Imagine the unit square in the plane to be a [i]carrom board[/i]. Assume the [i]striker[/i] is just a point, moving with no friction (so it goes forever), and that when it hits an edge, the angle of reflection is equal to the angle of incidence, as in real life. If the striker ever hits a corner it falls into the pocket and disappears. The trajectory of the striker is completely determined by its starting point $(x,y)$ and its initial velocity $\overrightarrow{(p,q)}$. If the striker eventually returns to its initial state (i.e. initial position and initial velocity), we define its [i]bounce number[/i] to be the number of edges it hits before returning to its initial state for the $1^{\text{st}}$ time. For example, the trajectory with initial state $[(.5,.5);\overrightarrow{(1,0)}]$ has bounce number $2$ and it returns to its initial state for the $1^{\text{st}}$ time in $2$ time units. And the trajectory with initial state $[(.25,.75);\overrightarrow{(1,1)}]$ has bounce number $4$. $\textbf{(a)}$ Suppose the striker has initial state $[(.5,.5);\overrightarrow{(p,q)}]$. If $p>q\geqslant 0$ then what is its velocity after it hits an edge for the $1^{\text{st}}$ time ? What if $q>p\geqslant 0$ ? $\textbf{(b)}$ Draw a trajectory with bounce number $5$ or justify why it is impossible. $\textbf{(c)}$ Consider the trajectory with initial state $[(x,y);\overrightarrow{(p,0)}]$ where $p$ is a positive integer. In how much time will the striker $1^{\text{st}}$ return to its initial state ? $\textbf{(d)}$ What is the bounce number for the initial state $[(x,y);\overrightarrow{(p,q)}]$ where $p,q$ are relatively prime positive integers, assuming the striker never hits a corner ?

The company has $100$ people. For any $k$, we can find a group of $k$ people such that there are two (different from them) strangers, each of them knows all of these $k$ people. At what maximum $k$ is this possible?

Let $n>1$ be a natural number. Let $U=\{1,2,...,n\}$, and define $A\Delta B$ to be the set of all those elements of $U$ which belong to exactly one of $A$ and $B$. Show that $|\mathcal{F}|\le 2^{n-1}$, where $\mathcal{F}$ is a collection of subsets of $U$ such that for any two distinct elements of $A,B$ of $\mathcal{F}$ we have $|A\Delta B|\ge 2$. Also find all such collections $\mathcal{F}$ for which the maximum is attained.

Prove the following assertion: The four altitudes of a tetrahedron $ABCD$ intersect in a point if and only if \[AB^2 + CD^2 = BC^2 + AD^2 = CA^2 + BD^2.\]

Consider a convex polygon $\mathcal{P}$ in space with perimeter $20$ and area $30$. What is the volume of the locus of points that are at most $1$ unit away from some point in the interior of $\mathcal{P}$?

Positive numbers $a$ and $b$ satisfy $a^{1999}+b^{2000}{\ge}a^{2000}+b^{2001}$.Prove that $a^{2000}+b^{2000}{\leq}2$. _____________________________________ Azerbaijan Land of the Fire :lol:

7) A train, originally of mass $M$, is traveling on a frictionless straight horizontal track with constant speed $v$. Snow starts to fall vertically and sticks to the train at a rate of $\rho$, where $\rho$ has units of kilograms per second. The train's engine keeps the train moving at constant speed $v$ as snow accumulates on the train. What is the rate at which the kinetic energy of the train and snow increases? A) $0$ B) $Mgv$ C) $\frac{1}{2}Mv^2$ D) $\frac{1}{2}pv^2$ E) $\rho v^2$

Let $ABC$ be an acute angled triangle with incenter $I$. Line perpendicular to $BI$ at $I$ meets $BA$ and $BC$ at points $P$ and $Q$ respectively. Let $D, E$ be the incenters of $\triangle BIA$ and $\triangle BIC$ respectively. Suppose $D,P,Q,E$ lie on a circle. Prove that $AB=BC$.

We throw a dice three times and the numbers are $x,y,z$ find out the possibility of $\binom{7}{x}<\binom{7}{y}<\binom{7}{z}$

Let $ABC$ be an acute triangle with $AB<BC$. Let $I$ be the incenter of $ABC$, and let $\omega$ be the circumcircle of $ABC$. The incircle of $ABC$ is tangent to the side $BC$ at $K$. The line $AK$ meets $\omega$ again at $T$. Let $M$ be the midpoint of the side $BC$, and let $N$ be the midpoint of the arc $BAC$ of $\omega$. The segment $NT$ intersects the circumcircle of $BIC$ at $P$. Prove that $PM\parallel AK$.

Find all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ such that \[f(x+y) + f(x)f(y) = f(xy) + (y+1)f(x) + (x+1)f(y)\] for all $x,y \in \mathbb{R}$.

Let $I$ be the incentre of triangle $ABC$. A circle containing the points $B$ and $C$ meets the segments $BI$ and $CI$ at points $P$ and $Q$ respectively. It is known that $BP\cdot CQ=PI\cdot QI$. Prove that the circumcircle of the triangle $PQI$ is tangent to the circumcircle of $ABC$. [i]Proposed by S. Berlov[/i]

An equilateral triangle $\Delta$ of side length $L>0$ is given. Suppose that $n$ equilateral triangles with side length 1 and with non-overlapping interiors are drawn inside $\Delta$, such that each unit equilateral triangle has sides parallel to $\Delta$, but with opposite orientation. (An example with $n=2$ is drawn below.) [asy] draw((0,0)--(1,0)--(1/2,sqrt(3)/2)--cycle,linewidth(0.5)); filldraw((0.45,0.55)--(0.65,0.55)--(0.55,0.55-sqrt(3)/2*0.2)--cycle,gray,linewidth(0.5)); filldraw((0.54,0.3)--(0.34,0.3)--(0.44,0.3-sqrt(3)/2*0.2)--cycle,gray,linewidth(0.5)); [/asy] Prove that \[n \leq \frac{2}{3} L^{2}.\]

An object in the plane moves from one lattice point to another. At each step, the object may move one unit to the right, one unit to the left, one unit up, or one unit down. If the object starts at the origin and takes a ten-step path, how many different points could be the final point? $ \textbf{(A) } 120 \qquad \textbf{(B) } 121 \qquad \textbf{(C) } 221 \qquad \textbf{(D) } 230 \qquad \textbf{(E) } 231$

Let $\triangle ABC$ have $AB=6$, $BC=7$, and $CA=8$, and denote by $\omega$ its circumcircle. Let $N$ be a point on $\omega$ such that $AN$ is a diameter of $\omega$. Furthermore, let the tangent to $\omega$ at $A$ intersect $BC$ at $T$, and let the second intersection point of $NT$ with $\omega$ be $X$. The length of $\overline{AX}$ can be written in the form $\tfrac m{\sqrt n}$ for positive integers $m$ and $n$, where $n$ is not divisible by the square of any prime. Find $100m+n$. [i]Proposed by David Altizio[/i]

Two tangents to a circle are drawn from a point $A$. The points of contact $B$ and $C$ divide the circle into arcs with lengths in the ratio $2:3$. What is the degree measure of $\angle BAC$? $ \textbf{(A)}\ 24 \qquad \textbf{(B)}\ 30 \qquad \textbf{(C)}\ 36 \qquad \textbf{(D)}\ 48 \qquad \textbf{(E)}\ 60 $