Let $n \geq 2$ be a natural number, and let $\left( a_{1};\;a_{2};\;...;\;a_{n}\right)$ be a permutation of $\left(1;\;2;\;...;\;n\right)$. For any integer $k$ with $1 \leq k \leq n$, we place $a_k$ raisins on the position $k$ of the real number axis. [The real number axis is the $x$-axis of a Cartesian coordinate system.] Now, we place three children A, B, C on the positions $x_A$, $x_B$, $x_C$, each of the numbers $x_A$, $x_B$, $x_C$ being an element of $\left\{1;\;2;\;...;\;n\right\}$. [It is not forbidden to place different children on the same place!] For any $k$, the $a_k$ raisins placed on the position $k$ are equally handed out to those children whose positions are next to $k$. [So, if there is only one child lying next to $k$, then he gets the raisin. If there are two children lying next to $k$ (either both on the same position or symmetric with respect to $k$), then each of them gets one half of the raisin. Etc..] After all raisins are distributed, a child is unhappy if he could have received more raisins than he actually has received if he had moved to another place (while the other children would rest on their places). For which $n$ does there exist a configuration $\left( a_{1};\;a_{2};\;...;\;a_{n}\right)$ and numbers $x_A$, $x_B$, $x_C$, such that all three children are happy?

A circle contains the points $(0, 11)$ and $(0, -11)$ on its circumference and contains all points $(x, y)$ with $x^2+y^2<1$ in its interior. Compute the largest possible radius of the circle.

Given six real numbers $a$, $b$, $c$, $x$, $y$, $z$ such that $0 < b-c < a < b+c$ and $ax + by + cz = 0$. What is the sign of the sum $ayz + bzx + cxy$ ?

Let $ S$ be the set of points in the Cartesian plane that satisfy \[ \Big|\big|{|x| \minus{} 2}\big| \minus{} 1\Big| \plus{} \Big|\big|{|y| \minus{} 2}\big| \minus{} 1\Big| \equal{} 1. \] If a model of $ S$ were built from wire of negligible thickness, then the total length of wire required would be $ a\sqrt {b},$ where $ a$ and $ b$ are positive integers and $ b$ is not divisible by the square of any prime number. Find $ a \plus{} b.$

Given triangle $ABC$. Point $M$ is the midpoint of side $BC$, and point $P$ is the projection of $B$ to the perpendicular bisector of segment $AC$. Line $PM$ meets $AB$ in point $Q$. Prove that triangle $QPB$ is isosceles.

given any 3 distinct points $X,Y,Z$on the integer coordinates of the x-axis,the following operation is allowed:A point say $X$ is reflected over another point say $Y$. Note that after each operation only one among three points is moved. we perform these operations till 2 out of the 3 points coincide. let $N=N(X,Y,Z)$ denote the minimum number of operations before we are forced to stop.(this could happen in different ways). show that there are at most $2^N$coordinates that point $X$ could end up if we are forced to stop after $N$operations

Point $O$ is the origin of a space. Points $A_1, A_2,\dots, A_n$ have nonnegative coordinates. Prove the following inequality: $$|\overrightarrow{OA_1}|+|\overrightarrow {OA_2}|+\dots+|\overrightarrow {OA_n}|\leq \sqrt{3}|\overrightarrow {OA_1}+\overrightarrow{OA_2}+\dots+\overrightarrow{OA_n}|$$ [I]Proposed by A. Khrabrov[/i]

A square $ABCD$ is divided into $(n - 1)^2$ congruent squares, with sides parallel to the sides of the given square. Consider the grid of all $n^2$ corners obtained in this manner. Determine all integers $n$ for which it is possible to construct a non-degenerate parabola with its axis parallel to one side of the square and that passes through exactly $n$ points of the grid.

Let $x, y, z$ be real numbers satisfying $x+y+z=0$ and $x^2+y^2+z^2=6$. Find the maximum value of \[ |(x-y)(y-z)(z-x) | \]

It is desired to construct a right triangle in the coordinate plane so that its legs are parallel to the $x$ and $y$ axes and so that the medians to the midpoints of the legs lie on the lines $y = 3x + 1$ and $y = mx + 2$. The number of different constants $m$ for which such a triangle exists is $ \textbf{(A)}\ 0\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ \text{more than 3} $

The taxicab distance between points $(x_1, y_1$) and $(x_2, y_2)$ is $|x_2 -x_1|+|y_2 -y_1|$. A regular octagon is positioned in the $xy$ plane so that one of its sides has endpoints $(0, 0)$ and $(1, 0)$. Let $S$ be the set of all points inside the octagon whose taxicab distance from some octagon vertex is at most $2/3$. The area of $S$ can be written as $m/n$ , where $m, n$ are positive integers and $gcd (m, n) = 1$. Find $100m + n$.

Let $M$, $N$ be the midpoints of diagonals $AC$, $BD$ of a right-angled trapezoid $ABCD$ ($\measuredangle A=\measuredangle D = 90^\circ$). The circumcircles of triangles $ABN$, $CDM$ meet the line $BC$ in the points $Q$, $R$. Prove that the distances from $Q$, $R$ to the midpoint of $MN$ are equal.

On a Cartesian plane 101 planes are drawn and all points of intersection are labeled. Is it possible, that for every line, 50 of the points have positive coordinates and 50 of the points have negative coordinates [I]Proposed by S. Ivanov[/i]

Find the largest value for $\frac{y}{x}$ for pairs of real numbers $(x,y)$ which satisfy \[(x-3)^2 + (y-3)^2 = 6.\] $\textbf{(A) }3 + 2 \sqrt 2\qquad \textbf{(B) } 2 + \sqrt 3\qquad \textbf{(C ) }3 \sqrt 3\qquad \textbf{(D) }6\qquad \textbf{(E) }6 + 2 \sqrt 3$

For a triangle $ ABC $ which $ \angle B \ne 90^{\circ} $ and $ AB \ne AC $, define $ P_{ABC} $ as follows ; Let $ I $ be the incenter of triangle $ABC$, and let $ D, E, F $ be the intersection points with the incircle and segments $ BC, CA, AB $. Two lines $ AB $ and $ DI $ meet at $ S $ and let $ T $ be the intersection point of line $ DE $ and the line which is perpendicular with $ DF $ at $ F $. The line $ ST $ intersects line $ EF $ at $ R$. Now define $ P_{ABC} $ be one of the intersection points of the incircle and the circle with diameter $ IR $, which is located in other side with $ A $ about $ IR $. Now think of an isosceles triangle $ XYZ $ such that $ XZ = YZ > XY $. Let $ W $ be the point on the side $ YZ $ such that $ WY < XY $ and Let $ K = P_{YXW} $ and $ L = P_{ZXW} $. Prove that $ 2 KL \le XY $.

Triangle $ ABC$ has $ \angle C \equal{} 60^{\circ}$ and $ BC \equal{} 4$. Point $ D$ is the midpoint of $ BC$. What is the largest possible value of $ \tan{\angle BAD}$? $ \textbf{(A)} \ \frac {\sqrt {3}}{6} \qquad \textbf{(B)} \ \frac {\sqrt {3}}{3} \qquad \textbf{(C)} \ \frac {\sqrt {3}}{2\sqrt {2}} \qquad \textbf{(D)} \ \frac {\sqrt {3}}{4\sqrt {2} \minus{} 3} \qquad \textbf{(E)}\ 1$

A rectangle, $HOMF$, has sides $HO=11$ and $OM=5$. A triangle $\Delta ABC$ has $H$ as orthocentre, $O$ as circumcentre, $M$ be the midpoint of $BC$, $F$ is the feet of altitude from $A$. What is the length of $BC$ ? [asy] unitsize(0.3 cm); pair F, H, M, O; F = (0,0); H = (0,5); O = (11,5); M = (11,0); draw(H--O--M--F--cycle); label("$F$", F, SW); label("$H$", H, NW); label("$M$", M, SE); label("$O$", O, NE); [/asy]

Triangle $ABC$ is isosceles ($AB=AC$). From $A$, we draw a line $\ell$ parallel to $BC$. $P,Q$ are on perpendicular bisectors of $AB,AC$ such that $PQ\perp BC$. $M,N$ are points on $\ell$ such that angles $\angle APM$ and $\angle AQN$ are $\frac\pi2$. Prove that \[\frac{1}{AM}+\frac1{AN}\leq\frac2{AB}\]

Two perpendicular lines pass through the point $F(1;1)$ of coordinate plane. One of them intersects hyperbola $y=\frac{1}{2x}$ at $A$ and $C$ ($C_x>A_x$), and the other one intersects the left part of hyperbola at $B$ and the right at $D$. Let $m=(C_x-A_x)(D_x-B_x)$ Find the area of non-convex quadraliteral $ABCD$ (in terms of $m$)

Prove that any convex pentagon whose vertices (no three of which are collinear) have integer coordinates must have area greater than or equal to $ \dfrac {5}{2} $.

A dartboard is the region B in the coordinate plane consisting of points $(x, y)$ such that $|x| + |y| \le 8$. A target T is the region where $(x^2 + y^2 - 25)^2 \le 49$. A dart is thrown at a random point in B. The probability that the dart lands in T can be expressed as $\frac{m}{n} \pi$, where $m$ and $n$ are relatively prime positive integers. What is $m + n$? $ \textbf{(A) }39 \qquad \textbf{(B) }71 \qquad \textbf{(C) }73 \qquad \textbf{(D) }75 \qquad \textbf{(E) }135 \qquad $

Let $X=\mathbb{Z}^3$ denote the set of all triples $(a,b,c)$ of integers. Define $f: X \to X$ by \[ f(a,b,c) = (a+b+c, ab+bc+ca, abc) . \] Find all triples $(a,b,c)$ such that \[ f(f(a,b,c)) = (a,b,c) . \]

There are given the integers $1 \le m < n$. Consider the set $M = \{ (x,y);x,y \in \mathbb{Z_{+}}, 1 \le x,y \le n \}$. Determine the least value $v(m,n)$ with the property that for every subset $P \subseteq M$ with $|P| = v(m,n)$ there exist $m+1$ elements $A_{i}= (x_{i},y_{i}) \in P, i = 1,2,...,m+1$, for which the $x_{i}$ are all distinct, and $y_{i}$ are also all distinct.

Let $ABCD$ be a convex quadrilateral.Let diagonals $AC$ and $BD$ intersect at $P$. Let $PE,PF,PG$ and $PH$ are altitudes from $P$ on the side $AB,BC,CD$ and $DA$ respectively. Show that $ABCD$ has a incircle if and only if $\frac{1}{PE}+\frac{1}{PG}=\frac{1}{PF}+\frac{1}{PH}.$

Two circles of radius $ 2$ are centered at $ (2,0)$ and at $ (0,2)$. What is the area of the intersection of the interiors of the two circles? $ \textbf{(A)}\ \pi \minus{} 2\qquad \textbf{(B)}\ \frac {\pi}{2}\qquad \textbf{(C)}\ \frac {\pi\sqrt {3}}{3}\qquad \textbf{(D)}\ 2(\pi \minus{} 2)\qquad \textbf{(E)}\ \pi$