Consider a trapezoid $ABCD$ with $BC||AD$ and $BC<AD$. Let the lines $AB$ and $CD$ meet at $X$. Let $\omega_1$ be the incircle of the triangle $XBC$, and let $\omega_2$ be the excircle of the triangle $XAD$ which is tangent to the segment $AD$ . Denote by $a$ and $d$ the lines tangent to $\omega_1$ , distinct from $AB$ and $CD$, and passing through $A$ and $D$, respectively. Denote by $b$ and $c$ the lines tangent to $\omega_2$ , distinct from $AB$ and $CD$, passing through $B$ and $C$ respectively. Assume that the lines $a,b,c$ and $d$ are distinct. Prove that they form a parallelogram.

We have six pairwise non-intersecting circles that the radius of each is at least one (no circle lies in the interior of any other circle). Prove that the radius of any circle intersecting all the six circles, is at least one. [i]Proposed by Mohammad Ali Abam - Morteza Saghafian[/i]

Nine distinct positive integers are arranged in a circle such that the product of any two non-adjacent numbers in the circle is a multiple of $n$ and the product of any two adjacent numbers in the circle is not a multiple of $n$, where $n$ is a fixed positive integer. Find the smallest possible value for $n$.

Let the trapezoids $A_iB_iC_iD_i$ ($i = 1, 2, 3$) be similar and have the same clockwise direction. Their angles at $A_i$ and $B_i$ are $60^o$ and the sides $A_1B_1$, $B_2C_2$ and $A_3D_3$ are parallel. The lines $B_iD_{i+1}$ and $C_iA_{i+1}$ intersect at the point $P_i$ (the indices are understood cyclically, i.e. $A_4 = A_1$ and $D_4 = D_1$). Prove that the points $P_1$, $P_2$ and $P_3$ lie on a line.

Let $\mathcal{F}$ be a family of (distinct) subsets of the set $\{1,2,\dots,n\}$ such that for all $A$, $B\in \mathcal{F}$,we have that $A^C\cup B\in \mathcal{F}$, where $A^C$ is the set of all members of ${1,2,\dots,n}$ that are not in $A$. Prove that every $k\in {1,2,\dots,n}$ appears in at least half of the sets in $\mathcal{F}$. [i]Stijn Cambie, Mohammad Javad Moghaddas Mehr[/i]

Let $A_1A_2...A_{2018}$ be regular $2018$-gon. Radius of it's circumcircle is $R$. Prove that: $$A_1A_{1008}-A_1A_{1006}+A_1A_{1004}-A_1A_{1002} + ... + A_1A_4 -A_1A_2=R$$

How many permutations $a_1, a_2, \ldots, a_n$ of $\{1, 2, . . ., n \}$ are sorted into increasing order by at most three repetitions of the following operation: Move from left to right and interchange $a_i$ and $a_{i+1}$ whenever $a_i > a_{i+1}$ for $i$ running from $1$ up to $n - 1 \ ?$

Let $n\geqslant 3$ be a fixed integer and $\omega=\cos\dfrac{2\pi}n+i\sin\dfrac{2\pi}n$. Show that for every $a\in\mathbb{C}$ and $r>0$, the number \[\sum\limits_{k=1}^n \dfrac{|a-r\omega^k|^2}{|a|^2+r^2}\] is an integer. Interpet this result geometrically. [i]Octavian Stănășilă[/i]

Let $ABCDEF$ be a convex hexagon and denote $U$,$V$,$W$,$X$,$Y$ and $Z$ be the midpoint of $AB$,$BC$,$CD$,$DE$,$EF$ and $FA$ respectively. Prove that the length of $UX$,$VY$,$WZ$ can be the length of each sides of some triangle.

Let $AD$ be the diameter of the circumcircle of an acute triangle $ABC$. The lines through $D$ parallel to $AB$ and $AC$ meet lines $AC$ and $AB$ in points $E$ and $F$, respectively. Lines $EF$ and $BC$ meet at $G$. Prove that $AD$ and $DG$ are perpendicular.

Rachel and Robert run on a circular track. Rachel runs counterclockwise and completes a lap every $ 90$ seconds, and Robert runs clockwise and completes a lap every $ 80$ seconds. Both start from the start line at the same time. At some random time between $ 10$ minutes and $ 11$ minutes after they begin to run, a photographer standing inside the track takes a picture that shows one-fourth of the track, centered on the starting line. What is the probability that both Rachel and Robert are in the picture? $ \textbf{(A)}\ \frac{1}{16}\qquad \textbf{(B)}\ \frac18\qquad \textbf{(C)}\ \frac{3}{16} \qquad \textbf{(D)}\ \frac14\qquad \textbf{(E)}\ \frac{5}{16}$

If real numbers $a>b>1$ satisfy the inequality$$(ab+1)^2+(a+b)^2\leq 2(a+b)(a^2-ab+b^2+1)$$what is the minimum possible value of $\dfrac{\sqrt{a-b}}{b-1}$

Find the maximum possible value of $9\sqrt{x} + 8\sqrt{y} + 5\sqrt{z}$ where $x, y,$ and $z$ are positive real numbers satisfying $9x + 4y + z = 128$.

Does there exist a convex pentagon from which a similar pentagon can be cut off by a straight line? (S Tokarev)

Let $\mathbb{R}[x]$ be the set of all polynomials with real coefficients. Find all functions $f: \mathbb{R}[x] \rightarrow \mathbb{R}[x]$ satisfying the following conditions: [list] [*] $f$ maps the zero polynomial to itself, [*] for any non-zero polynomial $P \in \mathbb{R}[x]$, $\text{deg} \, f(P) \le 1+ \text{deg} \, P$, and [*] for any two polynomials $P, Q \in \mathbb{R}[x]$, the polynomials $P-f(Q)$ and $Q-f(P)$ have the same set of real roots. [/list] [i]Proposed by Anant Mudgal, Sutanay Bhattacharya, Pulkit Sinha[/i]

A circle of radius $r$ has chords $\overline{AB}$ of length $10$ and $\overline{CD}$ of length $7$. When $\overline{AB}$ and $\overline{CD}$ are extended through $B$ and $C$, respectively, they intersect at $P$, which is outside the circle. If $\angle APD = 60^{\circ}$ and $BP = 8$, then $r^{2} =$ $ \textbf{(A)}\ 70\qquad\textbf{(B)}\ 71\qquad\textbf{(C)}\ 72\qquad\textbf{(D)}\ 73\qquad\textbf{(E)}\ 74 $

Consider $\vartriangle ABC$ right at $B, F$ a point such that $B - F - C$ and $AF$ bisects $\angle BAC$, $I$ a point such that $A - I - F$ and CI bisect $\angle ACB$, and $E$ a point such that $A- E - C$ and $AF \perp EI$. If $AB = 4$ and $\frac{AI}{IF}={4}{3}$ , determine $AE$. Notation: $A-B-C$ means than points $A,B,C$ are collinear in that order i.e. $ B$ lies between $ A$ and $C$.

Let $S = \{1,2,..., 2017\}$. Find the maximal $n$ with the property that there exist $n$ distinct subsets of $S$ such that for no two subsets their union equals $S$. Proposed by Gerhard Woeginger

Let ${a_1, a_2, . . . , a_n,}$ and ${b_1, b_2, . . . , b_n}$ be real numbers with ${a_1, a_2, . . . , a_n}$ distinct. Show that if the product ${(a_i + b_1)(a_i + b_2) \cdot \cdot \cdot (a_i + b_n)}$ takes the same value for every ${ i = 1, 2, . . . , n, }$ , then the product ${(a_1 + b_j)(a_2 + b_j) \cdot \cdot \cdot (a_n + b_j)}$ also takes the same value for every ${j = 1, 2, . . . , n, }$ .

Patricia has a rectangular painting that she wishes to frame. The frame must also be rectangular and will extend $3\text{ cm}$ outward from each of the four sides of the painting. When the painting is framed, the area of the frame not covered by the painting is $108\text{ cm}^2$. What is the perimeter of the painting alone (without the frame)?

Let $S$ be the smallest set of positive integers such that a) $2$ is in $S,$ b) $n$ is in $S$ whenever $n^2$ is in $S,$ and c) $(n+5)^2$ is in $S$ whenever $n$ is in $S.$ Which positive integers are not in $S?$ (The set $S$ is ``smallest" in the sense that $S$ is contained in any other such set.)

Given three infinite arithmetic progressions of natural numbers such that each of the numbers 1,2,3,4,5,6,7 and 8 belongs to at least one of them, prove that the number 1980 also belongs to at least one of them.

Let $A B C$ be an acut-angles triangle of incenter $I$, circumcenter $O$ and inradius $r.$ Let $\omega$ be the inscribed circle of the triangle $A B C$. $A_1$ is the point of $\omega$ such that $A IA_1O$ is a convex trapezoid of bases $A O$ and $IA_1$. Let $\omega_1$ be the circle of radius $r$ which goes through $A_1$, tangent to the line $A B$ and is different from $\omega$ . Let $\omega_2$ be the circle of radius $r$ which goes through $A_1$, is tangent to the line $A C$ and is different from $\omega$ . Circumferences $\omega_1$ and $\omega_2$ they are cut at points $A_1$ and $A_2$. Similarly are defined points $B_2$ and $C_2$. Prove that the lines $A A_2, B B_2$ and $CC2$ they are concurrent.

Let $z$ be a complex non-zero number such that $Re(z),Im(z)\in \mathbb{Z}$. Prove that $z$ is uniquely representable as $a_0+a_1(1+i)+a_2(1+i)^2+\dots+a_n(1+i)^n$ where $n\geq 0$ and $a_j \in \{0,1\}$ and $a_n=1$. Time allowed for this problem was 1 hour.

In triangle $ ABC$ the medians $ \overline{AD}$ and $ \overline{CE}$ have lengths 18 and 27, respectively, and $ AB \equal{} 24$. Extend $ \overline{CE}$ to intersect the circumcircle of $ ABC$ at $ F$. The area of triangle $ AFB$ is $ m\sqrt {n}$, where $ m$ and $ n$ are positive integers and $ n$ is not divisible by the square of any prime. Find $ m \plus{} n$.