Let $ABC$ be a triangle with circumcircle $\Omega$, and let $P$ be a point on the arc $BC$ of $\Omega$ not containing $A$. Let $\omega_B$ and $\omega_C$ be circles respectively passing through $B$ and $C$ and such that both of them are tangent to line $AP$ at point $P$. Let $R$, $R_B$, $R_C$ be the radii of $\Omega$, $\omega_B$, and $\omega_C$, respectively. Prove that if $h$ is the distance from $A$ to line $BC$, then \[ \frac{R_B+R_C}{R} \le \frac{BC}{h}. \]

Let $ABCDEF$ be a convex hexagon that does not have two parallel sides, such that $\angle AF B = \angle F DE, \angle DF E = \angle BDC$ and $\angle BFC = \angle ADF.$ Prove that the lines $ AB, FC$ and $DE$ are concurrent if and only if the lines $ AF, BE$ and $CD$ are concurrent.

There are $2020$ points on the coordinate plane {$A_i = (x_i, y_i):i = 1, ..., 2020$}, satisfying $$0=x_1<x_2<...<x_{2020}$$ $$0=y_{2020}<y_{2019}<...<y_1$$ Let $O=(0, 0)$ be the origin, $OA_1A_2...A_{2020}$ forms a polygon $C$. Now, you want to blacken the polygon $C$. Every time you can choose a point $(x,y)$ with $x, y > 0$, and blacken the area {$(x', y'): 0\leq x' \leq x, 0\leq y' \leq y$}. However, you have to pay $xy$ dollars for doing so. Prove that you could blacken the whole polygon $C$ by using $4|C|$ dollars. Here, $|C|$ stands for the area of the polygon $C$. [i]Proposed by me[/i]

Let $M$ be the midpoint of the side $AC$ of $ \triangle ABC$. Let $P\in AM$ and $Q\in CM$ be such that $PQ=\frac{AC}{2}$. Let $(ABQ)$ intersect with $BC$ at $X\not= B$ and $(BCP)$ intersect with $BA$ at $Y\not= B$. Prove that the quadrilateral $BXMY$ is cyclic. [i]F. Ivlev, F. Nilov[/i]

Let $ABC$ be a triangle with $AB = AC \neq BC$ and let $I$ be its incentre. The line $BI$ meets $AC$ at $D$, and the line through $D$ perpendicular to $AC$ meets $AI$ at $E$. Prove that the reflection of $I$ in $AC$ lies on the circumcircle of triangle $BDE$.

Let $\Gamma$ be a circle of center $O$, and $\delta$. be a line in the plane of $\Gamma$, not intersecting it. Denote by $A$ the foot of the perpendicular from $O$ onto $\delta$., and let $M$ be a (variable) point on $\Gamma$. Denote by $\gamma$ the circle of diameter $AM$ , by $X$ the (other than M ) intersection point of $\gamma$ and $\Gamma$, and by $Y$ the (other than $A$) intersection point of $\gamma$ and $\delta$. Prove that the line $XY$ passes through a fixed point.

Given an acute triangle $T$ with sides $a,b,c$, find the tetrahedra with base $T$ whose all faces are acute triangles of the same area.

In a cyclic quadrilateral $ ABCD$ the sides $ AB$ and $ AD$ are equal, $ CD>AB\plus{}BC$. Prove that $ \angle ABC>120^\circ$.

If a given equilateral triangle $\Delta$ of side length $a$ lies in the union of five equilateral triangles of side length $b$, show that there exist four equilateral triangles of side length $b$ whose union contains $\Delta$.

Let $ABC$ be a triangle. On the sides $BC$, $CA$, $AB$ of the triangle, construct outwardly three squares with centres $O_a$, $O_b$, $O_c$ respectively. Let $\omega$ be the circumcircle of $\vartriangle O_aO_bO_c$. Given that $A$ lies on $\omega$, prove that the centre of $\omega$ lies on the perimeter of $\vartriangle ABC$. [i]Sam Bealing, United Kingdom[/i]

Let $\Omega$ be a circle, and $B, C$ are two fixed points on $\Omega$. Given a third point $A$ on $\Omega$, let $X$ and $Y$ denote the feet of the altitudes from $B$ and $C$, respectively, in the triangle $ABC$. Prove that there exists a fixed circle $\Gamma$ such that $XY$ is tangent to $\Gamma$ regardless of the choice of the point $A$.

[b]p1. [/b] In basketball, teams can score $1, 2$, or $3$ points each time. Suppose that Duke basketball have scored $8$ points so far. What is the total number of possible ways (ordered) that they have scored? For example, $(1, 2, 2, 2, 1)$,$(1, 1, 2, 2, 2)$ are two different ways. [b]p2.[/b] All the positive integers that are coprime to $2021$ are grouped in increasing order, such that the nth group contains $2n - 1$ numbers. Hence the first three groups are $\{1\}$, $\{2, 3, 4\}$, $\{5, 6, 7, 8, 9\}$. Suppose that $2022$ belongs to the $k$th group. Find $k$. [b]p3.[/b] Let $A = (0, 0)$ and $B = (3, 0)$ be points in the Cartesian plane. If $R$ is the set of all points $X$ such that $\angle AXB \ge 60^o$ (all angles are between $0^o$ and $180^o$), find the integer that is closest to the area of $R$. [b]p4.[/b] What is the smallest positive integer greater than $9$ such that when its left-most digit is erased, the resulting number is one twenty-ninth of the original number? [b]p5. [/b] Jonathan is operating a projector in the cartesian plane. He sets up $2$ infinitely long mirrors represented by the lines $y = \tan(15^o)x$ and $y = 0$, and he places the projector at $(1, 0)$ pointed perpendicularly to the $x$-axis in the positive $y$ direction. Jonathan furthermore places a screen on one of the mirrors such that light from the projector reflects off the mirrors a total of three times before hitting the screen. Suppose that the coordinates of the screen is $(a, b)$. Find $10a^2 + 5b^2$. [b]p6.[/b] Dr Kraines has a cube of size $5 \times 5 \times 5$, which is made from $5^3$ unit cubes. He then decides to choose $m$ unit cubes that have an outside face such that any two different cubes don’t share a common vertex. What is the maximum value of $m$? [b]p7.[/b] Let $a_n = \tan^{-1}(n)$ for all positive integers $n$. Suppose that $$\sum_{k=4}^{\infty}(-1)^{\lfloor \frac{k}{2} \rfloor +1} \tan(2a_k)$$ is equals to $a/b$ , where $a, b$ are relatively prime. Find $a + b$. [b]p8.[/b] Rishabh needs to settle some debts. He owes $90$ people and he must pay \$ $(101050 + n)$ to the $n$th person where $1 \le n \le 90$. Rishabh can withdraw from his account as many coins of values \$ $2021$ and \$ $x$ for some fixed positive integer $x$ as is necessary to pay these debts. Find the sum of the four least values of $x$ so that there exists a person to whom Rishabh is unable to pay the exact amount owed using coins. [b]p9.[/b] A frog starts at $(1, 1)$. Every second, if the frog is at point $(x, y)$, it moves to $(x + 1, y)$ with probability $\frac{x}{x+y}$ and moves to $(x, y + 1)$ with probability $\frac{y}{x+y}$ . The frog stops moving when its $y$ coordinate is $10$. Suppose the probability that when the frog stops its $x$-coordinate is strictly less than $16$, is given by $m/n$ where $m, n$ are positive integers that are relatively prime. Find $m + n.$ [b]p10.[/b] In the triangle $ABC$, $AB = 585$, $BC = 520$, $CA = 455$. Define $X, Y$ to be points on the segment $BC$. Let $Z \ne A$ be the intersection of $AY$ with the circumcircle of $ABC$. Suppose that $XZ$ is parallel to $AC$ and the circumcircle of $XYZ$ is tangent to the circumcircle of $ABC$ at $Z$. Find the length of $XY$ . PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $\Delta ABC$ be a triangle with orthocenter $H$ and midpoints $M_a,M_b$, and $M_c$ of $BC$, $AC$, and $AB$ respectively. A circle with center $H$ intersects the lines $M_bM_a$, $M_bM_c$, and $M_cM_a$ in points $U_1,U_2,V_1,V_2,W_1,W_2$ respectively. Prove that $CU_1=CU_2=AV_1=AV_2=BW_1=BW_2$.

Let $ABC$ be a triangle with circumcenter $O$ and circumcircle $\omega$. Let $D$ be the foot of the altitude from $A$ to $\overline{BC}$. Let $P$ and $Q$ be points on the circumcircles of triangles $AOB$ and $AOC$, respectively, such that $A$, $P$, and $Q$ are collinear. Prove that if the circumcircle of triangle $OPQ$ is tangent to $\omega$ at $T$, then $\angle BTD=\angle CAP$. [i]Tiger Zhang[/i]

In the plane there are given the lines $\ell_1$, $\ell_2$, the circle $\mathcal{C}$ with its center on the line $\ell_1$ and a second circle $\mathcal{C}_1$ which is tangent to $\ell_1$, $\ell_2$ and $\mathcal{C}$. Find the locus of the tangent point between $\mathcal{C}$ and $\mathcal{C}_1$ while the center of $\mathcal{C}$ is variable on $\ell_1$. [i]Mircea Becheanu[/i]

Which of all planes, the one and the same body diagonal of a cube with the edge length $a$, cuts out a cut figure with the smallest area from the cube? Calculate the area of such a cut figure. [hide=original wording]Welche von allen Ebenen, die eine und dieselbe Korperdiagonale eines Wurfels mit der Kantenlange a enthalten, schneiden aus den W¨urfel eine Schnittfigur kleinsten Flacheninhaltes heraus? Berechnen Sie den Fl¨acheninhalt solch einer Schnittfigur![/hide]

Let $\triangle A_1B_1C_1$ be an equilateral triangle of area $60$. Chloe constructs a new triangle $\triangle A_2B_2C_2$ as follows. First, she flips a coin. If it comes up heads, she constructs point $A_2$ such that $B_1$ is the midpoint of $\overline{A_2C_1}$. If it comes up tails, she instead constructs $A_2$ such that $C_1$ is the midpoint of $\overline{A_2B_1}$. She performs analogous operations on $B_2$ and $C_2$. What is the expected value of the area of $\triangle A_2B_2C_2$?

If circular arcs $ AC$ and $ BC$ have centers at $ B$ and $ A$, respectively, then there exists a circle tangent to both $ \stackrel{\frown}{AC}$ and $ \stackrel{\frown}{BC}$, and to $ \overline{AB}$. If the length of $ \stackrel{\frown}{BC}$ is $ 12$, then the circumference of the circle is [asy]unitsize(4cm); defaultpen(fontsize(8pt)+linewidth(.8pt)); dotfactor=3; pair O=(0,.375); pair A=(-.5,0); pair B=(.5,0); pair C=shift(-.5,0)*dir(60); draw(Arc(A,1,0,60)); draw(Arc(B,1,120,180)); draw(A--B); draw(Circle(O,.375)); dot(A); dot(B); dot(C); label("$A$",A,SW); label("$B$",B,SE); label("$C$",C,N);[/asy]$ \textbf{(A)}\ 24 \qquad \textbf{(B)}\ 25 \qquad \textbf{(C)}\ 26 \qquad \textbf{(D)}\ 27 \qquad \textbf{(E)}\ 28$

Let $P$ and $P^{\prime }$ be two parallelograms with equal area, and let their sidelengths be $a,$ $b$ and $a^{\prime },$ $b^{\prime }.$ Assume that $a^{\prime }\leq a\leq b\leq b^{\prime },$ and moreover, it is possible to place the segment $b^{\prime }$ such that it completely lies in the interior of the parallelogram $P.$ Show that the parallelogram $P$ can be partitioned into four polygons such that these four polygons can be composed again to form the parallelogram $% P^{\prime }.$

Let $m$ and $n$ be positive integers. A sequence of points $(A_0,A_1,\ldots,A_n)$ on the Cartesian plane is called [i]interesting[/i] if $A_i$ are all lattice points, the slopes of $OA_0,OA_1,\cdots,OA_n$ are strictly increasing ($O$ is the origin) and the area of triangle $OA_iA_{i+1}$ is equal to $\frac{1}{2}$ for $i=0,1,\ldots,n-1$. Let $(B_0,B_1,\cdots,B_n)$ be a sequence of points. We may insert a point $B$ between $B_i$ and $B_{i+1}$ if $\overrightarrow{OB}=\overrightarrow{OB_i}+\overrightarrow{OB_{i+1}}$, and the resulting sequence $(B_0,B_1,\ldots,B_i,B,B_{i+1},\ldots,B_n)$ is called an [i]extension[/i] of the original sequence. Given two [i]interesting[/i] sequences $(C_0,C_1,\ldots,C_n)$ and $(D_0,D_1,\ldots,D_m)$, prove that if $C_0=D_0$ and $C_n=D_m$, then we may perform finitely many [i]extensions[/i] on each sequence until the resulting two sequences become identical.

Consider in a plane $ P$ the points $ O,A_1,A_2,A_3,A_4$ such that \[ \sigma(OA_iA_j) \geq 1 \quad \forall i, j \equal{} 1, 2, 3, 4, i \neq j.\] where $ \sigma(OA_iA_j)$ is the area of triangle $ OA_iA_j.$ Prove that there exists at least one pair $ i_0, j_0 \in \{1, 2, 3, 4\}$ such that \[ \sigma(OA_iA_j) \geq \sqrt{2}.\]

In a cyclic quadrilateral $ABCD$, $AB=a$, $BC=b$, $CD=c$, $\angle ABC = 120^\circ$ and $\angle ABD = 30^\circ$. Prove that (1) $c \ge a + b$; (2) $|\sqrt{c + a} - \sqrt{c + b} | = \sqrt{c - a - b}$.

The segments $BF$ and $CN$ are the altitudes in the acute-angled triangle $ABC$. The line $OI$, which connects the centers of the circumscribed and inscribed circles of triangle $ABC$, is parallel to the line $FN$. Find the length of the altitude $AK$ in the triangle $ABC$ if the radii of its circumscribed and inscribed circles are $R$ and $r$, respectively. (Grigory Filippovsky)

Let ${ABC}$ be a triangle. Line [i]l[/i] is parallel to ${BC}$ and it respectively intersects side ${AB}$ at point ${D}$, side ${AC}$ at point ${E}$, and the circumcircle of the triangle ${ABC}$ at points ${F}$ and ${G}$, where points ${F,D,E,G}$ lie in this order on [i]l[/i]. The circumcircles of triangles ${FEB}$ and ${DGC}$ intersect at points ${P}$ and ${Q}$. Prove that points ${A, P,Q}$ are collinear. (Slovakia)

Sheila is making a regular-hexagon-shaped sign with side length $ 1$. Let $ABCDEF$ be the regular hexagon, and let $R, S,T$ and U be the midpoints of $FA$, $BC$, $CD$ and $EF$, respectively. Sheila splits the hexagon into four regions of equal width: trapezoids $ABSR$, $RSCF$ , $FCTU$, and $UTDE$. She then paints the middle two regions gold. The fraction of the total hexagon that is gold can be written in the form $m/n$ , where m and n are relatively prime positive integers. Compute $m + n$. [img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvYS9lLzIwOTVmZmViZjU3OTMzZmRlMzFmMjM1ZWRmM2RkODMyMTA0ZjNlLnBuZw==&rn=MjAyMCBCTVQgSW5kaXZpZHVhbCAxMy5wbmc=[/img]