There are two very strict laws in the country of Militaria. (i) Anyone who is shorter than $80\%$ (or more) of his “neighbours” (i.e. men living at distance less then $r$ from him) is freed from the military service. (ii) Anyone who is taller than $80\%$ (or more) of his “neighbours” (i.e. men living at distance less then $R$ from him) is allowed to serve in the police. A nice thing is that each man $X$ may choose his own (possibly different) positive numbers $r = r(X)$ and $R = R(X)$. Can it happen that $90\%$ (or more) of the men in Militaria are free from the army and, at the same time, $90\%$ (or more) of the men in Militaria are allowed to serve in the police? (The places of living of the men are fixed points in the plane.) (N Konstantinov)

Two parallel lines $r,s$ and two points $P \in r$ and $Q \in s$ are given in a plane. Consider all pairs of circles $(C_P, C_Q)$ in that plane such that $C_P$ touches $r$ at $P$ and $C_Q$ touches $s$ at $Q$ and which touch each other externally at some point $T$. Find the locus of $T$.

Five points $A,B,C,P,Q$ are chosen so that $A,B,C$ aren't collinear. The following length conditions hold: $\frac{AP}{BP}=\frac{AQ}{BQ}=\frac{21}{20}$ and $\frac{BP}{CP}=\frac{BQ}{CQ}=\frac{20}{19}$. Prove that line $PQ$ goes through the circumcentre of $\triangle ABC$.

Let $M$ be a set of $2017$ positive integers. For any subset $A$ of $M$ we define $f(A) := \{x\in M\mid \text{ the number of the members of }A\,,\, x \text{ is multiple of, is odd }\}$. Find the minimal natural number $k$, satisfying the condition: for any $M$, we can color all the subsets of $M$ with $k$ colors, such that whenever $A\neq f(A)$, $A$ and $f(A)$ are colored with different colors.

The first $32$ perfect squares, $1$, $4$, $9$, $16$, $25$, $\ldots$, $961$, $1024$ are combined together into one large number by appending their digits in succession, forming the number $N = 1491625\ldots9611024$. How many digits does $N$ have? $\textbf{(A) }84\qquad\textbf{(B) }85\qquad\textbf{(C) }86\qquad\textbf{(D) }87\qquad\textbf{(E) }88$

Let $K > 0$ be an integer. An integer $k \in [0,K]$ is randomly chosen. A sequence of integers is defined starting on $k$ and ending on $0$, where each nonzero term $t$ is followed by $t$ minus the largest Lucas number not exceeding $t$. The probability that $4$, $5$, or $6$ is in this sequence approaches $\tfrac{a - b \sqrt c}{d}$ for arbitrarily large $K$, where $a$, $b$, $c$, $d$, are positive integers, $\gcd(a,b,d) = 1$, and $c$ is squarefree. Find $a + b + c + d$. [i](Lucas numbers are defined as the members of the infinite integer sequence $2$, $1$, $3$, $4$, $7$, $\ldots$ where each term is the sum of the two before it.)[/i] [i]Proposed by Evan Chang[/i]

$56$ lines are drawn on a plane such that no three of them are concurrent. If the lines intersect at exactly $594$ points, what is the maximum number of them that could have the same slope?

The incircle of an acute-angled triangle $ABC$ touches $AB, BC, CA$ at points $C_1, A_1, B_1$ respectively. Points $A_2, B_2$ are the midpoints of the segments $B_1C_1, A_1C_1$ respectively. Let $P$ be a common point of the incircle and the line $CO$, where $O$ is the circumcenter of triangle $ABC.$ Let also $A'$ and $B'$ be the second common points of $PA_2$ and $PB_2$ with the incircle. Prove that a common point of $AA'$ and $BB'$ lies on the altitude of the triangle dropped from the vertex $C.$

There are $360$ permutations of the letters in $MMATHS.$ When ordered alphabetically, starting from $AHMMST,$ $MMATHS$ is in the $n$th permutation. What is $n$?

Let $\alpha,\beta, \gamma$ be the angles of a triangle. Prove that $\cos\alpha, + \cos\beta + \cos\gamma \le \frac{3}{2} $ and find the cases of equality.

Three different circles of equal radii intersect in point $Q$. The circle $C$ touches all of them. Prove that $Q$ is the center of $C$.

The number of points common to the graphs of \[(x-y+2)(3x+y-4)=0\text{ and }(x+y-2)(2x-5y+7)=0\] is: $\textbf{(A) }2\qquad \textbf{(B) }4\qquad \textbf{(C) }6\qquad \textbf{(D) }16\qquad \textbf{(E) }\text{infinite}$

For positive real numbers $a, b, c$ such that $a+b+c=3$, show that: \[\sqrt{a} + \sqrt{b} + \sqrt{c} \ge ab+bc+ca.\]

Let $I$ be the incenter of a triangle $ABC$. The points $X, Y$ lie on the prolongations of the lines $IB, IC$ after $I$ so that $\angle IAX=\angle IBA$ and $\angle IAY=\angle ICA$. Show that the line through the midpoints of $IA$ and $XY$ passes through the circumcenter of $ABC$.

A triple $(a,b,c)$ of positive integers is called [i]quasi-Pythagorean[/i] if there exists a triangle with lengths of the sides $a,b,c$ and the angle opposite to the side $c$ equal to $120^{\circ}$. Prove that if $(a,b,c)$ is a quasi-Pythagorean triple then $c$ has a prime divisor bigger than $5$.

Let $ABC$ be an acute-angled triangle in which $\angle ABC$ is the largest angle. Let $O$ be its circumcentre. The perpendicular bisectors of $BC$ and $AB$ meet $AC$ at $X$ and $Y$ respectively. The internal angle bisectors of $\angle AXB$ and $\angle BYC$ meet $AB$ and $BC$ at $D$ and $E$ respectively. Prove that $BO$ is perpendicular to $AC$ if $DE$ is parallel to $AC$.

Let $ABC$ be an acute triangle. Circle $k$ with diameter $AB$ intersects $AC$ and $BC$ again at $M$ and $N$ respectively. The tangents to $k$ at $M$ and $N$ meet at point $P$. Given that $CP=MN$, determine $\angle ACB$.

In the expansion of $\left(a-\dfrac{1}{\sqrt{a}}\right)^7$ the coefficient of $a^{-\dfrac{1}{2}}$ is: $\textbf{(A)}\ -7 \qquad \textbf{(B)}\ 7 \qquad \textbf{(C)}\ -21 \qquad \textbf{(D)}\ 21 \qquad \textbf{(E)}\ 35$

Find all $(m,n) \in \mathbb{Z}^2$ that we can color each unit square of $m \times n$ with the colors black and white that for each unit square number of unit squares that have the same color with it and have at least one common vertex (including itself) is even.

Let $\pi$ be a randomly chosen permutation of the numbers from $1$ through $2012$. Find the probability that$$ \pi (\pi(2012)) = 2012.$$

Let $n\geq 2$ be a positive integer. On an $n\times n$ board, $n$ rooks are placed in such a manner that no two attack each other. All rooks move at the same time and are only allowed to move in a square adjacent to the one in which they are located. Determine all the values ​​of $n$ for which there is a placement of the rooks so that, after a move, the rooks still do not attack each other. [i]Note: Two squares are adjacent if they share a common side.[/i]

In triangle $\triangle ABC$ points $M,N$ lie on $BC$ st : $\angle BAM= \angle MAN= \angle NAC$ . Points $P,Q$ are on the angle bisector of $BAC$, on the same side of $BC$ as A , st : $$\frac{1}{3} \angle BAC = \frac{1}{2} \angle BPC = \angle BQC$$ Let $E = AM \cap CQ$ and $F = AN \cap BQ$ . Prove that the common tangents to $(EPF), (EQF)$ and the circumcircle of $\triangle ABC$ , are concurrent.

Let be $n$ positive integer than calculate: $1\cdot 1!+2\cdot2!+...+n\cdot n!$

The points $P$ and $Q$ lie on the diagonals $AC$ and $BD$, respectively, of a quadrilateral $ABCD$ such that $\frac{AP}{AC} + \frac{BQ}{BD} =1$. The line $PQ$ meets the sides $AD$ and $BC$ at points $M$ and $N$. Prove that the circumcircles of the triangles $AMP$, $BNQ$, $DMQ$, and $CNP$ are concurrent.

The incircle of a triangle $ABC$ touches the sides $BC$, $CA$, $AB$ at the points $D$, $E$, $F$ respectively. Let the line $AD$ intersect this incircle of triangle $ABC$ at a point $X$ (apart from $D$). Assume that this point $X$ is the midpoint of the segment $AD$, this means, $AX = XD$. Let the line $BX$ meet the incircle of triangle $ABC$ at a point $Y$ (apart from $X$), and let the line $CX$ meet the incircle of triangle $ABC$ at a point $Z$ (apart from $X$). Show that $EY = FZ$.