the services of a same company, while for $n < 11$ this is not anymore necessarily true. [i]Dan Schwarz[/i]

Point $O$ is interior to triangle $ABC$. Through $O$, draw three lines $DE \parallel BC, FG \parallel CA$, and $HI \parallel AB$, where $D, G$ are on $AB$, $I, F$ are on $BC$ and $E, H$ are on $CA$. Denote by $S_1$ the area of hexagon $DGHEFI$, and $S_2$ the area of triangle $ABC$. Prove that $S_1 \geq \frac 23 S_2.$

Let triangle $ABC$ be such that its circumradius is $R = 1.$ Let $r$ be the inradius of $ABC$ and let $p$ be the inradius of the orthic triangle $A'B'C'$ of triangle $ABC.$ Prove that \[ p \leq 1 - \frac{1}{3 \cdot (1+r)^2}. \] [hide="Similar Problem posted by Pascual2005"] Let $ABC$ be a triangle with circumradius $R$ and inradius $r$. If $p$ is the inradius of the orthic triangle of triangle $ABC$, show that $\frac{p}{R} \leq 1 - \frac{\left(1+\frac{r}{R}\right)^2}{3}$. [i]Note.[/i] The orthic triangle of triangle $ABC$ is defined as the triangle whose vertices are the feet of the altitudes of triangle $ABC$. [b]SOLUTION 1 by mecrazywong:[/b] $p=2R\cos A\cos B\cos C,1+\frac{r}{R}=1+4\sin A/2\sin B/2\sin C/2=\cos A+\cos B+\cos C$. Thus, the ineqaulity is equivalent to $6\cos A\cos B\cos C+(\cos A+\cos B+\cos C)^2\le3$. But this is easy since $\cos A+\cos B+\cos C\le3/2,\cos A\cos B\cos C\le1/8$. [b]SOLUTION 2 by Virgil Nicula:[/b] I note the inradius $r'$ of a orthic triangle. Must prove the inequality $\frac{r'}{R}\le 1-\frac 13\left( 1+\frac rR\right)^2.$ From the wellknown relations $r'=2R\cos A\cos B\cos C$ and $\cos A\cos B\cos C\le \frac 18$ results $\frac{r'}{R}\le \frac 14.$ But $\frac 14\le 1-\frac 13\left( 1+\frac rR\right)^2\Longleftrightarrow \frac 13\left( 1+\frac rR\right)^2\le \frac 34\Longleftrightarrow$ $\left(1+\frac rR\right)^2\le \left(\frac 32\right)^2\Longleftrightarrow 1+\frac rR\le \frac 32\Longleftrightarrow \frac rR\le \frac 12\Longleftrightarrow 2r\le R$ (true). Therefore, $\frac{r'}{R}\le \frac 14\le 1-\frac 13\left( 1+\frac rR\right)^2\Longrightarrow \frac{r'}{R}\le 1-\frac 13\left( 1+\frac rR\right)^2.$ [b]SOLUTION 3 by darij grinberg:[/b] I know this is not quite an ML reference, but the problem was discussed in Hyacinthos messages #6951, #6978, #6981, #6982, #6985, #6986 (particularly the last message). [/hide]

Let $a,b,c\in\mathbb R_+$ satisfy that $$\sqrt{(1+a)(1+b)(1+c)}=\sqrt{(ab-a-b+1)(1+c)}+\sqrt{(bc-b-c+1)(1+a)}+\sqrt{(ca-c-a+1)(1+b)}.$$ Find the value range of $a+b+c.$

Let $P$ be a polynomial with each root is real and each coefficient is either $1$ or $-1$. The degree of $P$ can be at most ? $ \textbf{(A)}\ 5 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ \text{None} $

Let $m<n$ be two positive integers, let $I$ and $J$ be two index sets such that $|I|=|J|=n$ and $|I\cap J|=m$, and let $u_k$, $k\in I\cup J$ be a collection of vectors in the Euclidean plane such that \[|\sum_{i\in I}u_i|=1=|\sum_{j\in J}u_j|.\] Prove that \[\sum_{k\in I\cup J}|u_k|^2\geq \frac{2}{m+n}\] and find the cases of equality.

Prove that $$\sum \limits_{k=1}^n \frac{1}{d(k)}>\sqrt{n+1}-1$$ For every $n\geq 1$, $d(n)$ is the number of divisors of $n$

Determine the set of integers $n$ for which $n^2+19n+92$ is a square.

Let $x_1, \ldots , x_{100}$ be nonnegative real numbers such that $x_i + x_{i+1} + x_{i+2} \leq 1$ for all $i = 1, \ldots , 100$ (we put $x_{101 } = x_1, x_{102} = x_2).$ Find the maximal possible value of the sum $S = \sum^{100}_{i=1} x_i x_{i+2}.$ [i]Proposed by Sergei Berlov, Ilya Bogdanov, Russia[/i]

If $a,b,c$ are three real numbers and \[ a + \dfrac{1}{b} = b + \dfrac{1}{c} = c + \dfrac{1}{a} = t \] for some real number $t$, prove that $abc + t = 0 .$

Find all non-decreasing functions $f:\mathbb R^+\cup\{0\}\rightarrow\mathbb R^+\cup\{0\}$ such that for each $x,y\in \mathbb R^+\cup\{0\}$ \[f\left(\frac{x+f(x)}2+y\right)=2x-f(x)+f(f(y)).\]

Let $a,b,c>0$ so that $a+b+c=1$. Then prove that $$(a+2ab+2ac+bc)^a(b+2bc+2ba+ca)^b(c+2ca+2cb+ab)^c\le1.$$ [i]Proposed by Marius Drăgan[/i]

Let $0 \le a$, $b \le 1$ be real numbers. Prove the following inequality: \[\sqrt{a^3b^3}+ \sqrt{(1-a^2)(1-ab)(1-b^2)} \le 1.\] [i](41th Austrian Mathematical Olympiad, regional competition, problem 1)[/i]

Let $x,y,z$ be positive real numbers. Prove that \[ \sum_{cyclic} \frac{xy}{xy+x^2+y^2} ~\le~ \sum_{cyclic} \frac{x}{2x+z} \] [i](Proposed by Šefket Arslanagić, Bosnia and Herzegovina)[/i]

Let $a$, $b$, $c \in (0,+\infty)$ . Then the following inequality is true:$$\sqrt{(a+b)(b+c)}+\sqrt{(b+c)(c+a)}+\sqrt{(c+a)(a+b)}+a+b+c\leq \left(ab+bc+ca\right)\left(\frac{1}{\sqrt{ab}}+\frac{1}{\sqrt{bc}}+\frac{1}{\sqrt{ca}}\right)$$

Let's consider the inequality $ a^3\plus{}b^3\plus{}c^3<k(a\plus{}b\plus{}c)(ab\plus{}bc\plus{}ca)$ where $ a,b,c$ are the sides of a triangle and $ k$ a real number. [b]a)[/b] Prove the inequality for $ k\equal{}1$. [b]b) [/b]Find the smallest value of $ k$ such that the inequality holds for all triangles.