The coefficient of $x^7$ in the expansion of $(\frac{x^2}{2}-\frac{2}{x})^8$ is: $ \textbf{(A)}\ 56\qquad\textbf{(B)}\ -56\qquad\textbf{(C)}\ 14\qquad\textbf{(D)}\ -14\qquad\textbf{(E)}\ 0 $

Let $ABC$ be a triangle with semiperimeter $s$ and inradius $r$. The semicircles with diameters $BC$, $CA$, $AB$ are drawn on the outside of the triangle $ABC$. The circle tangent to all of these three semicircles has radius $t$. Prove that \[\frac{s}{2}<t\le\frac{s}{2}+\left(1-\frac{\sqrt{3}}{2}\right)r. \] [i]Alternative formulation.[/i] In a triangle $ABC$, construct circles with diameters $BC$, $CA$, and $AB$, respectively. Construct a circle $w$ externally tangent to these three circles. Let the radius of this circle $w$ be $t$. Prove: $\frac{s}{2}<t\le\frac{s}{2}+\frac12\left(2-\sqrt3\right)r$, where $r$ is the inradius and $s$ is the semiperimeter of triangle $ABC$. [i]Proposed by Dirk Laurie, South Africa[/i]

Let $a,b,c,d,e,f$ be six real numbers with sum 10, such that \[ (a-1)^2+(b-1)^2+(c-1)^2+(d-1)^2+(e-1)^2+(f-1)^2 = 6. \] Find the maximum possible value of $f$. [i]Cyprus[/i]

Given integer $n\geq 2$ and positive real number $a$, find the smallest real number $M=M(n,a)$, such that for any positive real numbers $x_1,x_2,\cdots,x_n$ with $x_1 x_2\cdots x_n=1$, the following inequality holds: \[\sum_{i=1}^n \frac {1}{a+S-x_i}\leq M\] where $S=\sum_{i=1}^n x_i$.

Two beasts, Rosencrans and Gildenstern, play a game. They have a circle with $n$ points ($n \ge 5$) on it. On their turn, each beast (starting with Rosencrans) draws a chord between a pair of points in such a way that any two chords have a shared point. (The chords either intersect or have a common endpoint.) For example, two potential legal moves for the second player are drawn below with dotted lines. [asy] unitsize(0.7cm); draw(circle((0,0),1)); dot((0,-1)); pair A = (-1/2,-(sqrt(3))/2); dot(A); pair B = ((sqrt(2))/2,-(sqrt(2))/2); dot(B); pair C = ((sqrt(3))/2,1/2); dot(C); draw(A--C); pair D = (-(sqrt(0.05)),sqrt(0.95)); dot(D); pair E = (-(sqrt(0.2)),sqrt(0.8)); dot(E); draw(B--E,dotted); draw(C--D,dotted); [/asy] The game ends when a player cannot draw a chord. The last beast to draw a chord wins. For which $n$ does Rosencrans win?

Let $\alpha$ be given positive real number, find all the functions $f: N^{+} \rightarrow R$ such that $f(k + m) = f(k) + f(m)$ holds for any positive integers $k$, $m$ satisfying $\alpha m \leq k \leq (\alpha + 1)m$.

Let $a=x_1\leq x_2\leq ...\leq x_n=b$. Prove the following inequality: $(x_1+x_2+...+x_n )(\frac{1}{x_1} +\frac{1}{x_2} +...+\frac{1}{x_n} )\leq \frac{(a+b)}{4ab} n^2$.

Find all monic polynomials $f(x)$ in $\mathbb Z[x]$ such that $f(\mathbb Z)$ is closed under multiplication. [i]By Mohsen Jamali[/i]

Prove that for all real numbers $x$ holds: $$\sin 5x = 16 \sin x \cdot \sin \left(x -\frac{\pi}{5} \right) \cdot \sin\left(x -\frac{2\pi}{5} \right) \sin \left(x +\frac{2\pi}{5} \right) $$

Prove that the following inequality is valid for the positive $x$: $$2^{x^{1/12}}+ 2^{x^{1/4}} \ge 2^{1 + x^{1/6} }$$

In a group of $n\geq 4$ persons, every three who know each other have a common signal. Assume that these signals are not repeatad and that there are $m\geq 1$ signals in total. For any set of four persons in which there are three having a common signal, the fourth person has a common signal with at most one of them. Show that there three persons who have a common signal, such that the number of persons having no signal with anyone of them does not exceed $[n+3-\frac{18m}{n}]$.

In a convex quadrilateral of area $1$, the sum of the lengths of all sides and diagonals is not less than $4+\sqrt 8$. Prove this.

In fig. the bisectors of the angles $\angle DAC$, $ \angle EBD$, $\angle ACE$, $\angle BDA$ and $\angle CEB$ intersect at one point. Prove that the bisectors of the angles $\angle TPQ$, $\angle PQR$, $\angle QRS$, $\angle RST$ and $\angle STP$ also intersect at one point. [img]https://cdn.artofproblemsolving.com/attachments/6/e/870e4f20bc7fdcb37534f04541c45b1cd5034a.png[/img]

The graph $y=x^2+ax+b$ intersects any of the two axes at points $A$, $B$, and $C$. The incenter of triangle $ABC$ lies on the line $y=x$. Prove that $a+b+1=0$.

A positive integer $N$ is given. Panda builds a tree on $N$ vertices, and writes a real number on each vertex, so that $1$ plus the number written on each vertex is greater or equal to the average of the numbers written on the neighboring vertices. Let the maximum number written be $M$ and the minimal number written $m$. Mink then gives Panda $M-m$ kilograms of bamboo. What is the maximum amount of bamboo Panda can get?

There are $6$ points marked on the plane. Find the greatest possible number of acute triangles with vertices at the marked points.

Let $S(n)$ denote the sum of digits of $n \in N$. Find all $n$ such that $S(n) = S(2n) = S(3n) =... = S(n^2)$

Let $ab+bc+ca=1$. Show that \[\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}\ge\sqrt{3}+\frac{ab}{a+b}+\frac{bc}{b+c}+\frac{ca}{c+a}\]

Assume that for pairwise distinct real numbers $a,b,c,d$ holds: $$ (a^2+b^2-1)(a+b)=(b^2+c^2-1)(b+c)=(c^2+d^2-1)(c+d).$$ Prove that $ a+b+c+d=0.$

Let $n$ be a positive integer. Find the number of permutations $a_1$, $a_2$, $\dots a_n$ of the sequence $1$, $2$, $\dots$ , $n$ satisfying $$a_1 \le 2a_2\le 3a_3 \le \dots \le na_n$$. Proposed by United Kingdom

How many primes $p$ are there such that $2p^4-7p^2+1$ is equal to square of an integer? $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ \text{Infinitely many} \qquad\textbf{(E)}\ \text{None of the preceding} $

Find the number of integers $n$ with $1 \le n \le 2017$ so that $(n-2)(n-0)(n-1)(n-7)$ is an integer multiple of $1001$.

Someone wrote six letters to six people and addressed six envelopes to them. How many ways can the letters be put into the envelopes so that none of the letters end up in the correct envelope?

The equation $2^{2x}-8\cdot 2^x+12=0$ is satisfied by: $\textbf{(A) }\log3\qquad \textbf{(B) }\tfrac12\log6\qquad \textbf{(C) }1+\log\tfrac34\qquad$ $\textbf{(D) }1+\tfrac{\log3}{\log2}\qquad \textbf{(E) }\text{none of these}$

How many four-digit positive integers $\overline{a_1a_2a_3a_4}$ have only nonzero digits and have the property that $|a_i-a_j| \neq 1$ for all $1 \leq i<j \leq 4?$ [i]Proposed by Kyle Lee[/i]