Find the minimum possible value of |m -n|, where $m$ and $n$ are integers satisfying $m + n = mn - 2021$.

Prove that there are no functions $f,g:\mathbb{R}\rightarrow \mathbb{R}$ such that $\forall x,y\in \mathbb{R}:$ $ f(x^2+g(y)) -f(x^2)+g(y)-g(x) \leq 2y$ and $f(x)\geq x^2$. [i]Proposed by Mohammad Ahmadi[/i]

Two spheres with radii $36$ and one sphere with radius $13$ are each externally tangent to the other two spheres and to two different planes $\mathcal{P}$ and $\mathcal{Q}$. The intersection of planes $\mathcal{P}$ and $\mathcal{Q}$ is the line $\ell$. The distance from line $\ell$ to the point where the sphere with radius $13$ is tangent to plane $\mathcal{P}$ is $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$. [img]https://imgur.com/1mfBNNL.png[/img]

Point $E$ is the midpoint of the base $AD$ of the trapezoid $ABCD$. Segments $BD$ and $CE$ intersect at point $F$. It is known that $AF$ is perpendicular to $BD$. Prove that $BC = FC$.

Let $S$ be a finite set of points on a plane, where no three points are collinear, and the convex hull of $S$, $\Omega$, is a $2016-$gon $A_1A_2\ldots A_{2016}$. Every point on $S$ is labelled one of the four numbers $\pm 1,\pm 2$, such that for $i=1,2,\ldots , 1008,$ the numbers labelled on points $A_i$ and $A_{i+1008}$ are the negative of each other. Draw triangles whose vertices are in $S$, such that any two triangles do not have any common interior points, and the union of these triangles is $\Omega$. Prove that there must exist a triangle, where the numbers labelled on some two of its vertices are the negative of each other.

Triangle $ ABC$ has $ \angle C \equal{} 60^{\circ}$ and $ BC \equal{} 4$. Point $ D$ is the midpoint of $ BC$. What is the largest possible value of $ \tan{\angle BAD}$? $ \textbf{(A)} \ \frac {\sqrt {3}}{6} \qquad \textbf{(B)} \ \frac {\sqrt {3}}{3} \qquad \textbf{(C)} \ \frac {\sqrt {3}}{2\sqrt {2}} \qquad \textbf{(D)} \ \frac {\sqrt {3}}{4\sqrt {2} \minus{} 3} \qquad \textbf{(E)}\ 1$

The vertices of a regular dodecagon are coloured either blue or red. Find the number of all possible colourings which do not contain monochromatic sub-polygons. [i]Vasile Pop[/i]

Five edges of a triangular pyramid are equal to $1$. Find the sixth edge if it is known that the radius of the ball circumscribed about this pyramid is equal to $1$.

For continuously differentiable function \(f : [0, 1] \to\mathbb{R}\) with \(f (1/2) = 0\), show that \[\left(\int_0^1 f(x)\mathrm{d}x\right)^2\leq \frac{1}{4}\int_0^1\left(f'(x)\right)^2\mathrm{d}x\]

a) Let $f,g:\mathbb{Z}\rightarrow\mathbb{Z}$ be one to one maps. Show that the function $h:\mathbb{Z}\rightarrow\mathbb{Z}$ defined by $h(x)=f(x)g(x)$, for all $x\in\mathbb{Z}$, cannot be a surjective function. b) Let $f:\mathbb{Z}\rightarrow\mathbb{Z}$ be a surjective function. Show that there exist surjective functions $g,h:\mathbb{Z}\rightarrow\mathbb{Z}$ such that $f(x)=g(x)h(x)$, for all $x\in\mathbb{Z}$.

Given a $20 \times 101$ square table with $20$ rows and $101$ columns. One wants to fill numbers $0$ and $1$ in the unit squares of the table satisfying the following conditions: $[\text{i}]$ Each square has exactly one number to be filled in. $[\text{ii}]$ Each column is filled with exactly two $1'$s. $[\text{iii}]$ Any two rows with no more than one column are filled with two $1'$s. $a.$ How many ways to fill the numbers satisfying the given conditions? $b.$ With a satisfied numbering way, we number the rows in order from top to bottom. A triple of row (distinct, unordered) $\{a; b; c\}$ is said to be [i]united[/i] if the sets of numbers in the three rows are $(a_1, a_2, . . . , a_{101}), (b_1, b_2, . . . . , b_{101}),$ and $(c_1, c_2, . . . . , c_{101})$ satisfied$$\sum\limits_{i = 1}^{101} {({a_i}{b_i} + {b_i}{c_i} + {c_i}{a_i})} = 3.$$ Prove that: there are at least $10$ [i]united[/i] sets.

Two lines with slopes $\dfrac{1}{2}$ and $2$ intersect at $(2,2)$. What is the area of the triangle enclosed by these two lines and the line $x+y=10 ?$ $\textbf{(A) } 4 \qquad\textbf{(B) } 4\sqrt{2} \qquad\textbf{(C) } 6 \qquad\textbf{(D) } 8 \qquad\textbf{(E) } 6\sqrt{2}$

In the game of [i]Ring Mafia[/i], there are $2019$ counters arranged in a circle. $673$ of these counters are mafia, and the remaining $1346$ counters are town. Two players, Tony and Madeline, take turns with Tony going first. Tony does not know which counters are mafia but Madeline does. On Tony’s turn, he selects any subset of the counters (possibly the empty set) and removes all counters in that set. On Madeline’s turn, she selects a town counter which is adjacent to a mafia counter and removes it. Whenever counters are removed, the remaining counters are brought closer together without changing their order so that they still form a circle. The game ends when either all mafia counters have been removed, or all town counters have been removed. Is there a strategy for Tony that guarantees, no matter where the mafia counters are placed and what Madeline does, that at least one town counter remains at the end of the game? [i]Proposed by Andrew Gu[/i]

Find all prime numbers $p$, for which there exist $x, y \in \mathbb{Q}^+$ and $n \in \mathbb{N}$, satisfying $x+y+\frac{p}{x}+\frac{p}{y}=3n$.

[b]p1.[/b] If $x$ is $20\%$ of $23$ and $y$ is $23\%$ of $20$, compute $xy$ . [b]p2.[/b] Pablo wants to eat a banana, a mango, and a tangerine, one at a time. How many ways can he choose the order to eat the three fruits? [b]p3.[/b] Let $a$, $b$, and $c$ be $3$ positive integers. If $a + \frac{b}{c} = \frac{11}{6}$ , what is the minimum value of $a + b + c$? [b]p4.[/b] A rectangle has an area of $12$. If all of its sidelengths are increased by $2$, its area becomes $32$. What is the perimeter of the original rectangle? [b]p5.[/b] Rohit is trying to build a $3$-dimensional model by using several cubes of the same size. The model’s front view and top view are shown below. Suppose that every cube on the upper layer is directly above a cube on the lower layer and the rotations are considered distinct. Compute the total number of different ways to form this model. [img]https://cdn.artofproblemsolving.com/attachments/b/b/40615b956f3d18313717259b12fcd6efb74cf8.png[/img] [b]p6.[/b] Priscilla has three octagonal prisms and two cubes, none of which are touching each other. If she chooses a face from these five objects in an independent and uniformly random manner, what is the probability the chosen face belongs to a cube? (One octagonal prism and cube are shown below.) [img]https://cdn.artofproblemsolving.com/attachments/0/0/b4f56a381c400cae715e70acde2cdb315ee0e0.png[/img] [b]p7.[/b] Let triangle $\vartriangle ABC$ and triangle $\vartriangle DEF$ be two congruent isosceles right triangles where line segments $\overline{AC}$ and $\overline{DF}$ are their respective hypotenuses. Connecting a line segment $\overline{CF}$ gives us a square $ACFD$ but with missing line segments $\overline{AC}$, $\overline{AD}$, and $\overline{DF}$. Instead, $A$ and $D$ are connected by an arc defined by the semicircle with endpoints $A$ and $D$. If $CF = 1$, what is the perimeter of the whole shape $ABCFED$ ? [img]https://cdn.artofproblemsolving.com/attachments/2/5/098d4f58fee1b3041df23ba16557ed93ee9f5b.png[/img] [b]p8.[/b] There are two moles that live underground, and there are five circular holes that the moles can hop out of. The five holes are positioned as shown in the diagram below, where $A$, $B$, $C$, $D$, and $E$ are the centers of the circles, $AE = 30$ cm, and congruent triangles $\vartriangle ABC$, $\vartriangle CBD$, and $\vartriangle CDE$ are equilateral. The two moles randomly choose exactly two of the five holes, hop out of the two chosen holes, and hop back in. What is the probability that the holes that the two moles hop out of have centers that are exactly $15$ cm apart? [img]https://cdn.artofproblemsolving.com/attachments/c/e/b46ba87b954a1904043020d7a211477caf321d.png[/img] [b]p9.[/b] Carson is planning a trip for $n$ people. Let $x$ be the number of cars that will be used and $y$ be the number of people per car. What is the smallest value of $n$ such that there are exactly $3$ possibilities for $x$ and $y$ so that $y$ is an integer, $x < y$, and exactly one person is left without a car? [b]p10.[/b] Iris is eating an ice cream cone, which consists of a hemisphere of ice cream with radius $r > 0$ on top of a cone with height $12$ and also radius $r$. Iris is a slow eater, so after eating one-third of the ice cream, she notices that the rest of the ice cream has melted and completely filled the cone. Assuming the ice cream did not change volume after it melted, what is the value of $r$? [b]p11.[/b] As Natasha begins eating brunch between $11:30$ AM and $12$ PM, she notes that the smaller angle between the minute and hour hand of the clock is $27$ degrees. What is the number of degrees in the smaller angle between the minute and hour hand when Natasha finishes eating brunch $20$ minutes later? [b]p12.[/b] On a regular hexagon $ABCDEF$, Luke the frog starts at point $A$, there is food on points $C$ and $E$ and there are crocodiles on points $B$ and $D$. When Luke is on a point, he hops to any of the five other vertices with equal probability. What is the probability that Luke will visit both of the points with food before visiting any of the crocodiles? [b]p13.[/b] $2023$ regular unit hexagons are arranged in a tessellating lattice, as follows. The first hexagon $ABCDEF$ (with vertices in clockwise order) has leftmost vertex $A$ at the origin, and hexagons $H_2$ and $H_3$ share edges $\overline{CD}$ and $\overline{DE}$ with hexagon $H_1$, respectively. Hexagon $H_4$ shares edges with both hexagons $H_2$ and $H_3$, and hexagons $H_5$ and $H_6$ are constructed similarly to hexagons H_2 and $H_3$. Hexagons $H_7$ to $H_{2022}$ are constructed following the pattern of hexagons $H_4$, $H_5$, $H_6$. Finally, hexagon H_{2023} is constructed, sharing an edge with both hexagons H2021 and H2022. Compute the perimeter of the resulting figure. [img]https://cdn.artofproblemsolving.com/attachments/1/d/eaf0d04676bac3e3c197b4686dcddd08fce9ac.png[/img] [b]p14.[/b] Aditya’s favorite number is a positive two-digit integer. Aditya sums the integers from $5$ to his favorite number, inclusive. Then, he sums the next $12$ consecutive integers starting after his favorite number. If the two sums are consecutive integers and the second sum is greater than the first sum, what is Aditya’s favorite number? [b]p15.[/b] The $100^{th}$ anniversary of BMT will fall in the year $2112$, which is a palindromic year. Compute the sum of all years from $0000$ to $9999$, inclusive, that are palindromic when written out as four-digit numbers (including leading zeros). Examples include $2002$, $1991$, and $0110$. [b]p16.[/b] Points $A$, $B$, $C$, $D$, and $E$ lie on line $r$, in that order, such that $DE = 2DC$ and $AB = 2BC$. Let $M$ be the midpoint of segment $\overline{AC}$. Finally, let point $P$ lie on $r$ such that $PE = x$. If $AB = 8x$, $ME = 9x$, and $AP = 112$, compute the sum of the two possible values of $CD$. [b]p17.[/b] A parabola $y = x^2$ in the xy-plane is rotated $180^o$ about a point $(a, b)$. The resulting parabola has roots at $x = 40$ and $x = 48$. Compute $a + b$. [b]p18.[/b] Susan has a standard die with values $1$ to $6$. She plays a game where every time she rolls the die, she permanently increases the value on the top face by $1$. What is the probability that, after she rolls her die 3 times, there is a face on it with a value of at least $7$? [b]p19.[/b] Let $N$ be a $6$-digit number satisfying the property that the average value of the digits of $N^4$ is $5$. Compute the sum of the digits of $N^4$. [b]p20.[/b] Let $O_1$, $O_2$, $...$, $O_8$ be circles of radius $1$ such that $O_1$ is externally tangent to $O_8$ and $O_2$ but no other circles, $O_2$ is externally tangent to $O_1$ and $O_3$ but no other circles, and so on. Let $C$ be a circle that is externally tangent to each of $O_1$, $O_2$, $...$, $O_8$. Compute the radius of $C$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Three circles $\omega_1,\omega_2,\omega_3$ are tangent to line $l$ at points $A,B,C$ ($B$ lies between $A,C$) and $\omega_2$ is externally tangent to the other two. Let $X,Y$ be the intersection points of $\omega_2$ with the other common external tangent of $\omega_1,\omega_3$. The perpendicular line through $B$ to $l$ meets $\omega_2$ again at $Z$. Prove that the circle with diameter $AC$ touches $ZX,ZY$. [i]Proposed by Iman Maghsoudi - Siamak Ahmadpour[/i]