In scalene $\triangle ABC$, $I$ is the incenter, $I_a$ is the $A$-excenter, $D$ is the midpoint of arc $BC$ of the circumcircle of $ABC$ not containing $A$, and $M$ is the midpoint of side $BC$. Extend ray $IM$ past $M$ to point $P$ such that $IM = MP$. Let $Q$ be the intersection of $DP$ and $MI_a$, and $R$ be the point on the line $MI_a$ such that $AR\parallel DP$. Given that $\frac{AI_a}{AI}=9$, the ratio $\frac{QM} {RI_a}$ can be expressed in the form $\frac{m}{n}$ for two relatively prime positive integers $m,n$. Compute $m+n$. [i]Ray Li.[/i] [hide="Clarifications"][list=1][*]"Arc $BC$ of the circumcircle" means "the arc with endpoints $B$ and $C$ not containing $A$".[/list][/hide]

Consider $ 8$ points that are a knight’s move away from the origin (i.e., the eight points $\{(2, 1)$ , $(2, -1)$ , $(1, 2)$ , $(1, -2)$ , $(-1, 2)$ , $(-1, -2)$ , $(-2, 1)$, $(-2, -1)\}$). Each point has probability $\frac12$ of being visible. What is the expected value of the area of the polygon formed by points that are visible? (If exactly $0, 1, 2$ points appear, this area will be zero.)

[u]Part 6 [/u] [b]p16.[/b] Le[b][/b]t $p(x)$ and $q(x)$ be polynomials with integer coefficients satisfying $p(1) = q(1)$. Find the greatest integer $n$ such that $\frac{p(2023)-q(2023)}{n}$ is an integer no matter what $p(x)$ and $q(x)$ are. [b]p17.[/b] Find all ordered pairs of integers $(m,n)$ that satisfy $n^3 +m^3 +231 = n^2m^2 +nm.$ [b]p18.[/b] Ben rolls the frustum-shaped piece of candy (shown below) in such a way that the lateral area is always in contact with the table. He rolls the candy until it returns to its original position and orientation. Given that $AB = 4$ and $BD =CD = 3$, find the length of the path traced by $A$. [u]Part 7 [/u] [b]p19.[/b] In their science class, Adam, Chris, Eddie and Sam are independently and randomly assigned an integer grade between $70$ and $79$ inclusive. Given that they each have a distinct grade, what is the expected value of the maximum grade among their four grades? [b]p20.[/b] Let $ABCD$ be a regular tetrahedron with side length $2$. Let point $E$ be the foot of the perpendicular from $D$ to the plane containing $\vartriangle ABC$. There exist two distinct spheres $\omega_1$ and $\omega_2$, centered at points $O_1$ and $O_2$ respectively, such that both $O_1$ and $O_2$ lie on $\overrightarrow{DE}$ and both spheres are tangent to all four of the planes $ABC$, $BCD$, $CDA$, and $DAB$. Find the sum of the volumes of $\omega_1$ and $\omega_2$. [b]p21.[/b] Evaluate $$\sum^{\infty}_{i=0}\sum^{\infty}_{j=0}\sum^{\infty}_{k=0} \frac{1}{(i + j +k +1)2^{i+j+k+1}}.$$ [u]Part 8 [/u] [b]p22.[/b] In $\vartriangle ABC$, let $I_A$, $I_B$ , and $I_C$ denote the $A$, $B$, and $C$-excenters, respectively. Given that $AB = 15$, $BC = 14$ and $C A = 13$, find $\frac{[I_A I_B I_C ]}{[ABC]}$ . [b]p23.[/b] The polynomial $x +2x^2 +3x^3 +4x^4 +5x^5 +6x^6 +5x^7 +4x^8 +3x^9 +2x^{10} +x^{11}$ has distinct complex roots $z_1, z_2, ..., z_n$. Find $$\sum^n_{k=1} |R(z^2n))|+|I(z^2n)|,$$ where $R(z)$ and $I(z)$ indicate the real and imaginary parts of $z$, respectively. Express your answer in simplest radical form. [b]p24.[/b] Given that $\sin 33^o +2\sin 161^o \cdot \sin 38^o = \sin n^o$ , compute the least positive integer value of $n$. [u]Part 9[/u] [b]p25.[/b] Submit a prime between $2$ and $2023$, inclusive. If you don’t, or if you submit the same number as another team’s submission, you will receive $0$ points. Otherwise, your score will be $\min \left(30, \lfloor 4 \cdot ln(x) \rfloor \right)$, where $x$ is the positive difference between your submission and the closest valid submission made by another team. [b]p26.[/b] Sam, Derek, Jacob, andMuztaba are eating a very large pizza with $2023$ slices. Due to dietary preferences, Sam will only eat an even number of slices, Derek will only eat a multiple of $3$ slices, Jacob will only eat a multiple of $5$ slices, andMuztaba will only eat a multiple of $7$ slices. How many ways are there for Sam, Derek, Jacob, andMuztaba to eat the pizza, given that all slices are identical and order of slices eaten is irrelevant? If your answer is $A$ and the correct answer is $C$, the number of points you receive will be: irrelevant? If your answer is $A$ and the correct answer is $C$, the number of points you receive will be: $$\max \left( 0, \left\lfloor 30 \left( 1-2\sqrt{\frac{|A-C|}{C}}\right)\right\rfloor \right)$$ [b]p27.[/b] Let $ \Omega_(k)$ denote the number of perfect square divisors of $k$. Compute $$\sum^{10000}_{k=1} \Omega_(k).$$ If your answer is $A$ and the correct answer is $C$, the number of points you recieve will be $$\max \left( 0, \left\lfloor 30 \left( 1-4\sqrt{\frac{|A-C|}{C}}\right)\right\rfloor \right)$$ PS. You should use hide for answers. Rounds 1-5 have been posted [url=https://artofproblemsolving.com/community/c3h3267911p30056982]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In a triangle $ABC$ points $A_1,B_1,C_1$ are the midpoints of $BC,CA,AB$ respectively,and $A_2,B_2,C_2$ are the midpoints of the altitudes from $A,B,C$ respectively. Show that the lines $A_1A_2,B_1B_2,C_1,C_2$ are concurrent.

A circle $\omega$ passes through the two vertices $B$ and $C$ of a triangle $ABC.$ Furthermore, $\omega$ intersects segment $AC$ in $D\ne C$ and segment $AB$ in $E\ne B.$ On the ray from $B$ through $D$ lies a point $K$ such that $|BK| = |AC|,$ and on the ray from $C$ through $E$ lies a point $L$ such that $|CL| = |AB|.$ Show that the circumcentre $O$ of triangle $AKL$ lies on $\omega$.

The triangle with side lengths $3, 5$, and $k$ has area $6$ for two distinct values of $k$: $x$ and $y$. Compute $|x^2 -y^2|$.

Points $K$, $L$, $M$ lie on the sides $BC$, $CA$, $AB$ of equilateral triangle $ABC$ respectively, and satisfy the conditions $KM=LM$, $\angle KML=90^\circ$, and $AM=BK$. Prove that $\angle CKL=90^\circ$.

Let $C$ be a closed convex curve with a continuously turning tangent and let $O$ be a point inside $C.$ For each point $P$ on $C$ we define $T(P)$ as follows: Draw the tangent to $C$ at $P$ and from $O$ drop the perpendicular to that tangent. Then $T(P)$ is the point at which $C$ intersects this perpendicular. Starting now with a point $P_{0}$ on $C$, define points $P_n$ by $P_n =T(P_{n-1}).$ Prove that the points $P_{n}$ approach a limit and characterize all possible limit points. (You may assume that $T$ is continuous.)

A circle is circumscribed around triangle $ABC$ and a circle is inscribed in it, which touches the sides of the triangle $BC,CA,AB$ at points $A_1,B_1,C_1$, respectively. The line $B_1C_1$ intersects the line $BC$ at the point $P$, and $M$ is the midpoint of the segment $PA_1$. Prove that the segments of the tangents drawn from the point $M$ to the inscribed and circumscribed circle are equal.

[i]25 problems for 30 minutes[/i] [b]p1.[/b] You and nine friends spend $4000$ dollars on tickets to attend the new Harry Styles concert. Unfortunately, six friends cancel last minute due to the u. You and your remaining friends still attend the concert and split the original cost of $4000$ dollars equally. What percent of the total cost does each remaining individual have to pay? [b]p2.[/b] Find the number distinct $4$ digit numbers that can be formed by arranging the digits of $2021$. [b]p3.[/b] On a plane, Darnay draws a triangle and a rectangle such that each side of the triangle intersects each side of the rectangle at no more than one point. What is the largest possible number of points of intersection of the two shapes? [b]p4.[/b] Joy is thinking of a two-digit number. Her hint is that her number is the sum of two $2$-digit perfect squares $x_1$ and $x_2$ such that exactly one of $x_i - 1$ and $x_i + 1$ is prime for each $i = 1, 2$. What is Joy's number? [b]p5.[/b] At the North Pole, ice tends to grow in parallelogram structures of area $60$. On the other hand, at the South Pole, ice grows in right triangular structures, in which each triangular and parallelogram structure have the same area. If every ice triangle $ABC$ has legs $\overline{AB}$ and $\overline{AC}$ that are integer lengths, how many distinct possible lengths are there for the hypotenuse $\overline{BC}$? [b]p6.[/b] Carlsen has some squares and equilateral triangles, all of side length $1$. When he adds up the interior angles of all shapes, he gets $1800^o$. When he adds up the perimeters of all shapes, he gets $24$. How many squares does he have? [b]p7.[/b] Vijay wants to hide his gold bars by melting and mixing them into a water bottle. He adds $100$ grams of liquid gold to $100$ grams of water. His liquefied gold bars have a density of $20$ g/ml and water has a density of $1$ g/ml. Given that the density of the mixture in g/mL can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, compute the sum $m + n$. (Note: density is mass divided by volume, gram (g) is unit of mass and ml is unit of volume. Further, assume the volume of the mixture is the sum of the volumes of the components.) [b]p8.[/b] Julius Caesar has epilepsy. Specifically, if he sees $3$ or more flashes of light within a $0.1$ second time frame, he will have a seizure. His enemy Brutus has imprisoned him in a room with $4$ screens, which flash exactly every $4$, $5$, $6$, and $7$ seconds, respectively. The screens all flash at once, and $105$ seconds later, Caesar opens his eyes. How many seconds after he opened his eyes will Caesar first get a seizure? [b]p9.[/b] Angela has a large collection of glass statues. One day, she was bored and decided to use some of her statues to create an entirely new one. She melted a sphere with radius $12$ and a cone with height of 18 and base radius of $2$. If Angela wishes to create a new cone with a base radius $2$, what would the the height of the newly created cone be? [b]p10.[/b] Find the smallest positive integer $N$ satisfying these properties: (a) No perfect square besides $1$ divides $N$. (b) $N$ has exactly $16$ positive integer factors. [b]p11.[/b] The probability of a basketball player making a free throw is $\frac15$. The probability that she gets exactly $2$ out of $4$ free throws in her next game can be expressed as $\frac{m}{n}$ for relatively prime positive integers m and n. Find $m + n$. [b]p12.[/b] A new donut shop has $1000$ boxes of donuts and $1000$ customers arriving. The boxes are numbered $1$ to $1000$. Initially, all boxes are lined up by increasing numbering and closed. On the first day of opening, the first customer enters the shop and opens all the boxes for taste testing. On the second day of opening, the second customer enters and closes every box with an even number. The third customer then "reverses" (if closed, they open it and if open, they close it) every box numbered with a multiple of three, and so on, until all $1000$ customers get kicked out for having entered the shop and reversing their set of boxes. What is the number on the sixth box that is left open? [b]p13.[/b] For an assignment in his math class, Michael must stare at an analog clock for a period of $7$ hours. He must record the times at which the minute hand and hour hand form an angle of exactly $90^o$, and he will receive $1$ point for every time he records correctly. What is the maximum number of points Michael can earn on his assignment? [b]p14.[/b] The graphs of $y = x^3 +5x^2 +4x-3$ and $y = -\frac15 x+1$ intersect at three points in the Cartesian plane. Find the sum of the $y$-coordinates of these three points. [b]p15.[/b] In the quarterfinals of a single elimination countdown competition, the $8$ competitors are all of equal skill. When any $2$ of them compete, there is exactly a $50\%$ chance of either one winning. If the initial bracket is randomized, the probability that two of the competitors, Daniel and Anish, face off in one of the rounds can be expressed as $\frac{p}{q}$ for relatively prime positive integers $p$, $q$. Find $p + q$. [b]p16.[/b] How many positive integers less than or equal to $1000$ are not divisible by any of the numbers $2$, $3$, $5$ and $11$? [b]p17.[/b] A strictly increasing geometric sequence of positive integers $a_1, a_2, a_3,...$ satisfies the following properties: (a) Each term leaves a common remainder when divided by $7$ (b) The first term is an integer from $1$ to $6$ (c) The common ratio is an perfect square Let $N$ be the smallest possible value of $\frac{a_{2021}}{a_1}$. Find the remainder when $N$ is divided by $100$. [b]p18.[/b] Suppose $p(x) = x^3 - 11x^2 + 36x - 36$ has roots $r, s,t$. Find %\frac{r^2 + s^2}{t}+\frac{s^2 + t^2}{r}+\frac{t^2 + r^2}{s}%. [b]p19.[/b] Let $a, b \le 2021$ be positive integers. Given that $ab^2$ and $a^2b$ are both perfect squares, let $G = gcd(a, b)$. Find the sum of all possible values of $G$. [b]p20.[/b] Jessica rolls six fair standard six-sided dice at the same time. Given that she rolled at least four $2$'s and exactly one $3$, the probability that all six dice display prime numbers can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$, $n$. What is $m + n$? [b]p21.[/b] Let $a, b, c$ be numbers such $a + b + c$ is real and the following equations hold: $$a^3 + b^3 + c^3 = 25$$ $$\frac{1}{ab}+\frac{1}{bc}+\frac{1}{ac}= 1$$ $$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{25}{9}$$ The value of $a + b + c$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$, $n$. Find $m + n$. [b]p22.[/b] Let $\omega$ be a circle and $P$ be a point outside $\omega$. Let line $\ell$ pass through $P$ and intersect $\omega$ at points $A,B$ and with $PA < PB$ and let $m$ be another line passing through $P$ intersecting $\omega$ at points $C,D$ with $PC < PD$. Let X be the intersection of $AD$ and $BC$. Given that $\frac{PC}{CD}=\frac23$, $\frac{PC}{PA}=\frac45$, and $\frac{[ABC]}{[ACD]}=\frac79$,the value of $\frac{[BXD]}{[BXA]}$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m, n$: Find $m + n$. [b]p23.[/b] Define the operation $a \circ b =\frac{a^2 + 2ab + a - 12}{b}$. Given that $1 \circ (2 \circ (3 \circ (... 2019 \circ (2020 \circ 2021)))...)$ can be expressed as $-\frac{a}{b}$ for some relatively prime positive integers $a,b$, compute $a + b$. [b]p24.[/b] Find the largest integer $n \le 2021$ for which $5^{n-3} | (n!)^4$ [b]p25.[/b] On the Cartesian plane, a line $\ell$ intersects a parabola with a vertical axis of symmetry at $(0, 5)$ and $(4, 4)$. The focus $F$ of the parabola lies below $\ell$, and the distance from $F$ to $\ell$ is $\frac{16}{\sqrt{17}}$. Let the vertex of the parabola be $(x, y)$. The sum of all possible values of $y$ can be expressed as $\frac{p}{q}$ for relatively prime positive integers $p, q$. Find $p + q$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Given isosceles $\triangle ABC$ with $AB = AC$ and $\angle BAC = 22^{\circ}$. On the side $BC$ point $D$ is chosen such that $BD = 2CD$. The foots of perpendiculars from $B$ to lines $AD$ and $AC$ are points $E$, $F$ respectively. Find with the proof value of the angle $\angle CEF$.

Consider a circle and a point $A$ outside it. We start moving from $A$ along a closed broken line consisting of segments of tangents to the circle (the segment itself should not necessarily be tangent to the circle) and terminate back at $A$. (On the links of the broken line are solid.) We label parts of the segments with a plus sign if we approach the circle and with a minus sign otherwise. Prove that the sum of the lengths of the segments of our path, with the signs given, is zero. [img]https://cdn.artofproblemsolving.com/attachments/3/0/8d682813cf7dfc88af9314498b9afcecdf77d2.png[/img]

Let $O$ be the circumcentre of a triangle $ABC$ with $AB > AC > BC$. Let $D$ be a point on the minor arc $BC$ of the circumcircle and let $E$ and $F$ be points on $AD$ such that $AB \perp OE$ and $AC \perp OF$ . The lines $BE$ and $CF$ meet at $P$. Prove that if $PB=PC+PO$, then $\angle BAC = 30^{\circ}$.

[u]Set 8[/u] [b]p22.[/b] For monic quadratic polynomials $P = x^2 + ax + b$ and $Q = x^2 + cx + d$, where $1 \le a, b, c, d \le 10$ are integers, we say that $P$ and $Q$ are friends if there exists an integer $1 \le n \le 10$ such that $P(n) = Q(n)$. Find the total number of ordered pairs $(P, Q)$ of such quadratic polynomials that are friends. [b]p23.[/b] A three-dimensional solid has six vertices and eight faces. Two of these faces are parallel equilateral triangles with side length $1$, $\vartriangle A_1A_2A_3$ and $\vartriangle B_1B_2B_3$. The other six faces are isosceles right triangles — $\vartriangle A_1B_2A_3$, $\vartriangle A_2B_3A_1$, $\vartriangle A_3B_1A_2$, $\vartriangle B_1A_2B_3$, $\vartriangle B_2A_3B_1$, $\vartriangle B_3A_1B_2$ — each with a right angle at the second vertex listed (so for instace $\vartriangle A_1B_2A_3$ has a right angle at $B_2$). Find the volume of this solid. [b]p24.[/b] The digits $0, 1, 2, 3, 4, 5, 6, 7, 8, 9$ are each colored red, blue, or green. Find the number of colorings such that any integer $ n \ge 2$ has that (a) If $n$ is prime, then at least one digit of $n$ is not blue. (b) If $n$ is composite, then at least one digit of $n$ is not green. PS. You should use hide for answers.

Among all triangles with a given perimeter, find the one with the maximal radius of its incircle.

Let $AB$ be a diameter of a circle and let $C$ be a point on the segement $AB$ such that $AC : CB = 6 : 7$. Let $D$ be a point on the circle such that $DC$ is perpendicular to $AB$. Let $DE$ be the diameter through $D$. If $[XYZ]$ denotes the area of the triangle $XYZ$, find $[ABD]/[CDE]$ to the nearest integer.

[b]9.[/b] Lep $p$ be a connected non-closed broken line without self-intersection in the plane $\varphi $. Prove that if $v$ is a non-zero vector in $\varphi $ and $p$ has a commom point with the broken line $p+v$, then $p$ has a common point with the broken line $p+\alpha v$ too, where $\alpha =\frac{1}{n}$ and $n$ is a positive integer. Does a similar statemente hold for other positive values of $\alpha$? ($p+v$ denotes the broken line obtained from $p$ through displacemente by the vector $v$.) [b](G. 1)[/b]

In a certain circle, the chord of a $d$-degree arc is 22 centimeters long, and the chord of a $2d$-degree arc is 20 centimeters longer than the chord of a $3d$-degree arc, where $d<120.$ The length of the chord of a $3d$-degree arc is $-m+\sqrt{n}$ centimeters, where $m$ and $n$ are positive integers. Find $m+n.$

Let $ABC$ be a triangle with $\angle B > \angle C$. Let $P$ and $Q$ be two different points on line $AC$ such that $\angle PBA = \angle QBA = \angle ACB $ and $A$ is located between $P$ and $C$. Suppose that there exists an interior point $D$ of segment $BQ$ for which $PD=PB$. Let the ray $AD$ intersect the circle $ABC$ at $R \neq A$. Prove that $QB = QR$.

[b]p1.[/b] The Evergreen School booked buses for a field trip. Altogether, $138$ people went to West Lake, while $115$ people went to East Lake. The buses all had the same number of seats and every bus has more than one seat. All seats were occupied and everybody had a seat. How many seats were on each bus? [b]p2.[/b] In New Scotland there are three kinds of coins: $1$ cent, $6$ cent, and $36$ cent coins. Josh has $99$ of the $36$-cent coins (and no other coins). He is allowed to exchange a $36$ cent coin for $6$ coins of $6$ cents, and to exchange a $6$ cent coin for $6$ coins of $1$ cent. Is it possible that after several exchanges Josh will have $500$ coins? [b]p3.[/b] Find all solutions $a, b, c, d, e, f, g, h$ if these letters represent distinct digits and the following multiplication is correct: $\begin{tabular}{ccccc} & & a & b & c \\ + & & & d & e \\ \hline & f & a & g & c \\ x & b & b & h & \\ \hline f & f & e & g & c \\ \end{tabular}$ [b]p4.[/b] Is it possible to find a rectangle of perimeter $10$ m and cut it in rectangles (as many as you want) so that the sum of the perimeters is $500$ m? [b]p5.[/b] The picture shows a maze with chambers (shown as circles) and passageways (shown as segments). A cat located in chamber $C$ tries to catch a mouse that was originally in the chamber $M$. The cat makes the first move, moving from chamber $C$ to one of the neighboring chambers. Then the mouse moves, then the cat, and so forth. At each step, the cat and the mouse can move to any neighboring chamber or not move at all. The cat catches the mouse by moving into the chamber currently occupied by the mouse. Can the cat get the mouse? [img]https://cdn.artofproblemsolving.com/attachments/9/9/25f61e1499ff1cfeea591cb436d33eb2cdd682.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Two circles of equal radius can tightly fit inside right triangle $ABC$, which has $AB=13$, $BC=12$, and $CA=5$, in the three positions illustrated below. Determine the radii of the circles in each case. [asy] size(400); defaultpen(linewidth(0.7)+fontsize(12)); picture p = new picture; pair s1 = (20,0), s2 = (40,0); real r1 = 1.5, r2 = 10/9, r3 = 26/7; pair A=(12,5), B=(0,0), C=(12,0); draw(p,A--B--C--cycle); label(p,"$B$",B,SW); label(p,"$A$",A,NE); label(p,"$C$",C,SE); add(p); add(shift(s1)*p); add(shift(s2)*p); draw(circle(C+(-r1,r1),r1)); draw(circle(C+(-3*r1,r1),r1)); draw(circle(s1+C+(-r2,r2),r2)); draw(circle(s1+C+(-r2,3*r2),r2)); pair D=s2+156/17*(A-B)/abs(A-B), E=s2+(169/17,0), F=extension(D,E,s2+A,s2+C); draw(incircle(s2+B,D,E)); draw(incircle(s2+A,D,F)); label("Case (i)",(6,-3)); label("Case (ii)",s1+(6,-3)); label("Case (iii)",s2+(6,-3));[/asy]

The convex quadrilateral $ABCD$ has area $1$, and $AB$ is produced to $E$, $BC$ to $F$, $CD$ to $G$ and $DA$ to $H$, such that $AB=BE$, $BC=CF$, $CD=DG$ and $DA=AH$. Find the area of the quadrilateral $EFGH$.

If the sum of the lengths of the six edges of a trirectangular tetrahedron $ PABC$ (i.e., $ \angle APB \equal{} \angle BPC \equal{} \angle CPA \equal{} 90^\circ$) is $ S$, determine its maximum volume.

In an acute triangle the feet of altitudes drawn from vertices $A$ and $B$ are $D$ and $E$, respectively. Let $M$ be the midpoint of side $AB$. Line $CM$ intersects the circumcircle of $CDE$ again in point $P$ and the circumcircle of $CAB$ again in point $Q$. Prove that $|MP| = |MQ|$.

Given a scalene acute triangle $ ABC$ with $ AC>BC$ let $ F$ be the foot of the altitude from $ C$. Let $ P$ be a point on $ AB$, different from $ A$ so that $ AF\equal{}PF$. Let $ H,O,M$ be the orthocenter, circumcenter and midpoint of $ [AC]$. Let $ X$ be the intersection point of $ BC$ and $ HP$. Let $ Y$ be the intersection point of $ OM$ and $ FX$ and let $ OF$ intersect $ AC$ at $ Z$. Prove that $ F,M,Y,Z$ are concyclic.