$AB,AC$ are tangent to a circle $(O)$, $B,C$ are the points of tangency. $Q$ is a point iside the angle $BAC$, on the ray $AQ$, take a point $P$ suc that $OP$ is perpendicular to $AQ$. The line $OP$ meets the circumcircles triangles $BPQ$ and $CPQ$ at $I, J$. Prove that $OI = OJ$. Hồ Quang Vinh

A $ 9\times9\times9$ cube is composed of twenty-seven $ 3\times3\times3$ cubes. The big cube is 'tunneled' as follows: First, the six $ 3\times3\times3$ cubes which make up the center of each face as well as the center of $ 3\times3\times3$ cube are removed. Second, each of the twenty remaining $ 3\times3\times3$ cubes is diminished in the same way. That is, the central facial unit cubes as well as each center cube are removed. [asy] import three; size(4.5cm); triple eye = (6, 9, 5); currentprojection = perspective(eye); real eps = 0.001; for(int i = 0; i < 3; ++i){ for(int j = 0; j < 3; ++j){ for(int k = 0; k < 3; ++k){ if(i == 1 && j == 1) continue; if(j == 1 && k == 1) continue; if(k == 1 && i == 1) continue; draw(shift(i, j, k) * scale(1 - eps, 1 - eps, 1 - eps) * unitcube, gray(0.9), nolight); draw(shift(i, j, k) * (X--(X + Y)--Y--(Y+Z)--Z--(Z + X)--cycle)); draw(shift(i, j, k) * (X + Y + Z--X + Y)); draw(shift(i, j, k) * (X + Y + Z--Y + Z)); draw(shift(i, j, k) * (X + Y + Z--Z + X)); } } } [/asy] The surface area of the final figure is $ \textbf{(A)}\ 384\qquad \textbf{(B)}\ 729\qquad \textbf{(C)}\ 864\qquad \textbf{(D)}\ 1024\qquad \textbf{(E)}\ 1056$

As shown in the figure, $\odot O$ is the circumcircle of $\vartriangle ABC$, $\odot J$ is inscribed in $\odot O$ and is tangent to $AB$, $AC$ at points $D$ and E respectively, line segment $FG$ and $\odot O$ are tangent to point $A$, and $AF =AG=AD$, the circumscribed circle of $\vartriangle AFB$ intersects $\odot J$ at point $S$. Prove that the circumscribed circle of $\vartriangle ASG$ is tangent to $\odot J$. [img]https://cdn.artofproblemsolving.com/attachments/a/a/62d44e071ea9903ebdd68b43943ba1d93b4138.png[/img]

Let $ABC$ a triangle. Let $D$ be a point on the circumcircle of this triangle and let $E , F$ be the feet of the perpendiculars from $A$ on $DB, DC$, respectively. Finally, let $N$ be the midpoint of $EF$. Let $M \ne N$ be the midpoint of the side $BC$ . Prove that the lines $NA$ and $NM$ are perpendicular.

Let $ABC$ be a triangle with incentre $I$ and circumcircle $\omega$. Let $D$ and $E$ be the second intersection points of $\omega$ with $AI$ and $BI$, respectively. The chord $DE$ meets $AC$ at a point $F$, and $BC$ at a point $G$. Let $P$ be the intersection point of the line through $F$ parallel to $AD$ and the line through $G$ parallel to $BE$. Suppose that the tangents to $\omega$ at $A$ and $B$ meet at a point $K$. Prove that the three lines $AE,BD$ and $KP$ are either parallel or concurrent. [i]Proposed by Irena Majcen and Kris Stopar, Slovenia[/i]

In the trapezoid $ABCD$, both $\angle B$ and $\angle C$ are right angles, and all four sides of the trapezoid are tangent to the same circle. If $\overline{AB} = x$ and $\overline{CD} = y$, find the area of $ABCD$ (with proof).

Given a sphere and a plane that has no common points with the sphere. Find the geometric locus of the centers of the circles of tangency with the sphere of those cones circumcribed on the sphere whose vertices lie on the given plane.

In a cyclic quadrilateral $ABCD$, the diagonals intersect at $E$. $F$ and $G$ are on chord $AC$ and chord $BD$ respectively such that $AF = BE$ and $DG = CE$. Prove that, $A, G, F, D$ lie on the same circle.

Let $P$ be a point in the interior of triangle $ABC$. Lines $AP$, $BP$, $CP$ intersect sides $BC$, $CA$, $AB$ at $L$, $M$, $N$, respec­tively. Prove that $$AP \cdot BP \cdot CP \ge 8PL \cdot PM \cdot PN.$$

Circle $\Gamma$ has radius $10$, center $O$, and diameter $\overline{AB}$. Point $C$ lies on $\Gamma$ such that $AC = 12$. Let $P$ be the circumcenter of $\vartriangle AOC$. Line $AP$ intersects $\Gamma$ at $Q$, where $Q$ is different from $A$. Then the value of $\frac{AP}{AQ}$ can be expressed in the form $\frac{m}{n}$, where m and $n$ are relatively prime positive integers. Compute $m + n$.

50 knights of King Arthur sat at the Round Table. A glass of white or red wine stood before each of them. It is known that at least one glass of red wine and at least one glass of white wine stood on the table. The king clapped his hands twice. After the first clap every knight with a glass of red wine before him took a glass from his left neighbour. After the second clap every knight with a glass of white wine (and possibly something more) before him gave this glass to the left neughbour of his left neighbour. Prove that some knight was left without wine. [i]Proposed by A. Khrabrov, incorrect translation from Hungarian[/i]

[u]Round 9[/u] [b]p25.[/b] A positive integer is called spicy if it is divisible by the sum if its digits. Find the number of spicy integers between $100$ and $200$ inclusive. [b]p26.[/b] Rectangle $ABCD$ has points $E$ and $F$ on sides $AB$ and $BC$, respectively. Given that $\frac{AE}{BE} = \frac{BF}{FC} =\frac12$, $\angle ADE = 30^o$, and $[DEF] = 25$, find the area of rectangle $ABCD$. [b]p27.[/b] Find the largest value of $n$ for which $3^n$ divides ${100 \choose 33}$. [u]Round 10[/u] [b]p28.[/b] Isosceles trapezoid $ABCD$ is inscribed in a circle such that $AB \parallel CD$, $AB = 2$, $CD = 4$, and $AC = 9$. What is the radius of the circle? [b]p29.[/b] Find the product of all possible positive integers $n$ less than $11$ such that in a group of $n$ people, it is possible for every person to be friends with exactly $3$ other people within the group. Assume that friendship is amutual relationship. [b]p30.[/b] Compute the infinite product $$\left( 1+ \frac{1}{2^1} \right) \left( 1+ \frac{1}{2^2} \right) \left( 1+ \frac{1}{2^4} \right) \left( 1+ \frac{1}{2^8} \right) \left( 1+ \frac{1}{2^{16}} \right) ...$$ [u]Round 11[/u] [b]p31.[/b] Find the sum of all possible values of $x y$ if $x +\frac{1}{y}= 12$ and $\frac{1}{x}+ y = 8$. [b]p32.[/b] Find the number of ordered pairs $(a,b)$, where $0 < a,b < 1999$, that satisfy $a^2 +b^2 \equiv ab$ (mod $1999$) [b]p33.[/b] Let $f :N\to Q$ be a function such that $f(1) =0$, $f (2) = 1$ and $f (n) = \frac{f(n-1)+f (n-2)}{2}$ . Evaluate $$\lim_{n\to \infty} f (n).$$ [u]Round 12[/u] [b]p34.[/b] Estimate the sumof the digits of $2018^{2018}$. The number of points you will receive is calculated using the formula $\max \,(0,15-\log_{10}(A-E))$, where $A$ is the true value and $E$ is your estimate. [b]p35.[/b] Let $C(m,n)$ denote the number of ways to tile an $m$ by $n$ rectangle with $1\times 2$ tiles. Estimate $\log_{10}(C(100, 2))$. The number of points you will recieve is calculated using the formula $\max \,(0,15- \log_{10}(A-E))$, where $A$ is the true value and $E$ is your estimate. [b]p36.[/b] Estimate $\log_2 {1000 \choose 500}$. The number of points you earn is equal to $\max \,(0,15-|A-E|)$, where $A$ is the true value and $E$ is your estimate. PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h3165983p28809209]here [/url] and 5-8 [url=https://artofproblemsolving.com/community/c3h3165992p28809294]here[/url].. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In the triangle $ABC$ let $B'$ and $C'$ be the midpoints of the sides $AC$ and $AB$ respectively and $H$ the foot of the altitude passing through the vertex $A$. Prove that the circumcircles of the triangles $AB'C'$,$BC'H$, and $B'CH$ have a common point $I$ and that the line $HI$ passes through the midpoint of the segment $B'C'.$

Circle $ O$ is inscribed in a trapzoid $ ABCD$, $ \angle{A}$ and $ \angle{B}$ are all acute angles. A line through $ O$ intersects $ AD$ at $ E$ and $ BC$ at $ F$, and satisfies the following conditions: (1) $ \angle{DEF}$ and $ \angle{CFE}$ are acute angles. (2) $ AE\plus{}BF\equal{}DE\plus{}CF$. Let $ AB\equal{}a$, $ BC\equal{}b$, $ CD\equal{}c$, then use $ a,b,c$ to express $ AE$.

13. Let $\triangle T B D$ be a triangle with $T B=6, B D=8$, and $D T=7$. Let $I$ be the incenter of $\triangle T B D$, and let $T I$ intersect the circumcircle of $\triangle T B D$ at $M \neq T$. Let lines $T B$ and $M D$ intersect at $Y$, and let lines $T D$ and $M B$ intersect at $X$. Let the circumcircles of $\triangle Y B M$ and $\triangle X D M$ intersect at $Z \neq M$. If the area of $\triangle Y B Z$ is $x$ and the area of $\triangle X D Z$ is $y$, then the ratio $\frac{x}{y}$ can be expressed as $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.

Let $AH$ be the altitude of a given triangle $ABC.$ The points $I_b$ and $I_c$ are the incenters of the triangles $ABH$ and $ACH$ respectively. $BC$ touches the incircle of the triangle $ABC$ at a point $L.$ Find $\angle LI_bI_c.$

Given a triangle $ABC$. The points $A$, $B$, $C$ divide the circumcircle $\Omega$ of the triangle $ABC$ into three arcs $BC$, $CA$, $AB$. Let $X$ be a variable point on the arc $AB$, and let $O_{1}$ and $O_{2}$ be the incenters of the triangles $CAX$ and $CBX$. Prove that the circumcircle of the triangle $XO_{1}O_{2}$ intersects the circle $\Omega$ in a fixed point.

Points $P, Q, R$ were marked on the sides $BC, CA, AB$, respectively. Let $a$ be tangent at point $A$ to the circumcircle of triangle $AQR$, $b$ be tangent at point $B$ to the circumcircle of the triangle BPR, $c$ be tangent at point $C$ to the circumscribed circle triangle $CPQ$. Let $X$ be the point of intersection of the lines $b$ and $c, Y$ be the point the intersection of lines $c$ and $a, Z$ is the point of intersection of lines $a$ and $b$. Prove that the lines $AX, BY, CZ$ intersect at one point if and only if the lines $AP, BQ, CR$ intersect at one point.

Let $a, b, c$ be the sides and $R$ be the circumradius of a triangle. Prove that \[\frac{1}{ab}+\frac{1}{bc}+\frac{1}{ca}\ge\frac{1}{R^2}.\]

The map of a town shows a plane divided into equal equilateral triangles. The sides of these triangles are streets and their vertices are intersections; $6$ streets meet at each junction. Two cars start simultaneously in the same direction and at the same speed from points $A$ and $B$ situated on the same street (the same side of a triangle). After any intersection an admissible route for each car is either to proceed in its initial direction or turn through $120^o$ to the right or to the left. Can these cars meet? (Either prove that these cars won’t meet or describe a route by which they will meet.) [img]https://cdn.artofproblemsolving.com/attachments/2/d/2c934bcd0c7fc3d9dca9cee0b6f015076abbdb.png[/img]

Let $ABC$ be an acute triangle.The lines $l_1$ and $l_2$ are perpendicular to $AB$ at the points $A$ and $B$, respectively.The perpendicular lines from the midpoint $M$ of $AB$ to the lines $AC$ and $BC$ intersect $l_1$ and $l_2$ at the points $E$ and $F$, respectively.If $D$ is the intersection point of the lines $EF$ and $MC$, prove that \[\angle ADB = \angle EMF.\]

The figure shows a parallelogram and the point $P$ of intersection of its diagonals is marked. Draw a straight line through $P$ so that it breaks the parallelogram into two parts, from which you can fold a rhombus. [img]https://1.bp.blogspot.com/-Df2tIBthcmI/X2ZwIx3R4vI/AAAAAAAAMhQ/8Zkxfq30H8MSCdc66tm33n6jt-QKfGMowCLcBGAsYHQ/s0/2009%2Boral%2Bmoscow%2Bj1.png[/img]

Given is an acute triangle $ABC$. Its heights $BB_1$ and $CC_1$ are extended past points $B_1$ and $C_1$. On these extensions, points $P$ and $Q$ are chosen, such that angle $PAQ$ is right. Let $AF$ be a height of triangle $APQ$. Prove that angle $BFC$ is a right angle.

Let $[ABC]$ be an equilateral triangle and $P$ be a point on $AC$ such that $\overline{PC}= 7$. The straight line that passes through $P$ and is perpendicular to $AC$, intersects $CB$ at point $M$ and intersects $AB$ at point $Q$. The midpoint $N$ of $[MQ]$ is such that $\overline{BN} = 14$. Determine the side of the triangle $[ABC]$.

On the plane we consider $70$ points $A_1,A_2,...,A_{70}$ with integer coodinates. Suppose each pooints has weight $1$ and the centers of gravity of the triangles $ A_1A_2A_3$, $A_2A_3A_4$, $..$., $A_{68}A_{69}A_{70}$, $A_{69}A_{70}A_{1}$, $A_{70}A_{1}A_{2}$ have integer coodinates. Prove that the centers of gravity of any triple $A_i,A_j,...,A_{k}$ has integer coodinates.