Given a $ \triangle ABC $ with circumcircle $ \Gamma. $ Let $ A' $ be the antipode of $ A $ in $ \Gamma $ and $ D $ be the point s.t. $ \triangle BCD $ is an equilateral triangle ($ A $ and $ D $ are on the opposite side of $ BC $). Let the perpendicular from $ A' $ to $ A'D $ cuts $ CA, $ $ AB $ at $ E, $ $ F, $ resp. and $ T $ be the point s.t. $ \triangle ETF $ is an isosceles triangle with base $ EF $ and base angle $ 30^{\circ} $ ($ A $ and $ T $ are on the opposite side of $ EF $). Prove that $ AT $ passes through the 9-point center of $ \triangle ABC. $ [i]Proposed by Telv Cohl[/i]

Given a cyclic quadrilateral $ABCD$, let the diagonals $AC$ and $BD$ meet at $E$ and the lines $AD$ and $BC$ meet at $F$. The midpoints of $AB$ and $CD$ are $G$ and $H$, respectively. Show that $EF$ is tangent at $E$ to the circle through the points $E$, $G$ and $H$. [i]Proposed by David Monk, United Kingdom[/i]

In triangle $ABC$, with $AB<AC$, $I$ is the incenter, $E$ is the intersection of $A$-excircle and $BC$. Point $F$ lies on the external angle bisector of $BAC$ such that $E$ and $F$ lieas on the same side of the line $AI$ and $\angle AIF=\angle AEB$. Point $Q$ lies on $BC$ such that $\angle AIQ=90$. Circle $\omega_b$ is tangent to $FQ$ and $AB$ at $B$, circle $\omega_c$ is tangent to $FQ$ and $AC$ at $C$ and both circles pass through the inside of triangle $ABC$. if $M$ is the Midpoint od the arc $BC$, which does not contain $A$, prove that $M$ lies on the radical axis of $\omega_b$ and $\omega_c$. Proposed by Amirmahdi Mohseni

A large square is divided into a small square surrounded by four congruent rectangles as shown. The perimeter of each of the congruent rectangles is 14. What is the area of the large square? [asy]unitsize(3mm); defaultpen(linewidth(.8pt)); draw((0,0)--(7,0)--(7,7)--(0,7)--cycle); draw((1,0)--(1,6)); draw((7,1)--(1,1)); draw((6,7)--(6,1)); draw((0,6)--(6,6));[/asy]$ \textbf{(A)}\ \ 49 \qquad \textbf{(B)}\ \ 64 \qquad \textbf{(C)}\ \ 100 \qquad \textbf{(D)}\ \ 121 \qquad \textbf{(E)}\ \ 196$

Two circles $\Gamma_1$ and $\Gamma_2$are given with centres $O_1$ and $O_2$ and common exterior tangents $\ell_1$ and $\ell_2$. The line $\ell_1$ intersects $\Gamma_1$ in $A$ and $\Gamma_2$ in $B$. Let $X$ be a point on segment $O_1O_2$, not lying on $\Gamma_1$ or $\Gamma_2$. The segment $AX$ intersects $\Gamma_1$ in $Y \ne A$ and the segment $BX$ intersects $\Gamma_2$ in $Z \ne B$. Prove that the line through $Y$ tangent to $\Gamma_1$ and the line through $Z$ tangent to $\Gamma_2$ intersect each other on $\ell_2$.

Point $P$ is taken interior to a square with side-length $a$ and such that is it equally distant from two consecutive vertices and from the side opposite these vertices. If $d$ represents the common distance, then $d$ equals: $\textbf{(A)}\ \dfrac{3a}{5} \qquad \textbf{(B)}\ \dfrac{5a}{8} \qquad \textbf{(C)}\ \dfrac{3a}{8} \qquad \textbf{(D)}\ \dfrac{a\sqrt{2}}{2} \qquad \textbf{(E)}\ \dfrac{a}{2}$

Given a set $ M$ of points $ (x,y)$ with integral coordinates satisfying $ x^2 + y^2\leq 10^{10}$. Two players play a game. One of them marks a point on his first move. After this, on each move the moving player marks a point, which is not yet marked and joins it with the previous marked point. Players are not allowed to mark a point symmetrical to the one just chosen. So, they draw a broken line. The requirement is that lengths of edges of this broken line must strictly increase. The player, which can not make a move, loses. Who have a winning strategy?

A round coin may be used to construct a circle passing through one or two given points on the plane. Given a line on the plane, show how to use this coin to construct two points such that they dene a line perpendicular to the given line. Note that the coin may not be used to construct a circle tangent to the given line.

Circles with centers $P, Q$ and $R$, having radii $1, 2$ and $3$, respectively, lie on the same side of line $l$ and are tangent to $l$ at $P', Q'$ and $R'$, respectively, with $Q'$ between $P'$ and $R'$. The circle with center $Q$ is externally tangent to each of the other two circles. What is the area of triangle $PQR$? $\textbf{(A) } 0\qquad \textbf{(B) } \sqrt{\frac{2}{3}}\qquad\textbf{(C) } 1\qquad\textbf{(D) } \sqrt{6}-\sqrt{2}\qquad\textbf{(E) }\sqrt{\frac{3}{2}}$

p1. Given $f(x)=\frac{1+x}{1-x}$ , for $x \ne 1$ . Defined $p @ q = \frac{p+q}{1+pq}$ for all positive rational numbers $p$ and $q$. Note the sequence with $a_1,a_2,a_3,...$ with $a_1=2 @3$, $a_{n}=a_{n-1}@ (n+2)$ for $n \ge 2$. Determine $f(a_{233})$ and $a_{233}$ p2. It is known that $ a$ and $ b$ are positive integers with $a > b > 2$. Is $\frac{2^a+1}{2^b-1}$ an integer? Write down your reasons. p3. Given a cube $ABCD.EFGH$ with side length $ 1$ dm. There is a square $PQRS$ on the diagonal plane $ABGH$ with points $P$ on $HG$ and $Q$ on $AH$ as shown in the figure below. Point $T$ is the center point of the square $PQRS$. The line $HT$ is extended so that it intersects the diagonal line $BG$ at $N$. Point $M$ is the projection of $N$ on $BC$. Determine the volume of the truncated prism $DCM.HGN$. [img]https://cdn.artofproblemsolving.com/attachments/f/6/22c26f2c7c66293ad7065a3c8ce3ac2ffd938b.png[/img] 4. Nine pairs of husband and wife want to take pictures in a three-line position with the background of the Palembang Ampera Bridge. There are $4$ people in the front row, $6$ people in the middle row, and $ 8$ people in the back row. They agreed that every married couple must be in the same row, and every two people next to each other must be a married couple or of the same sex. Specify the number of different possible arrangements of positions. p5. p5. A hotel provides four types of rooms with capacity, rate, and number of rooms as presented in the following table. [b] type of room, capacity of persons/ room, day / rate (Rp.), / number of rooms [/b][img]https://cdn.artofproblemsolving.com/attachments/3/c/e9e1ed86887e692f9d66349a82eaaffc730b46.jpg[/img] A group of four families wanted to stay overnight at the hotel. Each family consists of husband and wife and their unmarried children. The number of family members by gender is presented in the following table. [b]family / man / woman/ total[/b] [img]https://cdn.artofproblemsolving.com/attachments/4/6/5961b130c13723dc9fa4e34b43be30c31ee635.jpg[/img] The group leader enforces the following provisions. I. Each husband and wife must share a room and may not share a room with other married couples. II. Men and women may not share the same room unless they are from the same family. III. At least one room is occupied by all family representatives (“representative room”) IV. Each family occupies at most $3$ types of rooms. V. No rooms are occupied by more than one family except representative rooms. You are asked to arrange a room for the group so that the total cost of lodging is as low as possible. Provide two possible alternative room arrangements for each family and determine the total cost.

11.4 Let $AA_1$ and $BB_1$ are altitudes of an acute non-isosceles triangle $ABC$, $A'$ is a midpoint of $BC$ and $B'$ is a midpoint of $AC$. A segement $A_1B_1$ intersects $A'B'$ at point $C'$. Prove that $CC'\perp HO$, where $H$ is a orthocenter and $O$ is a circumcenter of $ABC$. ([i]L. Emel'yanov[/i])

Given two circles $\Gamma_1$ and $\Gamma_2$ which intersect at points $A$ and $B$. A line through $A$ intersects $\Gamma_1$ and $\Gamma_2$ at points $C$ and $D$, respectively. Let $M$ be the midpoint of arc $BC$ in $\Gamma_1$ ,which does not contains $A$, and $N$ is the midpoint of the arc $BD$ in $\Gamma_2$, which does not contain $A$. If $K$ is the midpoint of $CD$, prove that $\angle MKN = 90^o.$

In a triangle $ABC$, let $H_a$, $H_b$ and $H_c$ denote the base points of the altitudes on the sides $BC$, $CA$ and $AB$, respectively. Determine for which triangles $ABC$ two of the lengths $H_aH_b$, $H_bH_c$ and $H_aH_c$ are equal.

A hexahedron $ABCDE$ is made of two regular congruent tetrahedra $ABCD$ and $ABCE.$ Prove that there exists only one isometry $\mathbf Z$ that maps points $A, B, C, D, E$ onto $B, C, A, E, D,$ respectively. Find all points $X$ on the surface of hexahedron whose distance from $\mathbf Z(X)$ is minimal.

Let $ H$ be the orthocenter of an acute-angled triangle $ ABC$. The circle $ \Gamma_{A}$ centered at the midpoint of $ BC$ and passing through $ H$ intersects the sideline $ BC$ at points $ A_{1}$ and $ A_{2}$. Similarly, define the points $ B_{1}$, $ B_{2}$, $ C_{1}$ and $ C_{2}$. Prove that the six points $ A_{1}$, $ A_{2}$, $ B_{1}$, $ B_{2}$, $ C_{1}$ and $ C_{2}$ are concyclic. [i]Author: Andrey Gavrilyuk, Russia[/i]

A regular tetrahedron has side $L$. What is the smallest $x$ such that the tetrahedron can be passed through a loop of twine of length $x$?

The angles of the triangle $ABC$ satisfy $\angle A / \angle C = \angle B / \angle A = 2$. The incenter is $O. K, L$ are the excenters of the excircles opposite $B$ and $A$ respectively. Show that triangles $ABC$ and $OKL$ are similar.

Let $h_a$, $h_b$ and $h_c$ be the heights and $r$ the inradius of a triangle. Prove that the triangle is equilateral if and only if $h_a + h_b + h_c = 9r$.

Let $ ABC$ be acute triangle. The circle with diameter $ AB$ intersects $ CA,\, CB$ at $ M,\, N,$ respectively. Draw $ CT\perp AB$ and intersects above circle at $ T$, where $ C$ and $ T$ lie on the same side of $ AB$. $ S$ is a point on $ AN$ such that $ BT = BS$. Prove that $ BS\perp SC$.

Show how to construct a line segment length $(a^4 + b^4)^{1/4}$ given segments lengths $a$ and $b$.

[u]Round 1[/u] [b]p1.[/b] What is the smallest number equal to its cube? [b]p2.[/b] Fhomas has $5$ red spaghetti and $5$ blue spaghetti, where spaghetti are indistinguishable except for color. In how many different ways can Fhomas eat $6$ spaghetti, one after the other? (Two ways are considered the same if the sequence of colors are identical) [b]p3.[/b] Jocelyn labels the three corners of a triangle with three consecutive natural numbers. She then labels each edge with the sum of the two numbers on the vertices it touches, and labels the center with the sum of all three edges. If the total sum of all labels on her triangle is $120$, what is the value of the smallest label? [u]Round 2[/u] [b]p4.[/b] Adam cooks a pie in the shape of a regular hexagon with side length $12$, and wants to cut it into right triangular pieces with angles $30^o$, $60^o$, and $90^o$, each with shortest side $3$. What is the maximum number of such pieces he can make? [b]p5.[/b] If $f(x) =\frac{1}{2-x}$ and $g(x) = 1-\frac{1}{x}$ , what is the value of $f(g(f(g(... f(g(f(2019))) ...))))$, where there are $2019$ functions total, counting both $f$ and $g$? [b]p6.[/b] Fhomas is buying spaghetti again, which is only sold in two types of boxes: a $200$ gram box and a $500$ gram box, each with a fixed price. If Fhomas wants to buy exactly $800$ grams, he must spend $\$8:80$, but if he wants to buy exactly 900 grams, he only needs to spend $\$7:90$! In dollars, how much more does the $500$ gram box cost than the $200$ gram box? [u]Round 3[/u] [b]p7.[/b] Given that $$\begin{cases} a + 5b + 9c = 1 \\ 4a + 2b + 3c = 2 \\ 7a + 8b + 6c = 9\end{cases}$$ what is $741a + 825b + 639c$? [b]p8.[/b] Hexagon $JAMESU$ has line of symmetry $MU$ (i.e., quadrilaterals $JAMU$ and $SEMU$ are reflections of each other), and $JA = AM = ME = ES = 1$. If all angles of $JAMESU$ are $135$ degrees except for right angles at $A$ and $E$, find the length of side $US$. [b]p9.[/b] Max is parked at the $11$ mile mark on a highway, when his pet cheetah, Min, leaps out of the car and starts running up the highway at its maximum speed. At the same time, Max starts his car and starts driving down the highway at $\frac12$ his maximum speed, driving all the way to the $10$ mile mark before realizing that his cheetah is gone! Max then immediately reverses directions and starts driving back up the highway at his maximum speed, nally catching up to Min at the $20$ mile mark. What is the ratio between Max's max speed and Min's max speed? [u]Round 4[/u] [b]p10.[/b] Kevin owns three non-adjacent square plots of land, each with side length an integer number of meters, whose total area is $2019$ m$^2$. What is the minimum sum of the perimeters of his three plots, in meters? [b]p11.[/b] Given a $5\times 5$ array of lattice points, how many squares are there with vertices all lying on these points? [b]p12.[/b] Let right triangle $ABC$ have $\angle A = 90^o$, $AB = 6$, and $AC = 8$. Let points $D,E$ be on side $AC$ such that $AD = EC = 2$, and let points $F,G$ be on side $BC$ such that $BF = FG = 3$. Find the area of quadrilateral $FGED$. PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h2949413p26408203]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Prove that for every positive integer $n \geq 3$ there exist two sets $A =\{ x_1, x_2,\ldots, x_n\}$ and $B =\{ y_1, y_2,\ldots, y_n\}$ for which [b]i)[/b] $A \cap B = \varnothing.$ [b]ii)[/b] $x_1+ x_2+\cdots+ x_n= y_1+ y_2+\cdots+ y_n.$ [b]ii)[/b] $x_1^2+ x_2^2+\cdots+ x_n^2= y_1^2+ y_2^2+\cdots+ y_n^2.$

Let \(\mathcal E\) be an ellipse with foci \(F_1\) and \(F_2\), and let \(P\) be a point on \(\mathcal E\). Suppose lines \(PF_1\) and \(PF_2\) intersect \(\mathcal E\) again at distinct points \(A\) and \(B\), and the tangents to \(\mathcal E\) at \(A\) and \(B\) intersect at point \(Q\). Show that the midpoint of \(\overline{PQ}\) lies on the circumcircle of \(\triangle PF_1F_2\). [i]Proposed by Karthik Vedula[/i]

A circle is divided into $2n$ equal arcs by $2n$ points. Find all $n>1$ such that these points can be joined in pairs using $n$ segments, all of different lengths and such that each point is the endpoint of exactly one segment.

Find all integers $n\geq 3$ for which every convex equilateral $n$-gon of side length $1$ contains an equilateral triangle of side length $1$. (Here, polygons contain their boundaries.)