$ABC$ is a triangle, $H$ its orthocenter, $I$ its incenter, $O$ its circumcenter and $\omega$ its circumcircle. Line $CI$ intersects circle $\omega$ at point $D$ different from $C$. Assume that $AB = ID$ and $AH = OH$. Find the angles of triangle $ABC$.

The diagram below shows equilateral $\triangle ABC$ with side length $2$. Point $D$ lies on ray $\overrightarrow{BC}$ so that $CD = 4$. Points $E$ and $F$ lie on $\overline{AB}$ and $\overline{AC}$, respectively, so that $E$, $F$, and $D$ are collinear, and the area of $\triangle AEF$ is half of the area of $\triangle ABC$. Then $\tfrac{AE}{AF}=\tfrac m n$, where $m$ and $n$ are relatively prime positive integers. Find $m + 2n$. [asy] import math; size(7cm); pen dps = fontsize(10); defaultpen(dps); dotfactor=4; pair A,B,C,D,E,F; B=origin; C=(2,0); D=(6,0); A=(1,sqrt(3)); E=(1/3,sqrt(3)/3); F=extension(A,C,E,D); draw(C--A--B--D,linewidth(1.1)); draw(E--D,linewidth(.7)); dot(A); dot(B); dot(C); dot(D); dot(E); dot(F); label("$A$",A,N); label("$B$",B,S); label("$C$",C,S); label("$D$",D,S); label("$E$",E,NW); label("$F$",F,NE); [/asy]

[b]p1.[/b] Solve: $INK + INK + INK + INK + INK + INK = PEN$ ($INK$ and $PEN$ are $3$-digit numbers, and different letters stand for different digits). [b]p2. [/b]Two people play a game. They put $3$ piles of matches on the table: the first one contains $1$ match, the second one $3$ matches, and the third one $4$ matches. Then they take turns making moves. In a move, a player may take any nonzero number of matches FROM ONE PILE. The player who takes the last match from the table loses the game. a) The player who makes the first move can win the game. What is the winning first move? b) How can he win? (Describe his strategy.) [b]p3.[/b] The planet Naboo is under attack by the imperial forces. Three rebellion camps are located at the vertices of a triangle. The roads connecting the camps are along the sides of the triangle. The length of the first road is less than or equal to $20$ miles, the length of the second road is less than or equal to $30$ miles, and the length of the third road is less than or equal to $45$ miles. The Rebels have to cover the area of this triangle with a defensive field. What is the maximal area that they may need to cover? [b]p4.[/b] Money in Wonderland comes in $\$5$ and $\$7$ bills. What is the smallest amount of money you need to buy a slice of pizza that costs $\$ 1$ and get back your change in full? (The pizza man has plenty of $\$5$ and $\$7$ bills.) For example, having $\$7$ won't do, since the pizza man can only give you $\$5$ back. [b]p5.[/b] (a) Put $5$ points on the plane so that each $3$ of them are vertices of an isosceles triangle (i.e., a triangle with two equal sides), and no three points lie on the same line. (b) Do the same with $6$ points. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $D$ be a point on the side $[AB]$ of the isosceles triangle $ABC$ such that $|AB|=|AC|$. The parallel line to $BC$ passing through $D$ intersects $AC$ at $E$. If $m(\widehat A) = 20^\circ$, $|DE|=1$, $|BC|=a$, and $|BE|=a+1$, then which of the followings is equal to $|AB|$? $ \textbf{(A)}\ 2a \qquad\textbf{(B)}\ a^2-a \qquad\textbf{(C)}\ a^2+1 \qquad\textbf{(D)}\ (a+1)^2 \qquad\textbf{(E)}\ a^2+a $

Let $ABCD$ be an inscribed quadrilateral in a circumference with center $O$ such that it lies inside $ABCD$ and $\angle{BAC} = \angle{ODA}$. Let $E$ be the intersection of $AC$ with $BD$. Lines $r$ and $s$ are drawn through $E$ such that $r$ is perpendicular to $BC$, and $s$ is perpendicular to $AD$. Let $P$ be the intersection of $r$ with $AD$, and $M$ the intersection of $s$ with $BC$. Let $N$ be the midpoint of $EO$. Prove that $M$, $N$, and $P$ lie on a line.

Let $ABC$ be a triangle with $AB = 13$, $BC = 14$, and $CA = 15$. Let $D$ be the foot of the altitude from $A$ to $BC$. The inscribed circles of triangles $ABD$ and $ACD$ are tangent to $AD$ at $P$ and $Q$, respectively, and are tangent to $BC$ at $X$ and $Y$ , respectively. Let $PX$ and $QY$ meet at $Z$. Determine the area of triangle $XY Z$.

In the coordinate space, define a square $S$, defined by the inequality $|x|\leq 1,\ |y|\leq 1$ on the $xy$-plane, with four vertices $A(-1,\ 1,\ 0),\ B(1,\ 1,\ 0),\ C(1,-1,\ 0), D(-1,-1,\ 0)$. Let $V_1$ be the solid by a rotation of the square $S$ about the line $BD$ as the axis of rotation, and let $V_2$ be the solid by a rotation of the square $S$ about the line $AC$ as the axis of rotation. (1) For a real number $t$ such that $0\leq t<1$, find the area of cross section of $V_1$ cut by the plane $x=t$. (2) Find the volume of the common part of $V_1$ and $V_2$.

Incircle $\omega$ of $ABC$ touch $AC$ at $B_1$. Point $E,F$ on the $\omega$ such that $\angle AEB_1=\angle B_1FC=90$. Tangents to $\omega$ at $E,F$ intersects in $D$, and $B$ and $D$ are on different sides for line $AC$. $M$- midpoint of $AC$. Prove, that $AE,CF,DM$ intersects at one point.

Given a circle $C$ and an exterior point $A$. For every point $P$ on the circle construct the square $APQR$ (in counterclock order). Determine the locus of the point $Q$ when $P$ moves on the circle $C$.

Given an acute triangle $ABC$ with altitudes $AA_1$, $BB_1$, and $CC_1$ ($A_1 \in BC$, $B_1 \in AC$, $C_1 \in AB$) and circumcircle $k$, the rays $B_1A_1$, $C_1B_1$, and $A_1C_1$ meet $k$ at points $A_2$, $B_2$, and $C_2$, respectively. Find the maximum possible value of \[ \sin \angle ABB_2 \cdot \sin \angle BCC_2 \cdot \sin \angle CAA_2 \] and all acute triangles $ABC$ for which it is achieved.

Two squares with side $12$ lie exactly on top of each other. One square is rotated around a corner point through an angle of $30$ degrees relative to the other square. Determine the area of the common piece of the two squares. [asy] unitsize (2 cm); pair A, B, C, D, Bp, Cp, Dp, P; A = (0,0); B = (-1,0); C = (-1,1); D = (0,1); Bp = rotate(-30)*(B); Cp = rotate(-30)*(C); Dp = rotate(-30)*(D); P = extension(C, D, Bp, Cp); fill(A--Bp--P--D--cycle, gray(0.8)); draw(A--B--C--D--cycle); draw(A--Bp--Cp--Dp--cycle); label("$30^\circ$", (-0.5,0.1), fontsize(10)); [/asy]

$4.$ How many line segments have both their endpoints located at the vertices of a given cube $?$

In a right triangle $ABC$ with a right angle $C$, let $AK$ and $BN$ be the angle bisectors. Let $D,E$ be the projections of $C$ on $AK, BN$ respectively. Prove that the length of the segment $DE$ is equal to the radius of the inscribed circle.

Around triangle $ABC$ with acute angle C is circumscribed a circle. On the arc $AB$, which does not contain point $C$, point $D$ is chosen. Point $D'$ is symmetric on point $D$ with respect to line $AB$. Straight lines $AD'$ and $BD'$ intersect segments $BC$ and $AC$ at points $E$ and $F$. Let point $C$ move along its arc $AB$. Prove that the center of the circumscribed circle of a triangle $CEF$ moves on a straight line.

In rhombus $NIMO$, $MN = 150\sqrt{3}$ and $\measuredangle MON = 60^{\circ}$. Denote by $S$ the locus of points $P$ in the interior of $NIMO$ such that $\angle MPO \cong \angle NPO$. Find the greatest integer not exceeding the perimeter of $S$. [i]Proposed by Evan Chen[/i]

Let $\omega$ be a circle with centre $O$, and let $\ell$ be a line that does not intersect $\omega$. Let $P$ be an arbitrary point on $\ell$. Let $A,B$ denote the tangent points of the tangent lines from $P$. Prove that $AB$ passes through a point being independent of choosing $P$.

Let $\triangle{ABC}$ be a triangle with integer side lengths and the property that $\angle{B} = 2\angle{A}$. What is the least possible perimeter of such a triangle? $ \textbf{(A) }13 \qquad \textbf{(B) }14 \qquad \textbf{(C) }15 \qquad \textbf{(D) }16 \qquad \textbf{(E) }17 \qquad $

Let $ABCDE$ be a convex pentagon such that $AB=BC=CD$, $\angle{EAB}=\angle{BCD}$, and $\angle{EDC}=\angle{CBA}$. Prove that the perpendicular line from $E$ to $BC$ and the line segments $AC$ and $BD$ are concurrent.

A trapezoid $ABCD$ ($AB || CF$,$AB > CD$) is circumscribed.The incircle of the triangle $ABC$ touches the lines $AB$ and $AC$ at the points $M$ and $N$,respectively.Prove that the incenter of the trapezoid $ABCD$ lies on the line $MN$.

Points $M$ and $N$ lie on the side $BC$ of the regular triangle $ABC$ ($M$ is between $B$ and $N$), and $\angle MAN=30^\circ.$ The circumcircles of triangles $AMC$ and $ANB$ meet at a point $K.$ Prove that the line $AK$ passes through the circumcenter of triangle $AMN.$

Find an acutangle triangle such that its sides and altitudes have integer length.

[b]p1.[/b] The number $2020$ is very special: the sum of its digits is equal to the product of its nonzero digits. How many such four digit numbers are there? (Numbers with only one nonzero digit, like $3000$, also count) [b]p2.[/b] A locker has a combination which is a sequence of three integers between $ 0$ and $49$, inclusive. It is known that all of the numbers in the combination are even. Let the total of a lock combination be the sum of the three numbers. Given that the product of the numbers in the combination is $12160$, what is the sum of all possible totals of the locker combination? [b]p3.[/b] Given points $A = (0, 0)$ and $B = (0, 1)$ in the plane, the set of all points P in the plane such that triangle $ABP$ is isosceles partitions the plane into $k$ regions. The sum of the areas of those regions that are bounded is $s$. Find $ks$. [b]p4.[/b] Three families sit down around a circular table, each person choosing their seat at random. One family has two members, while the other two families have three members. What is the probability that every person sits next to at least one person from a different family? [b]p5.[/b] Jacob and Alexander are walking up an escalator in the airport. Jacob walks twice as fast as Alexander, who takes $18$ steps to arrive at the top. Jacob, however, takes $27$ steps to arrive at the top. How many of the upward moving escalator steps are visible at any point in time? [b]p6.[/b] Points $A, B, C, D, E$ lie in that order on a circle such that $AB = BC = 5$, $CD = DE = 8$, and $\angle BCD = 150^o$ . Let $AD$ and $BE$ intersect at $P$. Find the area of quadrilateral $PBCD$. [b]p7.[/b] Ivan has a triangle of integers with one number in the first row, two numbers in the second row, and continues up to eight numbers in the eighth row. He starts with the first $8$ primes, $2$ through $19$, in the bottom row. Each subsequent row is filled in by writing the least common multiple of two adjacent numbers in the row directly below. For example, the second last row starts with$ 6, 15, 35$, etc. Let P be the product of all the numbers in this triangle. Suppose that P is a multiple of $a/b$, where $a$ and $b$ are positive integers and $a > 1$. Given that $b$ is maximized, and for this value of $b, a$ is also maximized, find $a + b$. [b]p8.[/b] Let $ABCD$ be a cyclic quadrilateral. Given that triangle $ABD$ is equilateral, $\angle CBD = 15^o$, and $AC = 1$, what is the area of $ABCD$? [b]p9.[/b] Let $S$ be the set of all integers greater than $ 1$. The function f is defined on $S$ and each value of $f$ is in $S$. Given that $f$ is nondecreasing and $f(f(x)) = 2x$ for all $x$ in $S$, find $f(100)$. [b]p10.[/b] An [i]origin-symmetric[/i] parallelogram $P$ (that is, if $(x, y)$ is in $P$, then so is $(-x, -y)$) lies in the coordinate plane. It is given that P has two horizontal sides, with a distance of $2020$ between them, and that there is no point with integer coordinates except the origin inside $P$. Also, $P$ has the maximum possible area satisfying the above conditions. The coordinates of the four vertices of P are $(a, 1010)$, $(b, 1010)$, $(-a, -1010)$, $(-b, -1010)$, where a, b are positive real numbers with $a < b$. What is $b$? [b]p11.[/b] What is the remainder when $5^{200} + 5^{50} + 2$ is divided by $(5 + 1)(5^2 + 1)(5^4 + 1)$? [b]p12.[/b] Let $f(n) = n^2 - 4096n - 2045$. What is the remainder when $f(f(f(... f(2046)...)))$ is divided by $2047$, where the function $f$ is applied $47$ times? [b]p13.[/b] What is the largest possible area of a triangle that lies completely within a $97$-dimensional hypercube of side length $1$, where its vertices are three of the vertices of the hypercube? [b]p14.[/b] Let $N = \left \lfloor \frac{1}{61} \right \rfloor + \left \lfloor\frac{3}{61} \right \rfloor+\left \lfloor \frac{3^2}{61} \right \rfloor+... +\left \lfloor\frac{3^{2019}}{61} \right \rfloor$. Given that $122N$ can be expressed as $3^a - b$, where $a, b$ are positive integers and $a$ is as large as possible, find $a + b$. Note: $\lfloor x \rfloor$ is defined as the greatest integer less than or equal to $x$. [b]p15.[/b] Among all ordered triples of integers $(x, y, z)$ that satisfy $x + y + z = 8$ and $x^3 + y^3 + z^3 = 134$, what is the maximum possible value of $|x| + |y| + |z|$? PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

An ellipse with foci $B_0,B_1$ intersects $AB_i$ at $C_i$ $(i=0,1)$. Let $P_0$ be a point on ray $AB_0$. $Q_0$ is a point on ray $C_1B_0$ such that $B_0P_0=B_0Q_0$; $P_1$ is on ray $B_1A$ such that $C_1Q_0=C_1P_1$; $Q_1$ is on ray $B_1C_0$ such that $B_1P_1=B_1Q_1$; $P_2$ is on ray $AB_0$ such that $C_0Q_1=C_0Q_2$. Prove that $P_0=P_2$ and that the four points $P_0,Q_0,Q_1,P_1$ are concyclic.

$BB_1$ and $CC_1$ are altitudes in $\Delta ABC$. Let $B_1 C_1$ intersect the circumscribed circle of $\Delta ABC$ in points $E$ and $F$. Let $k$ be a circle passing through $E$ and $F$ in such way that the center of $k$ lies on the arc $\widehat{BAC}$. We denote with $M$ the middle point of $BC$. $X$ and $Y$ are the points on $k$ for which $MX$ and $MY$ are tangent to $k$. Let $EX\cap FY=S_1,EY\cap FX=S_2,BX\cap CY=U,$ and $BY\cap CX=V$. Prove that $S_1 S_2$ and $UV$ intersect in the orthocenter of $\Delta ABC$.

An equilateral triangle has side length $ 6$. What is the area of the region containing all points that are outside the triangle and not more than $ 3$ units from a point of the triangle? $ \textbf{(A)}\ 36\plus{}24\sqrt{3} \qquad \textbf{(B)}\ 54\plus{}9\pi \qquad \textbf{(C)}\ 54\plus{}18\sqrt{3}\plus{}6\pi \qquad \textbf{(D)}\ \left(2\sqrt{3}\plus{}3\right)^2\pi \\ \textbf{(E)}\ 9\left(\sqrt{3}\plus{}1\right)^2\pi$