Let $ABCDEF$ be a convex hexagon such that $\angle ACE = \angle BDF$ and $\angle BCA = \angle EDF$. Let $A_1=AC\cap FB$, $B_1=BD\cap AC$, $C_1=CE\cap BD$, $D_1=DF\cap CE$, $E_1=EA\cap DF$, and $F_1=FB\cap EA$. Suppose $B_1, C_1, D_1, F_1$ lie on the same circle $\Gamma$. The circumcircles of $\triangle BB_1F_1$ and $ED_1F_1$ meet at $F_1$ and $P$. The line $F_1P$ meets $\Gamma$ again at $Q$. Prove that $B_1D_1$ and $QC_1$ are parrallel. (Here, we use $l_1\cap l_2$ to denote the intersection point of lines $l_1$ and $l_2$)

Let $\omega,\omega_1,\omega_2$ be three mutually tangent circles such that $\omega_1,\omega_2$ are externally tangent at $P$, $\omega_1,\omega$ are internally tangent at $A$, and $\omega,\omega_2$ are internally tangent at $B$. Let $O,O_1,O_2$ be the centers of $\omega,\omega_1,\omega_2$, respectively. Given that $X$ is the foot of the perpendicular from $P$ to $AB$, prove that $\angle{O_1XP}=\angle{O_2XP}$. [i]David Yang.[/i]

Let $I$ be an incenter of $\triangle ABC$. Denote $D, \ S \neq A$ intersections of $AI$ with $BC, \ O(ABC)$ respectively. Let $K, \ L$ be incenters of $\triangle DSB, \ \triangle DCS$. Let $P$ be a reflection of $I$ with the respect to $KL$. Prove that $BP \perp CP$.

In the following diagram, $O$ is the center of the circle. If three angles $\alpha, \beta$ and $\gamma$ be equal, find $\alpha.$ [asy] unitsize(40); import graph; size(300); real lsf = 0.5; pen dp = linewidth(0.7) + fontsize(10); defaultpen(dp); pen ds = black; pen ttttff = rgb(0.2,0.2,1); pen ffttww = rgb(1,0.2,0.4); pen qqwuqq = rgb(0,0.39,0); draw(circle((0,0),2.33),ttttff+linewidth(2.8pt)); draw((-1.95,-1.27)--(0.64,2.24),ffttww+linewidth(2pt)); draw((0.64,2.24)--(1.67,-1.63),ffttww+linewidth(2pt)); draw((-1.95,-1.27)--(1.06,0.67),ffttww+linewidth(2pt)); draw((1.67,-1.63)--(-0.6,0.56),ffttww+linewidth(2pt)); draw((-0.6,0.56)--(1.06,0.67),ffttww+linewidth(2pt)); pair parametricplot0_cus(real t){ return (0.6*cos(t)+0.64,0.6*sin(t)+2.24); } draw(graph(parametricplot0_cus,-2.2073069497794027,-1.3111498158746024)--(0.64,2.24)--cycle,qqwuqq); pair parametricplot1_cus(real t){ return (0.6*cos(t)+-0.6,0.6*sin(t)+0.56); } draw(graph(parametricplot1_cus,0.06654165390165974,0.9342857038103908)--(-0.6,0.56)--cycle,qqwuqq); pair parametricplot2_cus(real t){ return (0.6*cos(t)+-0.6,0.6*sin(t)+0.56); } draw(graph(parametricplot2_cus,-0.766242589858673,0.06654165390165967)--(-0.6,0.56)--cycle,qqwuqq); dot((0,0),ds); label("$O$", (-0.2,-0.38), NE*lsf); dot((0.64,2.24),ds); label("$A$", (0.72,2.36), NE*lsf); dot((-1.95,-1.27),ds); label("$B$", (-2.2,-1.58), NE*lsf); dot((1.67,-1.63),ds); label("$C$", (1.78,-1.96), NE*lsf); dot((1.06,0.67),ds); label("$E$", (1.14,0.78), NE*lsf); dot((-0.6,0.56),ds); label("$D$", (-0.92,0.7), NE*lsf); label("$\alpha$", (0.48,1.38),NE*lsf); label("$\beta$", (-0.02,0.94),NE*lsf); label("$\gamma$", (0.04,0.22),NE*lsf); clip((-8.84,-9.24)--(-8.84,8)--(11.64,8)--(11.64,-9.24)--cycle); [/asy]

Let $BM$ be the median of right-angled triangle $ABC (\angle B = 90^{\circ})$. The incircle of triangle $ABM$ touches sides $AB, AM$ in points $A_{1},A_{2}$; points $C_{1}, C_{2}$ are defined similarly. Prove that lines $A_{1}A_{2}$ and $C_{1}C_{2}$ meet on the bisector of angle $ABC$.

Let $ABC$ be a triangle and let $X$ and $Y$ be points on the $A$-symmedian such that $AX = XB$ and $AY = YC$. Let $BX$ and $CY$ meet at $Z$. Let the $Z$-excircle of triangle $XYZ$ touch $ZX$ and $ZY$ at $E$ and $F$. Show that $A$, $E$, $F$ are collinear.

Let the intersections of $\odot O_1$ and $\odot O_2$ be $A$ and $B$. Point $R$ is on arc $AB$ of $\odot O_1$ and $T$ is on arc $AB$ on $\odot O_2$. $AR$ and $BR$ meet $\odot O_2$ at $C$ and $D$; $AT$ and $BT$ meet $\odot O_1$ at $Q$ and $P$. If $PR$ and $TD$ meet at $E$ and $QR$ and $TC$ meet at $F$, then prove: $AE \cdot BT \cdot BR = BF \cdot AT \cdot AR$.

Prove that the polynomial $z^{2n} + z^n + 1\ (n \in \mathbb{N})$ is divisible by the polynomial $z^2 + z + 1$ if and only if $n$ is not a multiple of $3$.

Let $ABCD$ be inscribed in a circle $\omega$. Let the line parallel to the tangent to $\omega$ at $A$ and passing $D$ meet $\omega$ at $E$. $F$ is a point on $\omega$ such that lies on the different side of $E$ wrt $CD$. If $AE \cdot AD \cdot CF = BE \cdot BC \cdot DF$ and $\angle CFD = 2\angle AFB$, Show that the tangent to $\omega$ at $A, B$ and line $EF$ concur at one point. ($A$ and $E$ lies on the same side of $CD$)

A quadrilateral $ABCD$ is such that $\angle A= \angle C=60^o$ and $\angle B=100^o$. Let $O_1$ and $O_2$ be the centers of the incircles of triangles $ABD$ and $CBD$ respectively. Find the angle between the lines $AO_2$ and $CO_1$.

[b]p1.[/b] Two robots race on the plane from $(0, 0)$ to $(a, b)$, where $a$ and $b$ are positive real numbers with $a < b$. The robots move at the same constant speed. However, the first robot can only travel in directions parallel to the lines $x = 0$ or $y = 0$, while the second robot can only travel in directions parallel to the lines $y = x$ or $y = -x$. Both robots take the shortest possible path to $(a, b)$ and arrive at the same time. Find the ratio $\frac{a}{b}$ . [b]p2.[/b] Suppose $x + \frac{1}{x} + y + \frac{1}{y} = 12$ and $x^2 + \frac{1}{x^2} + y^2 + \frac{1}{y^2} = 70$. Compute $x^3 + \frac{1}{x^3} + y^3 + \frac{1}{y^3}$. [b]p3.[/b] Find the largest non-negative integer $a$ such that $2^a$ divides $$3^{2^{2018}}+ 3.$$ [b]p4.[/b] Suppose $z$ and $w$ are complex numbers, and $|z| = |w| = z \overline{w}+\overline{z}w = 1$. Find the largest possible value of $Re(z + w)$, the real part of $z + w$. [b]p5.[/b] Two people, $A$ and $B$, are playing a game with three piles of matches. In this game, a move consists of a player taking a positive number of matches from one of the three piles such that the number remaining in the pile is equal to the nonnegative difference of the numbers of matches in the other two piles. $A$ and $B$ each take turns making moves, with $A$ making the first move. The last player able to make a move wins. Suppose that the three piles have $10$, $x$, and $30$ matches. Find the largest value of $x$ for which $A$ does not have a winning strategy. [b]p6.[/b] Let $A_1A_2A_3A_4A_5A_6$ be a regular hexagon with side length $1$. For $n = 1$,$...$, $6$, let $B_n$ be a point on the segment $A_nA_{n+1}$ chosen at random (where indices are taken mod $6$, so $A_7 = A_1$). Find the expected area of the hexagon $B_1B_2B_3B_4B_5B_6$. [b]p7.[/b] A termite sits at the point $(0, 0, 0)$, at the center of the octahedron $|x| + |y| + |z| \le 5$. The termite can only move a unit distance in either direction parallel to one of the $x$, $y$, or $z$ axes: each step it takes moves it to an adjacent lattice point. How many distinct paths, consisting of $5$ steps, can the termite use to reach the surface of the octahedron? [b]p8.[/b] Let $$P(x) = x^{4037} - 3 - 8 \cdot \sum^{2018}_{n=1}3^{n-1}x^n$$ Find the number of roots $z$ of $P(x)$ with $|z| > 1$, counting multiplicity. [b]p9.[/b] How many times does $01101$ appear as a not necessarily contiguous substring of $0101010101010101$? (Stated another way, how many ways can we choose digits from the second string, such that when read in order, these digits read $01101$?) [b]p10.[/b] A perfect number is a positive integer that is equal to the sum of its proper positive divisors, that is, the sum of its positive divisors excluding the number itself. For example, $28$ is a perfect number because $1 + 2 + 4 + 7 + 14 = 28$. Let $n_i$ denote the ith smallest perfect number. Define $$f(x) =\sum_{i|n_x}\sum_{j|n_i}\frac{1}{j}$$ (where $\sum_{i|n_x}$ means we sum over all positive integers $i$ that are divisors of $n_x$). Compute $f(2)$, given there are at least $50 $perfect numbers. [b]p11.[/b] Let $O$ be a circle with chord $AB$. The perpendicular bisector to $AB$ is drawn, intersecting $O$ at points $C$ and $D$, and intersecting $AB$ at the midpoint $E$. Finally, a circle $O'$ with diameter $ED$ is drawn, and intersects the chord $AD$ at the point $F$. Given $EC = 12$, and $EF = 7$, compute the radius of $O$. [b]p12.[/b] Suppose $r$, $s$, $t$ are the roots of the polynomial $x^3 - 2x + 3$. Find $$\frac{1}{r^3 - 2}+\frac{1}{s^3 - 2}+\frac{1}{t^3 - 2}.$$ [b]p13.[/b] Let $a_1$, $a_2$,..., $a_{14}$ be points chosen independently at random from the interval $[0, 1]$. For $k = 1$, $2$,$...$, $7$, let $I_k$ be the closed interval lying between $a_{2k-1}$ and $a_{2k}$ (from the smaller to the larger). What is the probability that the intersection of $I_1$, $I_2$,$...$, $I_7$ is nonempty? [b]p14.[/b] Consider all triangles $\vartriangle ABC$ with area $144\sqrt3$ such that $$\frac{\sin A \sin B \sin C}{ \sin A + \sin B + \sin C}=\frac14.$$ Over all such triangles $ABC$, what is the smallest possible perimeter? [b]p15.[/b] Let $N$ be the number of sequences $(x_1,x_2,..., x_{2018})$ of elements of $\{1, 2,..., 2019\}$, not necessarily distinct, such that $x_1 + x_2 + ...+ x_{2018}$ is divisible by $2018$. Find the last three digits of $N$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

D is the a point on BC，I1 is the heart of a triangle ABD, I2 is the heart of a triangle ACD，O1 is the Circumcenter of triangle AI1D, O2 is the Circumcenter of the triangle AI2D，P is the intersection point of O1I2 and O2I1，Prove: PD is perpendicular to BC.

A point $D$ is chosen on the side $AC$ of a triangle $ABC$ with $\angle C < \angle A < 90^\circ$ in such a way that $BD=BA$. The incircle of $ABC$ is tangent to $AB$ and $AC$ at points $K$ and $L$, respectively. Let $J$ be the incenter of triangle $BCD$. Prove that the line $KL$ intersects the line segment $AJ$ at its midpoint.

Let $ABC$ be a triangle, and $M$ an interior point such that $\angle MAB=10^\circ$, $\angle MBA=20^\circ$, $\angle MAC=40^\circ$ and $\angle MCA=30^\circ$. Prove that the triangle is isosceles.

Let $ABC$ be an acute-angled triangle with circumcircle $\omega$. A circle $\Gamma$ is internally tangent to $\omega$ at $A$ and also tangent to $BC$ at $D$. Let $AB$ and $AC$ intersect $\Gamma$ at $P$ and $Q$ respectively. Let $M$ and $N$ be points on line $BC$ such that $B$ is the midpoint of $DM$ and $C$ is the midpoint of $DN$. Lines $MP$ and $NQ$ meet at $K$ and intersect $\Gamma$ again at $I$ and $J$ respectively. The ray $KA$ meets the circumcircle of triangle $IJK$ again at $X\neq K$. Prove that $\angle BXP = \angle CXQ$. [i]Kian Moshiri, United Kingdom[/i]

Let $K$ denote the circumference of a circular disk of radius $1$, and let $k$ denote a circular arc that joins two points $a,b$ on $K$ and lies otherwise in the given circular disc. Suppose that $k$ divides the circular disk into two parts of equal area. Prove that the length of $k$ exceeds $2.$

Let $AL$ be the inner bisector of triangle $ABC$. The circle centered at $B$ with radius $BL$ meets the ray $AL$ at points $L$ and $E$, and the circle centered at $C$ with radius $CL$ meets the ray $AL$ at points $L$ and $D$. Show that $AL^2 = AE\times AD$. [i](Proposed by Mykola Moroz)[/i]

The base $AB$ of a trapezium $ABCD$ is longer than the base $CD$, and $\angle ADC$ is a right angle. The diagonals $AC$ and $BD$ are perpendicular. Let $E$ be the foot of the altitude from $D$ to the line $BC$. Prove that $$\frac{AE}{BE} =\frac{ AC \cdot CD}{AC^2 - CD^2}$$ .

Transform by inversion two concentric and coplanar circles into two equal.

Given is triangle $ABC$ with incenter $I$ and $A$-excenter $J$. Circle $\omega_b$ centered at point $O_b$ passes through point $B$ and is tangent to line $CI$ at point $I$. Circle $\omega_c$ with center $O_c$ passes through point $C$ and touches line $BI$ at point $I$. Let $O_bO_c$ and $IJ$ intersect at point $K$. Find the ratio $IK/KJ$.

Several hikers travel at fixed speeds along a straight road. It is known that over some period of time, the sum of their pairwise distances is monotonically decreasing. Show that there is a hiker, the sum of whose distances to the other hikers is monotonically decreasing over the same period. [i]A. Shapovalov[/i]

[b]p1.[/b] An animal farm has geese and pigs with a total of $30$ heads and $84$ legs. Find the number of pigs and geese on this farm. [b]p2.[/b] What is the maximum number of $1 \times 1$ squares of a $7 \times 7$ board that can be colored black in such a way that the black squares don’t touch each other even at their corners? Show your answer on the figure below and explain why it is not possible to get more black squares satisfying the given conditions. [img]https://cdn.artofproblemsolving.com/attachments/d/5/2a0528428f4a5811565b94061486699df0577c.png[/img] [b]p3.[/b] Decide whether it is possible to divide a regular hexagon into three equal not necessarily regular hexagons? A regular hexagon is a hexagon with equal sides and equal angles. [img]https://cdn.artofproblemsolving.com/attachments/3/7/5d941b599a90e13a2e8ada635e1f1f3f234703.png[/img] [b]p4.[/b] A rectangle is subdivided into a number of smaller rectangles. One observes that perimeters of all smaller rectangles are whole numbers. Is it possible that the perimeter of the original rectangle is not a whole number? [b]p5.[/b] Place parentheses on the left hand side of the following equality to make it correct. $$ 4 \times 12 + 18 : 6 + 3 = 50$$ [b]p6.[/b] Is it possible to cut a $16\times 9$ rectangle into two equal parts which can be assembled into a square? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ABCD$ be a quadrilateral with $\overline{AB}\parallel\overline{CD}$, $AB=16$, $CD=12$, and $BC<AD$. A circle with diameter $12$ is inside of $ABCD$ and tangent to all four sides. Find $BC$.

The area of polygon $ ABCDEF$ is 52 with $ AB\equal{}8$, $ BC\equal{}9$ and $ FA\equal{}5$. What is $ DE\plus{}EF$? [asy]defaultpen(linewidth(0.8));pair a=(0,9), b=(8,9), c=(8,0), d=(4,0), e=(4,4), f=(0,4); draw(a--b--c--d--e--f--cycle); draw(shift(0,-.25)*a--shift(.25,-.25)*a--shift(.25,0)*a); draw(shift(-.25,0)*b--shift(-.25,-.25)*b--shift(0,-.25)*b); draw(shift(-.25,0)*c--shift(-.25,.25)*c--shift(0,.25)*c); draw(shift(.25,0)*d--shift(.25,.25)*d--shift(0,.25)*d); draw(shift(.25,0)*f--shift(.25,.25)*f--shift(0,.25)*f); label("$A$", a, NW); label("$B$", b, NE); label("$C$", c, SE); label("$D$", d, SW); label("$E$", e, SW); label("$F$", f, SW); label("5", (0,6.5), W); label("8", (4,9), N); label("9", (8, 4.5), E);[/asy] $ \textbf{(A)}\ 7\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 11 $

Let $ABC$ be an acute-angled triangle with circumference $\Omega$. Let the angle bisectors of $\angle B$ and $\angle C$ intersect $\Omega$ again at $M$ and $N$. Let $I$ be the intersection point of these angle bisectors. Let $M'$ and $N'$ be the respective reflections of $M$ and $N$ in $AC$ and $AB$. Prove that the center of the circle passing through $I$, $M'$, $N'$ lies on the altitude of triangle $ABC$ from $A$. [i]Proposed by Victor Domínguez and Ariel García[/i]