Divide the unit square into $9$ equal squares by means of two pairs of lines parallel to the sides (see figure). Now remove the central square. Treat the remaining $8$ squares the same way, and repeat the process $n$ times. (a) How many squares of side length $1/3^n$ remain? (b) What is the sum of the areas of the removed squares as $n$ becomes infinite? [center][img]https://cdn.artofproblemsolving.com/attachments/7/d/3e6e68559919583c24d4457f946bc4cef3922f.png[/img][/center]

Define a sequence $\{a_n\}_{n \geq 1}$ recursively by $a_1=1$, $a_2=2$, and for all integers $n \geq 2$, $a_{n+1}=(n+1)^{a_n}$. Determine the number of integers $k$ between $2$ and $2020$, inclusive, such that $k+1$ divides $a_k - 1$. [i]Proposed by Taiki Aiba[/i]

There is a token in one of the nodes of a hexagon with side $n$, divided into regular triangles (see figure). Two players take turns moving it to one of the neighboring nodes, and it is forbidden to go to a node that the token has already visited. The one who loses who can't make a move. Who wins with the right game? [img]https://cdn.artofproblemsolving.com/attachments/2/f/18314fe7f9f4cd8e783037a8e5642e17f4e1be.png[/img]

Let $a$, $b$, $c$ be the side lengths of a triangle. Prove that $$ \sqrt[3]{(a^2+bc)(b^2+ca)(c^2+ab)} > \frac{a^2+b^2+c^2}{2}. $$

$ m$ and $ n$ are positive integers. In a $ 8 \times 8$ chessboard, $ (m,n)$ denotes the number of grids a Horse can jump in a chessboard ($ m$ horizontal $ n$ vertical or $ n$ horizontal $ m$ vertical ). If a $ (m,n) \textbf{Horse}$ starts from one grid, passes every grid once and only once, then we call this kind of Horse jump route a $ \textbf{H Route}$. For example, the $ (1,2) \textbf{Horse}$ has its $ \textbf{H Route}$. Find the smallest positive integer $ t$, such that from any grid of the chessboard, the $ (t,t\plus{}1) \textbf{Horse}$ does not has any $ \textbf{H Route}$.

For what values of $n$ is it possible to paint the edges of a prism whose base is an $n$-gon so that there are edges of all three colours at each vertex and all the faces (including the upper and lower bases) have edges of all three colours? (AV Shapovelov)

Lukas is playing pool on a table shaped like an equilateral triangle. The pockets are at the corners of the triangle and are labeled $C$, $H$, and $T$. Each side of the table is $16$ feet long. Lukas shoots a ball from corner $C$ of the table in such a way that on the second bounce, the ball hits $2$ feet away from him along side $CH$. a. How many times will the ball bounce before hitting a pocket? b. Which pocket will the ball hit? c. How far will the ball travel before hitting the pocket?

For a given natural number $k{}$, a convex polygon is called $k{}$[i]-triangular[/i] if it is the intersection of some $k{}$ triangles. [list=a] [*]What is the largest $n{}$ for which there exist a $k{}$-triangular $n{}$-gon? [*]What is the largest $n{}$ for which any convex $n{}$-gon is $k{}$-triangular? [/list] [i]Proposed by P. Kozhevnikov[/i]

The numbers from $1$ to $2000$ are placed on the vertices of a regular polygon with $2000$ sides, one on each vertex, so that the following is true: If four integers $A, B, C, D$ satisfy that $1 \leq A<B<C<D \leq 2000$, then the segment that joins the vertices of the numbers $A$ and $B$ and the segment that joins the vertices of $C$ and $D$ do not intersect inside the polygon. Prove that there exists a perfect square such that the number diametrically opposite to it is not a perfect square.

$ABC$ is a triangle, and $E$ and $F$ are points on the segments $BC$ and $CA$ respectively, such that $\frac{CE}{CB}+\frac{CF}{CA}=1$ and $\angle CEF=\angle CAB$. Suppose that $M$ is the midpoint of $EF$ and $G$ is the point of intersection between $CM$ and $AB$. Prove that triangle $FEG$ is similar to triangle $ABC$.

A connected graph with at least one vertex of an odd degree is given. Show that one can color the edges of the graph red and blue in such a way that, for each vertex, the absolute difference between the numbers of red and blue edges at that vertex does not exceed 1.

The keys of a safe with five locks are cloned and distributed among eight people such that any of five of eight people can open the safe. What is the least total number of keys? $ \textbf{a)}\ 18 \qquad\textbf{b)}\ 20 \qquad\textbf{c)}\ 22 \qquad\textbf{d)}\ 24 \qquad\textbf{e)}\ 25 $

Let $ABC$ be a triangle and $M$ the midpoint of $AB$. Let circumcircles of triangles $CMO$ and $ABC$ intersect at $K$ where $O$ is the circumcenter of $ABC$. Let $P$ be the intersection of lines $OM$ and $CK$. Prove that $\angle{PAK} = \angle{MCB}$.

Given positive integers $m$ and $n$. Let $a_{i,j} ( 1 \le i \le m, 1 \le j \le n)$ be non-negative real numbers, such that $$ a_{i,1} \ge a_{i,2} \ge \cdots \ge a_{i,n} \text{ and } a_{1,j} \ge a_{2,j} \ge \cdots \ge a_{m,j} $$ holds for all $1 \le i \le m$ and $1 \le j \le n$. Denote $$ X_{i,j}=a_{1,j}+\cdots+a_{i-1,j}+a_{i,j}+a_{i,j-1}+\cdots+a_{i,1},$$ $$ Y_{i,j}=a_{m,j}+\cdots+a_{i+1,j}+a_{i,j}+a_{i,j+1}+\cdots+a_{i,n}.$$ Prove that $$ \prod_{i=1}^{m} \prod_{j=1}^{n} X_{i,j} \ge \prod_{i=1}^{m} \prod_{j=1}^{n} Y_{i,j}.$$

[b]p1.[/b] What is the area of a triangle with side lengths $ 6$, $ 8$, and $10$? [b]p2.[/b] Let $f(n) = \sqrt{n}$. If $f(f(f(n))) = 2$, compute $n$. [b]p3.[/b] Anton is buying AguaFina water bottles. Each bottle costs $14 $dollars, and Anton buys at least one water bottle. The number of dollars that Anton spends on AguaFina water bottles is a multiple of $10$. What is the least number of water bottles he can buy? [b]p4.[/b] Alex flips $3$ fair coins in a row. The probability that the first and last flips are the same can be expressed in the form $m/n$ for relatively prime positive integers $m$ and $n$. Compute $m + n$. [b]p5.[/b] How many prime numbers $p$ satisfy the property that $p^2 - 1$ is not a multiple of $6$? [b]p6.[/b] In right triangle $\vartriangle ABC$ with $AB = 5$, $BC = 12$, and $CA = 13$, point $D$ lies on $\overline{CA}$ such that $AD = BD$. The length of $CD$ can then be expressed in the form $m/n$ for relatively prime positive integers $m$ and $n$. Compute $m + n$. [b]p7.[/b] Vivienne is deciding on what courses to take for Spring $2021$, and she must choose from four math courses, three computer science courses, and five English courses. Vivienne decides that she will take one English course and two additional courses that are either computer science or math. How many choices does Vivienne have? [b]p8.[/b] Square $ABCD$ has side length $2$. Square $ACEF$ is drawn such that $B$ lies inside square $ACEF$. Compute the area of pentagon $AFECD$. [b]p9.[/b] At the Boba Math Tournament, the Blackberry Milk Team has answered $4$ out of the first $10$ questions on the Boba Round correctly. If they answer all $p$ remaining questions correctly, they will have answered exactly $\frac{9p}{5}\%$ of the questions correctly in total. How many questions are on the Boba Round? [b]p10.[/b] The sum of two positive integers is $2021$ less than their product. If one of them is a perfect square, compute the sum of the two numbers. [b]p11.[/b] Points $E$ and $F$ lie on edges $\overline{BC}$ and $\overline{DA}$ of unit square $ABCD$, respectively, such that $BE =\frac13$ and $DF =\frac13$ . Line segments $\overline{AE}$ and $\overline{BF}$ intersect at point $G$. The area of triangle $EFG$ can be written in the form $m/n$ , where $m$ and $n$ are relatively prime positive integers. Compute $m+n$. [b]p12.[/b] Compute the number of positive integers $n \le 2020$ for which $n^{k+1}$ is a factor of $(1+2+3+· · ·+n)^k$ for some positive integer $k$. [b]p13.[/b] How many permutations of $123456$ are divisible by their last digit? For instance, $123456$ is divisible by $6$, but $561234$ is not divisible by $4$. [b]p14.[/b] Compute the sum of all possible integer values for $n$ such that $n^2 - 2n - 120$ is a positive prime number. [b]p15. [/b]Triangle $\vartriangle ABC$ has $AB =\sqrt{10}$, $BC =\sqrt{17}$, and $CA =\sqrt{41}$. The area of $\vartriangle ABC$ can be expressed in the form $m/n$ for relatively prime positive integers $m$ and $n$. Compute $m + n$. [b]p16.[/b] Let $f(x) = \frac{1 + x^3 + x^{10}}{1 + x^{10}}$ . Compute $f(-20) + f(-19) + f(-18) + ...+ f(20)$. [b]p17.[/b] Leanne and Jing Jing are walking around the $xy$-plane. In one step, Leanne can move from any point $(x, y)$ to $(x + 1, y)$ or $(x, y + 1)$ and Jing Jing can move from $(x, y)$ to $(x - 2, y + 5)$ or $(x + 3, y - 1)$. The number of ways that Leanne can move from $(0, 0)$ to $(20, 20)$ is equal to the number of ways that Jing Jing can move from $(0, 0)$ to $(a, b)$, where a and b are positive integers. Compute the minimum possible value of $a + b$. [b]p18.[/b] Compute the number positive integers $1 < k < 2021$ such that the equation $x +\sqrt{kx} = kx +\sqrt{x}$ has a positive rational solution for $x$. [b]p19.[/b] In triangle $\vartriangle ABC$, point $D$ lies on $\overline{BC}$ with $\overline{AD} \perp \overline{BC}$. If $BD = 3AD$, and the area of $\vartriangle ABC$ is $15$, then the minimum value of $AC^2$ is of the form $p\sqrt{q} - r$, where $p, q$, and $r$ are positive integers and $q$ is not divisible by the square of any prime number. Compute $p + q + r$. [b]p20. [/b]Suppose the decimal representation of $\frac{1}{n}$ is in the form $0.p_1p_2...p_j\overline{d_1d_2...d_k}$, where $p_1, ... , p_j$ , $d_1,... , d_k$ are decimal digits, and $j$ and $k$ are the smallest possible nonnegative integers (i.e. it’s possible for $j = 0$ or $k = 0$). We define the [i]preperiod [/i]of $\frac{1}{n}$ to be $j$ and the [i]period [/i]of $\frac{1}{n}$ to be $k$. For example, $\frac16 = 0.16666...$ has preperiod $1$ and period $1$, $\frac17 = 0.\overline{142857}$ has preperiod $0$ and period $6$, and $\frac14 = 0.25$ has preperiod $2$ and period $0$. What is the smallest positive integer $n$ such that the sum of the preperiod and period of $\frac{1}{n}$ is $ 8$? PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Given a circle $\Gamma$ and the positive numbers $h$ and $m$, construct with straight edge and compass a trapezoid inscribed in $\Gamma$, such that it has altitude $h$ and the sum of its parallel sides is $m$.