Given are $4004$ distinct points, which lie in the interior of a convex polygon of area $1$. Prove that there exists a convex polygon of area $\frac{1}{2003}$, included in the given polygon, such that it does not contain any of the given points in its interior.

$(\text{a})$ Do there exist 14 consecutive positive integers each of which is divisible by one or more primes $p$ from the interval $2\le p \le 11$? $(\text{b})$ Do there exist 21 consecutive positive integers each of which is divisible by one or more primes $p$ from the interval $2\le p \le 13$?

Let there be an infinite sequence $ a_{k} $ with $ k\geq 1 $ defined by: $ a_{k+2} = a_{k} + 14 $ and $ a_{1} = 12 $ , $ a_{2} = 24 $. [b]a)[/b] Does $2012$ belong to the sequence? [b]b)[/b] Prove that the sequence doesn't contain perfect squares.

Is it possible to cut a square into five squares?

Given a circle $\Omega$ tangent to side $AB$ of angle $\angle BAC$ and lying outside this angle. We consider circles $w$ inscribed in angle $BAC$. The internal tangent of $\Omega$ and $w$, different from $AB$, touches $w$ at a point $K$. Let $L$ be the point of tangency of $w$ and $AC$. Prove that all such lines $KL$ pass through a fixed point without depending on the choice of circle $w$.

A teacher wants her $N$ students to know each other, so she creates various clubs of three people, so that each student can participate in several clubs. The clubs are formed in such a way that if $A$ and $B$ are two people, then there is a single club such that $A$ and $B$ are two of its three members. [b](1)[/b] Show that there is no way for the teacher to form the clubs if $N = 11$. [b](2)[/b] Show that the teacher can do it if $N = 9$.

Let $A$ be the sum of seven $7\text{’s}$. Let $B$ be the sum of seven $A\text{’s}$. What is $B$?

Let $p>7$ be a prime and let $A$ be subset of $\{0,1, \ldots, p-1\}$ with size at least $\frac{p-1}{2}$. Show that for each integer $r$, there exist $a, b, c, d \in A$, not necessarily distinct, such that $ab-cd \equiv r \pmod p$.

Let $n$ be a positive integer and let $p$ be a prime number. Prove that if $a$, $b$, $c$ are integers (not necessarily positive) satisfying the equations \[ a^n + pb = b^n + pc = c^n + pa\] then $a = b = c$. [i]Proposed by Angelo Di Pasquale, Australia[/i]

Let $B$ and $C$ be two fixed points on a circle centered at $O$ that are not diametrically opposed. Let $A$ be a variable point on the circle distinct from $B$ and $C$ and not belonging to the perpendicular bisector of $BC$. Let $H$ be the orthocenter of $\triangle ABC$, and $M$ and $N$ be the midpoints of the segments $BC$ and $AH$, respectively. The line $AM$ intersects the circle again at $D$, and finally, $NM$ and $OD$ intersect at $P$. Determine the locus of points $P$ as $A$ moves around the circle.

In the game of Colonel Blotto, you have 100 troops to distribute among 10 castles. Submit a 10-tuple $(x_1, x_2, \dots x_{10})$ of nonnegative integers such that $x_1 + x_2 + \dots + x_{10} = 100$, where each $x_i$ represent the number of troops you want to send to castle $i$. Your troop distribution will be matched up against each opponent's and you will win 10 points for each castle that you send more troops to (if you send the same number, you get 5 points, and if you send fewer, you get none). Your aim is to score the most points possible averaged over all opponents. For example, if team $A$ submits $(90,10,0,\dots,0)$, team B submits $(11,11,11,11,11,11,11,11,11,1)$, and team C submits $(10,10,10,\dots 10)$, then team A will win 10 points against team B and 15 points against team C, while team B wins 90 points against team C. Team A averages 12.5 points, team B averages 90 points, and team C averages 47.5 points. [i]2017 CCA Math Bonanza Lightning Round #5.4[/i]

For an arbitrary positive integer $m$, not divisible by $3$, consider the permutation $x \mapsto 3x \pmod{m}$ on the set $\{ 1,2,\dotsc ,m-1\}$. This permutation can be decomposed into disjointed cycles; for instance, for $m=10$ the cycles are $(1\mapsto 3\to 9,\mapsto 7,\mapsto 1)$, $(2\mapsto 6\mapsto 8\mapsto 4\mapsto 2)$ and $(5\mapsto 5)$. For which integers $m$ is the number of cycles odd?

Let $ABC$ be an acute triangle with circumcircle $\omega$. The altitudes $AD$, $BE$ and $CF$ of the triangle $ABC$ intersect at point $H$. A point $K$ is chosen on the line $EF$ such that $KH\parallel BC$. Prove that the reflection of $H$ in $KD$ lies on $\omega$.

Consider fractions $\frac{a}{b}$ where $a$ and $b$ are positive integers. (a) Prove that for every positive integer $n$, there exists such a fraction $\frac{a}{b}$ such that $\sqrt{n} \le \frac{a}{b} \le \sqrt{n+1}$ and $b \le \sqrt{n}+1$. (b) Show that there are infinitely many positive integers $n$ such that no such fraction $\frac{a}{b}$ satisfies $\sqrt{n} \le \frac{a}{b} \le \sqrt{n+1}$ and $b \le \sqrt{n}$.

For which natural numbers $n{}$ it is possible to mark several cells of an $n\times n$ board in such a way that there are an even number of marked cells in all rows and columns, and an odd number on all diagonals whose length is more than one cell? [i]Proposed by S. Berlov[/i]

Acute triangle $ABP$, where $AB > BP$, has altitudes $BH$, $PQ$, and $AS$. Let $C$ denote the intersection of lines $QS$ and $AP$, and let $L$ denote the intersection of lines $HS$ and $BC$. If $HS = SL$ and $HL$ is perpendicular to $BC$, find the value of $\frac{SL}{SC}$.

The straight line $l_{1}$ with equation $x-2y+10 = 0$ meets the circle with equation $x^2 + y^2 = 100$ at B in the first quadrant. A line through B, perpendicular to $l_{1}$ cuts the y-axis at P (0, t). Determine the value of $t$.

Fix an integer $n \ge 1$. On a $m\times n$ chess board, what is the minimum value of $m$ such that $n$ queens can be placed on the chessboard without any two attacking each other? (A queen can attack vertically, horizontally, or diagonally across any distance.)

Let $\Delta$ be a convex figure in a plane $\mathcal P$. Given a point $A\in\mathcal P$, to each pair $(M,N)$ of points in $\Delta$ we associate the point $m\in\mathcal P$ such that $\overrightarrow{Am}=\frac{\overrightarrow{MN}}2$ and denote by $\delta_A(\Delta)$ the set of all so obtained points $m$. (a) i. Prove that $\delta_A(\Delta)$ is centrally symmetric. ii. Under which conditions is $\delta_A(\Delta)=\Delta$? iii. Let $B,C$ be points in $\mathcal P$. Find a transformation which sends $\delta_B(\Delta)$ to $\delta_C(\Delta)$. (b) Determine $\delta_A(\Delta)$ if i. $\Delta$ is a set in the plane determined by two parallel lines. ii. $\Delta$ is bounded by a triangle. iii. $\Delta$ is a semi-disk. (c) Prove that in the cases $b.2$ and $b.3$ the lengths of the boundaries of $\Delta$ and $\delta_A(\Delta)$ are equal.

Prove that we can find an infinite set of positive integers of the from $2^n-3$ (where $n$ is a positive integer) every pair of which are relatively prime.

Let $ABC$ be a triangle, $I$ its incenter, and $\Gamma$ its circumcircle. Let $D$ be the second point of intersection of $AI$ with $\Gamma$. The line parallel to $BC$ through $I$ intersects $AB$ and $AC$ at $P$ and $Q$, respectively. The lines $PD$ and $QD$ intersect $BC$ at $E$ and $F$, respectively. Prove that triangles $IEF$ and $ABC$ are similar.

In a convex quadrilateral $ABCD$, $BD$ is the angle bisector of $\angle{ABC}$. The circumcircle of $ABC$ intersects $CD,AD$ in $P,Q$ respectively and the line through $D$ parallel to $AC$ cuts $AB,AC$ in $R,S$ respectively. Prove that point $P,Q,R,S$ lie on a circle.

$ABC$ is an obtuse triangle satisfying $\angle A>90^\circ$, and its circumcenter $O$ and circumcircle $\omega_1$. Let $\omega_2$ be a circle passing $C$ with center $B$. $\omega_2$ meets $BC$ at $D$. $\omega_1$ meets $AD$ and $\omega_2$ at $E$ and $F(\neq C)$, respectively. $AF$ meets $\omega_2$ at $G(\neq F)$. Prove that the intersection of $CE$ and $BG$ lies on the circumcircle of $AOB$.

Given two circles of radii $r$ and $r'$ exterior to each other, construct a line parallel to a given line and intersecting the two circles in chords with the sum of lengths $\ell$.

Let $ABCD$ be a parallelogram and let $K$ be a point on line $AD$ such that $BK=AB$. Suppose that $P$ is an arbitrary point on $AB$, and the perpendicular bisector of $PC$ intersects the circumcircle of triangle $APD$ at points $X$, $Y$. Prove that the circumcircle of triangle $ABK$ passes through the orthocenter of triangle $AXY$. [i]Proposed by Iman Maghsoudi[/i]