Let $ABC$ be a triangle. The incircle of triangle $ABC$ touches the side $BC$ at $A^{\prime}$, and the line $AA^{\prime}$ meets the incircle again at a point $P$. Let the lines $CP$ and $BP$ meet the incircle of triangle $ABC$ again at $N$ and $M$, respectively. Prove that the lines $AA^{\prime}$, $BN$ and $CM$ are concurrent.

$ABCDA'B'C'D'$ is a cube (with $ABCD$ and $A'B'C'D'$ faces, and $AA', BB', CC', DD'$ edges). $L$ is a line which intersects or is parallel to the lines $AA', BC$ and $DB'$. $L$ meets the line $BC$ at $M$ (which may be the point at infinity). Let $m = |BM|$. The plane $MAA'$ meets the line $B'C'$ at $E$. Show that $|B'E| = m$. The plane $MDB'$ meets the line $A'D'$ at $F$. Show that $|D'F| = m$. Hence or otherwise show how to construct the point $P$ at the intersection of $L$ and the plane $A'B'C'D'$. Find the distance between $P$ and the line $A'B'$ and the distance between $P$ and the line $A'D'$ in terms of $m$. Find a relation between these two distances that does not depend on $m$. Find the locus of $M$. Let $S$ be the envelope of the line $L$ as $M$ varies. Find the intersection of $S$ with the faces of the cube.

In the tetrahedron $ABCD$, the point $E$ is the projection of the point $D$ on the plane $(ABC)$. Prove that the following statements are equivalent: a) $C = E$ or $CE \parallel AB$ b) For each point M belonging to the segment $CD$, the following equation is satisfied $$S^2_{\vartriangle ABM}= \frac{CM^2}{CD^2}\cdot S^2_{\vartriangle ABD}+\left(1- \frac{CM^2}{CD^2} \right)S^2_{\vartriangle ABC}$$ where $S_{\vartriangle XYZ}$ means the area of ​​triangle $XYZ$.

What are the possible areas of a hexagon with all angles equal and sides $1, 2, 3, 4, 5$, and $6$, in some order?

Let $\triangle ABC$ have $AB=6$, $BC=7$, and $CA=8$, and denote by $\omega$ its circumcircle. Let $N$ be a point on $\omega$ such that $AN$ is a diameter of $\omega$. Furthermore, let the tangent to $\omega$ at $A$ intersect $BC$ at $T$, and let the second intersection point of $NT$ with $\omega$ be $X$. The length of $\overline{AX}$ can be written in the form $\tfrac m{\sqrt n}$ for positive integers $m$ and $n$, where $n$ is not divisible by the square of any prime. Find $100m+n$. [i]Proposed by David Altizio[/i]

Find the union of all projections of a given line segment $AB$ to all lines passing through a given point $O$.

Let $ABC$ be an acute, non isosceles triangle with $O,H$ are circumcenter and orthocenter, respectively. Prove that the nine-point circles of $AHO,BHO,CHO$ has two common points.

A circle of length $10$ is inscribed in a convex polygon with perimeter $15$. What part of the area of this polygon is occupied by the resulting circle?

Prove that there exists a positive real number $N$ such that for arbitrary real numbers $a,b>N$ it is possible to cover the perimeter of a rectangle with side lengths $a$ and $b$ using non-overlapping unit disks (the unit disks can be tangent to each other). [i]Submitted by Benedek Váli, Budapest[/i]

Let $ABC$ be a triangle. A point $D$ lies on line $BC$ and points $E,F$ are taken on $AC,AB$ such that $DE \parallel AB$ and $DF\parallel AC$. Let $G = (AEF) \cap (ABC) \neq A$ and $I = (DEF) \cap BC\neq D$. Let $H$ and $O$ denote the orthocenter and the circumcenter of triangle $DEF$. Prove that $A,O,I$ are collinear if and only if $G,H,I$ are collinear. [i]Proposed by Kaan Bilge[/i]

[u]Round 9[/u] [b]p25.[/b] Let $a$, $b$, and $c$ be positive numbers with $a +b +c = 4$. If $a,b,c \le 2$ and $$M =\frac{a^3 +5a}{4a^2 +2}+\frac{b^3 +5b}{4b^2 +2}+\frac{c^3 +5c}{4c^2 +2},$$ then find the maximum possible value of $\lfloor 100M \rfloor$. [b]p26.[/b] In $\vartriangle ABC$, $AB = 15$, $AC = 16$, and $BC = 17$. Points $E$ and $F$ are chosen on sides $AC$ and $AB$, respectively, such that $CE = 1$ and $BF = 3$. A point $D$ is chosen on side $BC$, and let the circumcircles of $\vartriangle BFD$ and $\vartriangle CED$ intersect at point $P \ne D$. Given that $\angle PEF = 30^o$, the length of segment $PF$ can be expressed as $\frac{m}{n}$ . Find $m+n$. [b]p27.[/b] Arnold and Barnold are playing a game with a pile of sticks with Arnold starting first. Each turn, a player can either remove $7$ sticks or $13$ sticks. If there are fewer than $7$ sticks at the start of a player’s turn, then they lose. Both players play optimally. Find the largest number of sticks under $200$ where Barnold has a winning strategy [u]Round 10[/u] [b]p28.[/b] Let $a$, $b$, and $c$ be positive real numbers such that $\log_2(a)-2 = \log_3(b) =\log_5(c)$ and $a +b = c$. What is $a +b +c$? [b]p29.[/b] Two points, $P(x, y)$ and $Q(-x, y)$ are selected on parabola $y = x^2$ such that $x > 0$ and the triangle formed by points $P$, $Q$, and the origin has equal area and perimeter. Find $y$. [b]p30.[/b] $5$ families are attending a wedding. $2$ families consist of $4$ people, $2$ families consist of $3$ people, and $1$ family consists of $2$ people. A very long row of $25$ chairs is set up for the families to sit in. Given that all members of the same family sit next to each other, let the number of ways all the people can sit in the chairs such that no two members of different families sit next to each other be $n$. Find the number of factors of $n$. [u]Round 11[/u] [b]p31.[/b] Let polynomial $P(x) = x^3 +ax^2 +bx +c$ have (not neccessarily real) roots $r_1$, $r_2$, and $r_3$. If $2ab = a^3 -20 = 6c -21$, then the value of $|r^3_1+r^3_2+r^3_3|$ can be written as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find the value of $m+n$. [b]p32.[/b] In acute $\vartriangle ABC$, let $H$, $I$ , $O$, and $G$ be the orthocenter, incenter, circumcenter, and centroid of $\vartriangle ABC$, respectively. Suppose that there exists a circle $\omega$ passing through $B$, $I$ , $H$, and $C$, the circumradius of $\vartriangle ABC$ is $312$, and $OG = 80$. Let $H'$, distinct from $H$, be the point on $\omega$ such that $\overline{HH'}$ is a diameter of $\omega$. Given that lines $H'O$ and $BC$ meet at a point $P$, find the length $OP$. [b]p33.[/b] Find the number of ordered quadruples $(x, y, z,w)$ such that $0 \le x, y, z,w \le 1000$ are integers and $$x!+ y! =2^z \cdot w!$$ holds (Note: $0! = 1$). [u]Round 12[/u] [b]p34.[/b] Let $Z$ be the product of all the answers from the teams for this question. Estimate the number of digits of $Z$. If your estimate is $E$ and the answer is $A$, your score for this problem will be $$\max \left( 0, \lceil 15- |A-E| \rceil \right).$$ Your answer must be a positive integer. [b]p35.[/b] Let $N$ be number of ordered pairs of positive integers $(x, y)$ such that $3x^2 -y^2 = 2$ and $x < 2^{75}$. Estimate $N$. If your estimate is $E$ and the answer is $A$, your score for this problem will be $$\max \left( 0, \lceil 15- 2|A-E| \rceil \right).$$ [b]p36.[/b] $30$ points are located on a circle. How many ways are there to draw any number of line segments between the points such that none of the line segments overlap and none of the points are on more than one line segment? (It is possible to draw no line segments). If your estimate is $E$ and the answer is $A$, your score for this problem will be $$\max \left( 0, \left \lceil 15- \ln \frac{A}{E} \right \rceil \right).$$ PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h3166472p28814057]here [/url] and 5-8 [url=https://artofproblemsolving.com/community/c3h3166476p28814111]here[/url].. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The ratio of the radii of two concentric circles is $1:3$. If $\overline{AC}$ is a diameter of the larger circle, $\overline{BC}$ is a chord of the larger circle that is tangent to the smaller circle, and $AB = 12$, then the radius of the larger circle is [asy] size(200); defaultpen(linewidth(0.7)+fontsize(10)); pair O=origin, A=3*dir(180), B=3*dir(140), C=3*dir(0); dot(O); draw(Arc(origin,1,0,360)); draw(Arc(origin,3,0,360)); draw(A--B--C--A); label("$A$", A, dir(O--A)); label("$B$", B, dir(O--B)); label("$C$", C, dir(O--C)); [/asy] $ \textbf{(A)}\ 13\qquad\textbf{(B)}\ 18\qquad\textbf{(C)}\ 21\qquad\textbf{(D)}\ 24\qquad\textbf{(E)}\ 26 $

Let $ABC$ be an isosceles triangle with $AB=AC.$ Let $ \Gamma $ be its circumcircle and let $O$ be the centre of $ \Gamma $ . let $CO$ meet $ \Gamma$ in $D .$ Draw a line parallel to $AC$ thrugh $D.$ Let it intersect $AB$ at $E.$ Suppose $AE : EB=2:1$ .Prove that $ABC$ is an equilateral triangle.

Using white cubes of side $1$, a prism (without holes) was assembled. The faces of the prism were painted black. It is known that the cubes left with exactly $4$ white faces are $20$ in total. Determine what the dimensions of the prism can be. Give all the possibilities.

In how many ways can we place the numbers from $1$ to $100$ in a $2\times 50$ rectangle (divided into $100$ unit squares) so that any two consecutive numbers are always placed in squares with a common side?

Let $P$ be a point in the interior of a triangle $ABC$ and let the rays $\overrightarrow{AP}, \overrightarrow{BP}$ and $\overrightarrow{CP}$ intersect the sides $BC, CA$ and $AB$ in $A_1,B_1$ and $C_1$, respectively. Let $D$ be the foot of the perpendicular from $A_1$ to $B_1C_1$. Show that \[\frac{CD}{BD}=\frac{B_1C}{BC_1} \cdot \frac{C_1A}{AB_1}.\]

A triangle $ ABC$ has an obtuse angle at $ B$. The perpindicular at $ B$ to $ AB$ meets $ AC$ at $ D$, and $ |CD| \equal{} |AB|$. Prove that $ |AD|^2 \equal{} |AB|.|BC|$ if and only if $ \angle CBD \equal{} 30^\circ$.

In $\triangle ABC$, the interior sides of which are mirrors, a laser is placed at point $A_1$ on side $BC$. A laser beam exits the point $A_1$, hits side $AC$ at point $B_1$, and then reflects off the side. (Because this is a laser beam, every time it hits a side, the angle of incidence is equal to the angle of reflection). It then hits side $AB$ at point $C_1$, then side $BC$ at point $A_2$, then side $AC$ again at point $B_2$, then side $AB$ again at point $C_2$, then side $BC$ again at point $A_3$, and finally, side $AC$ again at point $B_3$. (a) Prove that $\angle B_3A_3C = \angle B_1A_1C$. (b) Prove that such a laser exists if and only if all the angles in $\triangle ABC$ are less than $90^{\circ}$.

Let $ABC$ be a triangle and let $P$ be a point on one of the sides of $ABC$. Construct a line passing through $P$ that divides triangle $ABC$ into two parts of equal area.

Let $A$ and $B$ be two given points, distance $1 $ apart. Determine a point $P$ on the line $AB$ such that $$\frac{1}{1 + AP}+\frac{1}{1 + BP}$$ is a maximum.

Let $ ABCD $ be a rectangle that has $ M $ on its $ BD $ diagonal. If $ N,P $ are the projections of $ M $ on $ AB, $ respectively, $ AD, $ what's the locus of the intersection between $ CP $ and $ DN? $

$\Delta ABC$ is such that $AC+BC=2$ and the sum of its altitude through $C$ and its base $AB$ is $CD+AB=\sqrt{5}$. Find the sides of the triangle.

You have to divide a square paper into three parts, by two straight cuts, so that by locating these parts properly, without gaps or overlaps, an obtuse triangle is formed. Indicate how to cut the square and how to assemble the triangle with the three parts.

Let $ABCD$ be a rectangle. Two perpendicular lines pass through point $B$. One of them meets segment $AD$ at point $K$, and the second one meets the extension of side $CD$ at point $L$. Let $F$ be the common point of $KL$ and $AC$. Prove that $BF\perp KL$.

On the sides of the right triangle, outside are constructed regular nonagons, which are constructed on one of the catheti and on the hypotenuse, with areas equal to $1602$ $cm^2$ and $2019$ $cm^2$, respectively. What is the area of the nonagon that is constructed on the other cathetus of this triangle? (Vladislav Kirilyuk)