Let $(X,d)$ be a nonempty connected metric space such that the limit of every convergent sequence, is a term of that sequence. Prove that $X$ has exactly one element.

Given a non-negative integer \( n \), determine the values of \( c \) for which the sequence of numbers \[ a_n = 4^n c + \frac{4^n - (-1)^n}{5} \] contains at least one perfect square.

Let $n$ be a natural number. We define sequences $\langle a_i\rangle$ and $\langle b_i\rangle$ of integers as follows. We let $a_0=1$ and $b_0=n$. For $i>0$, we let $$\left( a_i,b_i\right)=\begin{cases} \left(2a_{i-1}+1,b_{i-1}-a_{i-1}-1\right) & \text{if } a_{i-1}<b_{i-1},\\ \left( a_{i-1}-b_{i-1}-1,2b_{i-1}+1\right) & \text{if } a_{i-1}>b_{i-1},\\ \left(a_{i-1},b_{i-1}\right) & \text{if } a_{i-1}=b_{i-1}.\end{cases}$$ Given that $a_k=b_k$ for some natural number $k$, prove that $n+3$ is a power of two.

[u]Down the Infinite Corridor[/u] Consider an isosceles triangle $T$ with base $10$ and height $12$. Define a sequence $\omega_1$, $\omega_2$,$...$of circles such that $\omega_1$ is the incircle of $T$ and $\omega_{i+1}$ is tangent to $\omega_i$ and both legs of the isosceles triangle for $i > 1$. [b]p1.[/b] Find the radius of $\omega_1$. [b]p2.[/b] Find the ratio of the radius of $\omega_{i+1}$ to the radius of $\omega_i$. [b]p3.[/b] Find the total area contained in all the circles.

[hide=D stands for Descartes, L stands for Leibniz]they had two problem sets under those two names[/hide] [b]L.10[/b] Given the following system of equations where $x, y, z$ are nonzero, find $x^2 + y^2 + z^2$. $$x + 2y = xy$$ $$3y + z = yz$$ $$3x + 2z = xz$$ [u]Set 4[/u] [b]L.16 / D.23[/b] Anson, Billiam, and Connor are looking at a $3D$ figure. The figure is made of unit cubes and is sitting on the ground. No cubes are floating; in other words, each unit cube must either have another unit cube or the ground directly under it. Anson looks from the left side and says, “I see a $5 \times 5$ square.” Billiam looks from the front and says the same thing. Connor looks from the top and says the same thing. Find the absolute difference between the minimum and maximum volume of the figure. [b]L.17[/b] The repeating decimal $0.\overline{MBMT}$ is equal to $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers, and $M, B, T$ are distinct digits. Find the minimum value of $q$. [b]L.18[/b] Annie, Bob, and Claire have a bag containing the numbers $1, 2, 3, . . . , 9$. Annie randomly chooses three numbers without replacement, then Bob chooses, then Claire gets the remaining three numbers. Find the probability that everyone is holding an arithmetic sequence. (Order does not matter, so $123$, $213$, and $321$ all count as arithmetic sequences.) [b]L.19[/b] Consider a set $S$ of positive integers. Define the operation $f(S)$ to be the smallest integer $n > 1$ such that the base $2^k$ representation of $n$ consists only of ones and zeros for all $k \in S$. Find the size of the largest set $S$ such that $f(S) < 2^{2019}$. [b]L.20 / D.25[/b] Find the largest solution to the equation $$2019(x^{2019x^{2019}-2019^2+2019})^{2019} = 2019^{x^{2019}+1}.$$ [u]Set 5[/u] [b]L.21[/b] Steven is concerned about his artistic abilities. To make himself feel better, he creates a $100 \times 100$ square grid and randomly paints each square either white or black, each with probability $\frac12$. Then, he divides the white squares into connected components, groups of white squares that are connected to each other, possibly using corners. (For example, there are three connected components in the following diagram.) What is the expected number of connected components with 1 square, to the nearest integer? [img]https://cdn.artofproblemsolving.com/attachments/e/d/c76e81cd44c3e1e818f6cf89877e56da2fc42f.png[/img] [b]L.22[/b] Let x be chosen uniformly at random from $[0, 1]$. Let n be the smallest positive integer such that $3^n x$ is at most $\frac14$ away from an integer. What is the expected value of $n$? [b]L.23[/b] Let $A$ and $B$ be two points in the plane with $AB = 1$. Let $\ell$ be a variable line through $A$. Let $\ell'$ be a line through $B$ perpendicular to $\ell$. Let X be on $\ell$ and $Y$ be on $\ell'$ with $AX = BY = 1$. Find the length of the locus of the midpoint of $XY$ . [b]L.24[/b] Each of the numbers $a_i$, where $1 \le i \le n$, is either $-1$ or $1$. Also, $$a_1a_2a_3a_4+a_2a_3a_4a_5+...+a_{n-3}a_{n-2}a_{n-1}a_n+a_{n-2}a_{n-1}a_na_1+a_{n-1}a_na_1a_2+a_na_1a_2a_3 = 0.$$ Find the number of possible values for $n$ between $4$ and $100$, inclusive. [b]L.25[/b] Let $S$ be the set of positive integers less than $3^{2019}$ that have only zeros and ones in their base $3$ representation. Find the sum of the squares of the elements of $S$. Express your answer in the form $a^b(c^d - 1)(e^f - 1)$, where $a, b, c, d, e, f$ are positive integers and $a, c, e$ are not perfect powers. PS. You should use hide for answers. D.1-15 / L1-9 problems have been collected [url=https://artofproblemsolving.com/community/c3h2790795p24541357]here [/url] and D.16-30/ L10-15 [url=https://artofproblemsolving.com/community/c3h2790818p24541688]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $a_1,a_2,...$ be an infinite sequence of integers that satisfies $a_{n+2}=a_{n+1}+a_n$ for all $n \ge 1$. There exists a positive integer $k$ such that $a_k=a_{k+2018}$. Prove that there exists a term of the sequence which is equal to zero.

Let $\{a_n\}^\infty_{n=0}$ be the sequence of real numbers satisfying $a_0=0$, $a_1=1$ and $$a_{n+2}=a_{n+1}+\frac{a_n}{2^n}$$for every $n\ge0$. Prove that $$\lim_{n\to\infty}a_n=1+\sum_{n=1}^\infty\frac1{2^{\frac{n(n-1)}2}\displaystyle\prod_{k=1}^n(2^k-1)}.$$

Four faces of tetrahedron $ABCD$ are congruent triangles whose angles form an arithmetic progression. If the lengths of the sides of the triangles are $a < b < c$, determine the radius of the sphere circumscribed about the tetrahedron as a function on $a, b$, and $c$. What is the ratio $c/a$ if $R = a \ ?$

Define $f (n) = n + 1$ if $n = p^k > 1$ is a power of a prime number, and $f (n) =p_1^{k_1}+... + p_r^{k_r}$ for natural numbers $n = p_1^{k_1}... p_r^{k_r}$ ($r > 1, k_i > 0$). Given $m > 1$, we construct the sequence $a_0 = m, a_{j+1} = f (a_j)$ for $j \ge 0$ and denote by $g(m)$ the smallest term in this sequence. For each $m > 1$, determine $g(m)$.

Natural numbers $1, 2, 3,.., 100$ are contained in the union of $N$ geometric progressions (not necessarily with integer denominations). Prove that $N \ge 31$

[b]p1.[/b] Elsie M. is fixing a watch with three gears. Gear $A$ makes a full rotation every $5$ minutes, gear $B$ makes a full rotation every $8$ minutes, and gear $C$ makes a full rotation every $12$ minutes. The gears continue spinning until all three gears are in their original positions at the same time. How many minutes will it take for the gears to stop spinning? [b]p2.[/b] Optimus has to pick $10$ distinct numbers from the set of positive integers $\{2, 3, 4,..., 29, 30\}$. Denote the numbers he picks by $\{a_1, a_2, ...,a_{10}\}$. What is the least possible value of $$d(a_1 ) + d(a_2) + ... + d(a_{10}),$$ where $d(n)$ denotes the number of positive integer divisors of $n$? For example, $d(33) = 4$ since $1$, $3$, $11$, and $33$ divide $33$. [b]p3.[/b] Michael is given a large supply of both $1\times 3$ and $1\times 5$ dominoes and is asked to arrange some of them to form a $6\times 13$ rectangle with one corner square removed. What is the minimum number of $1\times 3$ dominoes that Michael can use? [img]https://cdn.artofproblemsolving.com/attachments/6/6/c6a3ef7325ecee417e37ec9edb5374aceab9fd.png[/img] [b]p4.[/b] Andy, Ben, and Chime are playing a game. The probabilities that each player wins the game are, respectively, the roots $a$, $b$, and $c$ of the polynomial $x^3 - x^2 + \frac{111}{400}x - \frac{9}{400} = 0$ with $a \le b \le c$. If they play the game twice, what is the probability of the same player winning twice? [b]p5.[/b] TongTong is doodling in class and draws a $3 \times 3$ grid. She then decides to color some (that is, at least one) of the squares blue, such that no two $1 \times 1$ squares that share an edge or a corner are both colored blue. In how many ways may TongTong color some of the squares blue? TongTong cannot rotate or reflect the board. [img]https://cdn.artofproblemsolving.com/attachments/6/0/4b4b95a67d51fda0f155657d8295b0791b3034.png[/img] [b]p6.[/b] Given a positive integer $n$, we define $f(n)$ to be the smallest possible value of the expression $$| \square 1 \square 2 ... \square n|,$$ where we may place a $+$ or a $-$ sign in each box. So, for example, $f(3) = 0$, since $| + 1 + 2 - 3| = 0$. What is $f(1) + f(2) + ... + f(2011)$? [b]p7.[/b] The Duke Men's Basketball team plays $11$ home games this season. For each game, the team has a $\frac34$ probability of winning, except for the UNC game, which Duke has a $\frac{9}{10}$ probability of winning. What is the probability that Duke wins an odd number of home games this season? [b]p8.[/b] What is the sum of all integers $n$ such that $n^2 + 2n + 2$ divides $n^3 + 4n^2 + 4n - 14$? [b]p9.[/b] Let $\{a_n\}^N_{n=1}$ be a finite sequence of increasing positive real numbers with $a_1 < 1$ such that $$a_{n+1} = a_n \sqrt{1 - a^2_1}+ a_1\sqrt{1 - a^2_n}$$ and $a_{10} = 1/2$. What is $a_{20}$? [b]p10.[/b] Three congruent circles are placed inside a unit square such that they do not overlap. What is the largest possible radius of one of these circles? PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let the sequence of random variables $ \{ X_m, \; m \geq 0\ \}, \; X_0=0$, be an infinite random walk on the set of nonnegative integers with transition probabilities \[ p_i=P(X_{m+1}=i+1 \mid X_m=i) >0, \; i \geq 0 \,\] \[ q_i=P(X_{m+1}=i-1 \mid X_m=i ) >0, \; i>0.\] Prove that for arbitrary $ k >0$ there is an $ \alpha_k > 1$ such that \[ P_n(k)=P \left ( \max_{0 \leq j \leq n} X_j =k \right)\] satisfies the limit relation \[ \lim_{L \rightarrow \infty} \frac 1L \sum_{n=1}^L P_n(k) \alpha_k ^n < \infty.\] [i]J. Tomko[/i]

Let $n > 1$ be a positive integer. A 2-dimensional grid, infinite in all directions, is given. Each 1 by 1 square in a given $n$ by $n$ square has a counter on it. A [i]move[/i] consists of taking $n$ adjacent counters in a row or column and sliding them each by one space along that row or column. A [i]returning sequence[/i] is a finite sequence of moves such that all counters again fill the original $n$ by $n$ square at the end of the sequence. [list] [*] Assume that all counters are distinguishable except two, which are indistinguishable from each other. Prove that any distinguishable arrangement of counters in the $n$ by $n$ square can be reached by a returning sequence. [*] Assume all counters are distinguishable. Prove that there is no returning sequence that switches two counters and returns the rest to their original positions.[/list] [i]Mitchell Lee and Benjamin Gunby.[/i]

$33$ counters are shown in the left grid below. Choose a counter to start at and remove it from the grid. At each subsequent step, choose a direction (up, down, left, or right), move along the grid line from your current position to the nearest counter in that direction, and remove that counter. You cannot choose a direction that reverses your previous one (e.g., left then right is not allowed). Your goal is to pick up all $33$ counters in a single sequence of steps. When you find the right sequence, write the numbers $1$ to $33$ on the counters so that $N$ is written on the $N$th counter you removed. A smaller example of a solved grid is shown to the right below. (Note that the final move from $8$ to $9$ is possible because counters $3, 4,$ and $5$ have been removed in earlier steps.) There is a unique solution, but you do not need to prove that your answer is the only one possible. You merely need to find an answer that satisfies the constraints above. (Note: In any other USAMTS problem, you need to provide a full proof. Only in this problem is an answer without justification acceptable.) [asy] unitsize(0.5cm); string[][] x = { {"-","-","0","0","0","0","0","0","0","0","0"}, {"0","0","-","0","0","0","0","0","0","0","0"}, {"0","-","0","0","-","0","0","0","0","0","0"}, {"-","0","0","-","0","-","0","0","0","0","0"}, {"-","-","-","-","0","-","0","0","0","0","0"}, {"0","0","0","-","0","-","-","-","-","0","0"}, {"0","0","0","0","-","0","0","0","0","-","0"}, {"0","0","0","0","0","0","0","-","-","0","0"}, {"0","0","0","0","0","0","-","0","0","0","-"}, {"0","0","0","0","0","0","-","-","-","-","-"}, {"0","0","0","0","0","0","0","0","0","0","-"}, {"0","0","0","0","0","0","0","0","0","0","-"}, {"0","0","0","0","0","0","0","0","0","0","-"}, {"0","0","0","0","0","0","0","0","0","0","-"}}; void drawcircle(int x, int y) { filldraw(circle((x,y),0.45),white); } for(int i = 0; i < 11; ++i) { draw((i,-0.7)--(i,13.7),dashed); } for(int i = 0; i < 14; ++i) { draw((-0.7,i)--(10.7,i),dashed); } drawcircle(10,0); drawcircle(10,1); drawcircle(10,2); drawcircle(10,3); drawcircle(10,4); drawcircle(10,5); drawcircle(9,4); drawcircle(8,4); drawcircle(7,4); drawcircle(6,4); drawcircle(6,5); drawcircle(7,6); drawcircle(8,6); drawcircle(9,7); drawcircle(8,8); drawcircle(7,8); drawcircle(6,8); drawcircle(5,8); drawcircle(4,7); drawcircle(5,9); drawcircle(5,10); drawcircle(4,11); drawcircle(3,10); drawcircle(3,9); drawcircle(3,8); drawcircle(2,9); drawcircle(1,9); drawcircle(0,9); drawcircle(0,10); drawcircle(1,11); drawcircle(2,12); drawcircle(1,13); drawcircle(0,13); for(int k = 0; k<14; ++k){ for(int l = 0; l<11; ++l){ if(x[k][l]!="0"){ label((x[k][l]),(l,-k+13),fontsize(10pt)); } } } [/asy]

Define the sequence $x_1, x_2, ...$ inductively by $x_1 = \sqrt{5}$ and $x_{n+1} = x_n^2 - 2$ for each $n \geq 1$. Compute $\lim_{n \to \infty} \frac{x_1 \cdot x_2 \cdot x_3 \cdot ... \cdot x_n}{x_{n+1}}$.