A [i]windmill[/i] is a closed line segment of unit length with a distinguished endpoint, the [i]pivot[/i]. Let $S$ be a finite set of $n$ points such that the distance between any two points of $S$ is greater than $c$. A configuration of $n$ windmills is [i]admissible[/i] if no two windmills intersect and each point of $S$ is used exactly once as a pivot. An admissible configuration of windmills is initially given to Geoff in the plane. In one operation Geoff can rotate any windmill around its pivot, either clockwise or counterclockwise and by any amount, as long as no two windmills intersect during the process. Show that Geoff can reach any other admissible configuration in finitely many operations, where (i) $c = \sqrt 3$, (ii) $c = \sqrt 2$. [i]Proposed by Michael Ren[/i]

Given a positive integer $n$, show that the set $\{1,2,...,n\}$ can be partitioned into $m$ sets, each with the same sum, if and only if m is a divisor of $\frac{n(n + 1)}{2}$ which does not exceed $\frac{n + 1}{2}$.

What is the ratio of the area of the shaded square to the area of the large square? (The figure is drawn to scale) [asy] draw((0,0)--(0,4)--(4,4)--(4,0)--cycle); draw((0,0)--(4,4)); draw((0,4)--(3,1)--(3,3)); draw((1,1)--(2,0)--(4,2)); fill((1,1)--(2,0)--(3,1)--(2,2)--cycle,black);[/asy] $ \text{(A)}\ \frac{1}{6}\qquad\text{(B)}\ \frac{1}{7}\qquad\text{(C)}\ \frac{1}{8}\qquad\text{(D)}\ \frac{1}{12}\qquad\text{(E)}\ \frac{1}{16} $

In $\triangle ABC$, $\angle C = 90^\circ, AC = 6$ and $BC = 8$. Points $D$ and $E$ are on $\overline{AB}$ and $\overline{BC}$, respectively, and $\angle BED = 90^\circ$. If $DE = 4$, then $BD =$ [asy] size(100); pathpen = linewidth(0.7); pointpen = black+linewidth(3); pair A = (0,0), C = (6,0), B = (6,8), D = (2*A+B)/3, E = (2*C+B)/3; D(D("A",A,SW)--D("B",B,NW)--D("C",C,SE)--cycle); D(D("D",D,NW)--D("E",E,plain.E)); D(rightanglemark(D,E,B,16)); D(rightanglemark(A,C,B,16));[/asy] $\mathbf{(A)}\;5\qquad \mathbf{(B)}\;\frac{16}{3}\qquad \mathbf{(C)}\; \frac{20}{3}\qquad \mathbf{(D)}\; \frac{15}{2}\qquad \mathbf{(E)}\; 8$

Given an expression $x^2 + ax + b$ where $a,b$ are integer coefficients. At any step, one can change the expression by adding either $1$ or $-1$ to only one of the two coefficients $a, b$. a) Suppose that the initial expression has $a =-7$ and $b = 19$. Show your modification steps to obtain a new expression that has zero value at some integer value of $x$. b) Starting from the initial expression as above, one gets the expression $x^2 - 17x + 9$ after $m$ modification steps. Prove that at a certain step $k$ with $k < m$, the obtained expression has zero value at some integer value of $x$.

Let $p(x)$ be a polynomial with integer coefficients such that both equations $p(x)=1$ and $p(x)=3$ have integer solutions. Can the equation $p(x)=2$ have two different integer solutions?

On a blackboard there are written the numbers $ 11 $ and $ 13. $ One [i]step[/i] means to sum two written numbers and write it. Show that: [b]a)[/b] after any number of steps, the number $ 86 $ will not be written. [b]b)[/b] after some number of steps, the number $ 2015 $ may be written.

Let $k$ be a positive integer and $a_1, a_2,... , a_k$ be nonnegative real numbers. Initially, there is a sequence of $n \geq k$ zeros written on a blackboard. At each step, Nicole chooses $k$ consecutive numbers written on the blackboard and increases the first number by $a_1$, the second one by $a_2$, and so on, until she increases the $k$-th one by $a_k$. After a positive number of steps, Nicole managed to make all the numbers on the blackboard equal. Prove that all the nonzero numbers among $a_1, a_2, . . . , a_k$ are equal.

A "[i]2-line[/i]" is the area between two parallel lines. Length of "2-line" is distance of two parallel lines. We have covered unit circle with some "2-lines". Prove sum of lengths of "2-lines" is at least 2.

Let $S_n=1-2+3-4+\cdots +(-1)^{n-1}n,\ n=1, 2, \cdots$. Then $S_{17}+S_{33}+S_{50}$ equals: $\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ -1 \qquad\textbf{(E)}\ -2$

Let $ABC$ be an acute angled triangle with $AB<AC$. Let $M$ be the midpoint of side $BC$. On side $AB$, consider a point $D$ such that, if segment $CD$ intersects median $AM$ at point $E$, then $AD=DE$. Prove that $AB=CE$.

Do there exist 2018 positive irreducible fractions, each with a different denominator, so that the denominator of the difference of any two (after reducing the fraction) is less than the denominator of any of the initial 2018 fractions? (6 points) Maxim Didin

Prove that the polynomial $x^n+4$ can be expressed as a product of two polynomials (each with degree less than $n$) with integer coefficients, if and only if $n$ is divisible by $4$.

Nick is a runner, and his goal is to complete four laps around a circuit at an average speed of 10 mph. If he completes the first three laps at a constant speed of only 9 mph, what speed does he need to maintain in miles per hour on the fourth lap to achieve his goal?

Let $ABCD$ be a convex quadrilateral satisfying $\angle ADB + \angle ACB = \angle CAB + \angle DBA = 30^o$, $AD = BC$. Prove that there exists a right-angled triangle with side lengths $AC$, $BD$, $CD$.

Let $ABCD$ be a rectangle with sides $AB,BC,CD$ and $DA$. Let $K,L$ be the midpoints of the sides $BC,DA$ respectivily. The perpendicular from $B$ to $AK$ hits $CL$ at $M$. Find $$\frac{[ABKM]}{[ABCL]}$$

Solve in $ M_2(\mathbb{C})$ the equation $ X^2\equal{}\left( \begin{array}{cc} 1 & 2 \\ 3 & 6 \end{array} \right)$

If $a,b,c,d$ are positive reals such that $a^2+b^2=c^2+d^2$ and $a^2+d^2-ad=b^2+c^2+bc$, find the value of $\frac{ab+cd}{ad+bc}$

The value of $$\left(1-\frac{1}{2^2-1}\right)\left(1-\frac{1}{2^3-1}\right)\left(1-\frac{1}{2^4-1}\right)\dots\left(1-\frac{1}{2^{29}-1}\right)$$ can be written as $\tfrac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $2m - n.$

7. There are 100 chess bishops on white squares of a $100 \times 100$ chess board. Some of them are white and some of them are black. They can move in any order and capture the bishops of opposing color. What number of moves is sufficient for sure to retain only one bishop on the chess board?

Let acute scalene $\Delta ABC$ have circumcircle $\omega$. Let $M$ be the midpoint of $BC$, define $X$ as the other intersection of $AM$ with $\omega$. Let $E,F$ be the feet of altitudes from $B,C$ to $AC, AB$ respectively. Let $Q$ be the second intersection of the circumcircle of $\Delta AEF$ and $\omega$. Let $Y\neq X$ be a point such that $MX=MY$ and $QMXY$ is cyclic. Finally, let $S$ be a point on $BC$ such that $\angle BAS=\angle MAC.$ Prove that the quadrilaterals $BFYS$ and $CEYS$ are cyclic. Proposed by Kanav Talwar and Malay Mahajan

Let $BL, AD$ be the bisector and the altitude correspondingly of an acute triangle ABC. They intersect at point $T$. It turned out that the altitude $LK$ of $\triangle ALB$ is divided in half by the line $AD$. Prove that $KT \perp BL$. [i]Proposed by Mariia Rozhkova[/i]

There are $15$ (not necessarily distinct) integers chosen uniformly at random from the range from $0$ to $999$, inclusive. Yang then computes the sum of their units digits, while Michael computes the last three digits of their sum. The probability of them getting the same result is $\frac mn$ for relatively prime positive integers $m,n$. Find $100m+n$ [i]Proposed by Yannick Yao[/i]

Lazy Linus wants to minimize his amount of laundry over the course of a week (seven days), so he decides to wear only three different T-shirts and three different pairs of pants for the week. However, he doesn't want to look dirty or boring, so he decides to wear each piece of clothing for either two or three (possibly nonconsecutive) days total, and he cannot wear the same outfit (which consists of one T-shirt and one pair of pants) on two different (not necessarily consecutive) days. How many ways can he choose the outfits for these seven days? [i]Proposed by Yannick Yao[/i]

Prove that the set $S=\{\lfloor n\pi\rfloor \mid n=0,1,2,3,\ldots\}$ contains arithmetic progressions of any finite length, but no infinite arithmetic progressions. [i]Vasile Pop[/i]