[u]Round 1[/u] [b]p1.[/b] Three snails – Alice, Bobby, and Cindy – were racing down a road. Whenever one snail passed another, it waved at the snail it passed. During the race, Alice waved $3$ times and was waved at twice. Bobby waved $4$ times and was waved at $3$ times. Cindy waved $5$ times. How many times was she waved at? [b]p2.[/b] Sherlock and Mycroft are playing Battleship on a $4\times 4$ grid. Mycroft hides a single $3\times 1$ cruiser somewhere on the board. Sherlock can pick squares on the grid and fire upon them. What is the smallest number of shots Sherlock has to fire to guarantee at least one hit on the cruiser? [b]p3.[/b] Thirty girls – $13$ of them in red dresses and $17$ in blue dresses – were dancing in a circle, hand-in-hand. Afterwards, each girl was asked if the girl to her right was in a blue dress. Only the girls who had both neighbors in red dresses or both in blue dresses told the truth. How many girls could have answered “Yes”? [b]p4.[/b] Herman and Alex play a game on a $5\times 5$ board. On his turn, a player can claim any open square as his territory. Once all the squares are claimed, the winner is the player whose territory has the longer border. Herman goes first. If both play their best, who will win, or will the game end in a draw? [img]https://cdn.artofproblemsolving.com/attachments/5/7/113d54f2217a39bac622899d3d3eb51ec34f1f.png[/img] [b]p5.[/b] Is it possible to find $2014$ distinct positive integers whose sum is divisible by each of them? [u]Round 2[/u] [b]p6.[/b] Hermione and Ron play a game that starts with 129 hats arranged in a circle. They take turns magically transforming the hats into animals. On each turn, a player picks a hat and chooses whether to change it into a badger or into a raven. A player loses if after his or her turn there are two animals of the same species right next to each other. Hermione goes first. Who loses? [b]p7.[/b] Three warring states control the corner provinces of the island whose map is shown below. [img]https://cdn.artofproblemsolving.com/attachments/e/a/4e2f436be1dcd3f899aa34145356f8c66cda82.png[/img] As a result of war, each of the remaining $18$ provinces was occupied by one of the states. None of the states was able to occupy any province on the coast opposite their corner. The states would like to sign a peace treaty. To do this, they each must send ambassadors to a place where three provinces, one controlled by each state, come together. Prove that they can always find such a place to meet. For example, if the provinces are occupied as shown here, the squares mark possible meeting spots. [img]https://cdn.artofproblemsolving.com/attachments/e/b/81de9187951822120fc26024c1c1fbe2138737.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A sextic funtion $ y \equal{} ax^6 \plus{} bx^5 \plus{} cx^4 \plus{} dx^3 \plus{} ex^2 \plus{} fx \plus{} g\ (a\neq 0)$ touches the line $ y \equal{} px \plus{} q$ at $ x \equal{} \alpha ,\ \beta ,\ \gamma \ (\alpha < \beta < \gamma ).$ Find the area of the region bounded by these graphs in terms of $ a,\ \alpha ,\ \beta ,\gamma .$ created by kunny

Let $ABC$ be an acute triangle with circumcircle $\Gamma$. Let $A_1,B_1$ and $C_1$ be respectively the midpoints of the arcs $BAC,CBA$ and $ACB$ of $\Gamma$. Show that the inradius of triangle $A_1B_1C_1$ is not less than the inradius of triangle $ABC$.

In a triangle $ABC$, the median from $B$ to $CA$ is perpendicular to the median from $C$ to $AB$. If the median from $A$ to $BC$ is $30$, determine $\frac{BC^2 + CA^2 + AB^2}{100}$.

On the sides $(AB)$, $(BC)$, $(CD)$ and $(DA)$ of the regular tetrahedron $ABCD$, one considers the points $M$, $N$, $P$, $Q$, respectively Prove that $$MN \cdot NP \cdot PQ \cdot QM \ge AM \cdot BN \cdot CP \cdot DQ.$$

Let $A_{1}A_{2} \dots A_{3n}$ be a closed broken line consisting of $3n$ lines segments in the Euclidean plane. Suppose that no three of its vertices are collinear, and for each index $i=1,2,\dots,3n$, the triangle $A_{i}A_{i+1}A_{i+2}$ has counterclockwise orientation and $\angle A_{i}A_{i+1}A_{i+2} = 60º$, using the notation $A_{3n+1} = A_{1}$ and $A_{3n+2} = A_{2}$. Prove that the number of self-intersections of the broken line is at most $\frac{3}{2}n^{2} - 2n + 1$

Triangle $ ABC$ is scalene with angle $ A$ having a measure greater than 90 degrees. Determine the set of points $ D$ that lie on the extended line $ BC$, for which \[ |AD|\equal{}\sqrt{|BD| \cdot |CD|}\] where $ |BD|$ refers to the (positive) distance between $ B$ and $ D$.

Three circles $k_1,k_2$ and $k_3$ go through the points $A$ and $B$. A secant through $A$ intersects the circles $k_1,k_2$ and $k_3$ again in the points $C,D$ resp. $E$. Prove that the ratio $|CD|:|DE|$ does not depend on the choice of the secant.

Square $ ABCD$ has sides of length $ 4$, and $ M$ is the midpoint of $ \overline{CD}$. A circle with radius $ 2$ and center $ M$ intersects a circle with raidus $ 4$ and center $ A$ at points $ P$ and $ D$. What is the distance from $ P$ to $ \overline{AD}$? [asy]unitsize(8mm); defaultpen(linewidth(.8pt)); dotfactor=4; draw(Circle((2,0),2)); draw(Circle((0,4),4)); clip(scale(4)*unitsquare); draw(scale(4)*unitsquare); filldraw(Circle((2,0),0.07)); filldraw(Circle((3.2,1.6),0.07)); label("$A$",(0,4),NW); label("$B$",(4,4),NE); label("$C$",(4,0),SE); label("$D$",(0,0),SW); label("$M$",(2,0),S); label("$P$",(3.2,1.6),N);[/asy]$ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ \frac {16}{5} \qquad \textbf{(C)}\ \frac {13}{4} \qquad \textbf{(D)}\ 2\sqrt {3} \qquad \textbf{(E)}\ \frac {7}{2}$

Let $T_m$ be a number of non-congruent triangles which perimeter is $m$ and all its sides are positive integers. Prove that: $a)$ $T_{1999} > T_{2000}$ $b)$ $T_{4n+1}=T_{4n-2}+n$, $(n \in \mathbb{N})$

Find maximal numbers of planes, such there are $6$ points and 1) $4$ or more points lies on every plane. 2) No one line passes through $4$ points.

[asy] fill(circle((4,0),4),grey); fill((0,0)--(8,0)--(8,-4)--(0,-4)--cycle,white); fill(circle((7,0),1),white); fill(circle((3,0),3),white); draw((0,0)--(8,0),black+linewidth(1)); draw((6,0)--(6,sqrt(12)),black+linewidth(1)); MP("A", (0,0), W); MP("B", (8,0), E); MP("C", (6,0), S); MP("D",(6,sqrt(12)), N); [/asy] In this diagram semi-circles are constructed on diameters $\overline{AB}$, $\overline{AC}$, and $\overline{CB}$, so that they are mutually tangent. If $\overline{CD} \bot \overline{AB}$, then the ratio of the shaded area to the area of a circle with $\overline{CD}$ as radius is: $\textbf{(A)}\ 1:2\qquad \textbf{(B)}\ 1:3\qquad \textbf{(C)}\ \sqrt{3}:7\qquad \textbf{(D)}\ 1:4\qquad \textbf{(E)}\ \sqrt{2}:6$

Let $AA _1$ and $CC_1$ be altitudes of an acute triangle $ABC$. Let $I$ and $J$ be the incenters of the triangles $AA_1C$ and $AC_1C$ respectively. The $C_1J$ and $A_1 I$ lines cut into $T$. Prove that lines $AT$ and $TC$ are perpendicular.

Let the incircle of triangle $ABC$ touch sides $BC, CA$ and $AB$ at the points $D, E$ and $F$ respectively and let $I$ be the center of that circle. Furthermore, let $P$ be the foot of the perpendicular from point $I$ to line $AD$ and let $M$ be the midpoint of $DE$. If $N$ is the intersection point of $PM$ and $AC$, prove that $DN \parallel EF$.

Let $ ABC$ be a triangle, line $ l$ cuts its sides $ BC,CA,AB$ at $ D,E,F$, respectively. Denote by $ O_{1},O_{2},O_{3}$ the circumcenters of triangle $ AEF,BFD,CDE$, respectively. Prove that the orthocenter of triangle $ O_{1}O_{2}O_{3}$ lies on line $ l$.

Let $ S$ be any point on the circumscribed circle of $ PQR.$ Then the feet of the perpendiculars from S to the three sides of the triangle lie on the same straight line. Denote this line by $ l(S, PQR).$ Suppose that the hexagon $ ABCDEF$ is inscribed in a circle. Show that the four lines $ l(A,BDF),$ $ l(B,ACE),$ $ l(D,ABF),$ and $ l(E,ABC)$ intersect at one point if and only if $ CDEF$ is a rectangle.

Given triangle $ ABC $ with points $ M $ and $ N $ are in the sides $ AB $ and $ AC $ respectively. If $ \dfrac{BM}{MA} +\dfrac{CN}{NA} = 1 $ , then prove that the centroid of $ ABC $ lies on $ MN $ .

A circle $\omega$ cuts the sides $BC,CA,AB$ of the triangle $ABC$ at $A_1$ and $A_2$; $B_1$ and $B_2$; $C_1$ and $C_2$, respectively. Let $P$ be the center of $\omega$. $A'$ is the circumcenter of the triangle $A_1A_2P$, $B'$ is the circumcenter of the triangle $B_1B_2P$, $C'$ is the circumcenter of the triangle $C_1C_2P$. Prove that $AA', BB'$ and $CC'$ concur.

Triangle $ ABC$ has $ AB\equal{}13$ and $ AC\equal{}15$, and the altitude to $ \overline{BC}$ has length $ 12$. What is the sum of the two possible values of $ BC$? $ \textbf{(A)}\ 15\qquad \textbf{(B)}\ 16\qquad \textbf{(C)}\ 17\qquad \textbf{(D)}\ 18\qquad \textbf{(E)}\ 19$

For real numbers $x,$ let $T_x$ be the triangle with vertices $(5, 5^3),$ $(8, 8^3),$ and $(x, x^3)$ in $\mathbb{R}^2.$ Over all $x$ in the interval $[5, 8],$ the area of the triangle $T_x$ is maximized at $x = \sqrt{n},$ for some positive integer $n.$ Compute $n.$

A $(3n + 1) \times (3n + 1)$ table $(n \in \mathbb{N})$ is given. Prove that deleting any one of its squares yields a shape cuttable into pieces of the following form and its rotations: ''L" shape formed by cutting one square from a $2 \times 2$ squares.

The perpendicular bisectors of the sides AB and BC of a triangle ABC meet the lines BC and AB at the points X and Z, respectively. The angle bisectors of the angles XAC and ZCA intersect at a point B'. Similarly, define two points C' and A'. Prove that the points A', B', C' lie on one line through the incenter I of triangle ABC. [i]Extension:[/i] Prove that the points A', B', C' lie on the line OI, where O is the circumcenter and I is the incenter of triangle ABC. Darij

Equilateral triangle $ABC$ has area $1$. $A'$, $B'$, and $C'$ are the midpoints of $BC$, $CA$, and $AB$, respectively. $A''$, $B''$, $C''$ are the midpoints of $B'C'$, $C'A'$, and $A'B'$, respectively. The area of trapezoid $BB''C''C$ can be written as $\frac{m}{n}$ for relative prime positive integers $m$ and $n$. Find $m+n$.

Two circles intersect at points $A$ and $B$. The line $\ell$ through A intersects the circles at $C$ and $D$, respectively. Let $M, N$ be the midpoints of arc $BC$ and arc $BD$. which does not contain $A$, and suppose that $K$ is the midpoint of the segment $CD$ . Prove that $\angle MKN=90^o$.

We are given two mutually tangent circles in the plane, with radii $r_1, r_2$. A line intersects these circles in four points, determining three segments of equal length. Find this length as a function of $r_1$ and $r_2$ and the condition for the solvability of the problem.