In a circular ring with radii $11r$ and $9r$, we put circles of radius $r$ which are tangent to the boundary circles and do not overlap. Determine the maximum number of circles that can be put this way. (You may use that $9.94<\sqrt{99}<9.95$)

Find all subsets $A$ of $\left\{ 1, 2, 3, 4, \ldots \right\}$, with $|A| \geq 2$, such that for all $x,y \in A, \, x \neq y$, we have that $\frac{x+y}{\gcd (x,y)}\in A$. [i]Dan Schwarz[/i]

Given positive integers $m,n$ and a $m\times n$ matrix $A$ with real entries. (1) Show that matrices $X = I_m + AA^T$ and $Y = I_n + A^T A$ are invertible. ($I_l$ is the $l\times l$ unit matrix.) (2) Evaluate the value of $\text{tr}(X^{-1}) - \text{tr}(Y^{-1})$.

Find all possible values of $k $ such that there exist a $k\times k$ table colored in $k$ colors such that there do not exist two cells in a column or a row with the same color or four cells made of intersecting two columns and two rows painted in exactly three colors.

The [i]rank[/i] of a rational number $q$ is the unique $k$ for which $q=\frac{1}{a_1}+\cdots+\frac{1}{a_k}$, where each $a_i$ is the smallest positive integer $q$ such that $q\geq \frac{1}{a_1}+\cdots+\frac{1}{a_i}$. Let $q$ be the largest rational number less than $\frac{1}{4}$ with rank $3$, and suppose the expression for $q$ is $\frac{1}{a_1}+\frac{1}{a_2}+\frac{1}{a_3}$. Find the ordered triple $(a_1,a_2,a_3)$.

A regular tetrahedron of edge length $1$ is orthogonally projected onto a plane. Find the largest possible area of its image.

One square is inscribed in a circle, and another square is circumscribed around the same circle so that its vertices lie on the extensions of the sides of the first (see figure). Find the angle between the sides of these squares. [img]https://3.bp.blogspot.com/-8eLBgJF9CoA/XTodHmW87BI/AAAAAAAAKY0/xsHTx71XneIZ8JTn0iDMHupCanx-7u4vgCK4BGAYYCw/s400/sharygin%2Boral%2B2017%2B10-11%2Bp1.png[/img]

[color=darkred]Let $\mathcal{C}$ be the set of integrable functions $f\colon [0,1]\to\mathbb{R}$ such that $0\le f(x)\le x$ for any $x\in [0,1]$ . Define the function $V\colon\mathcal{C}\to\mathbb{R}$ by \[V(f)=\int_0^1f^2(x)\ \text{d}x-\left(\int_0^1f(x)\ \text{d}x\right)^2\ ,\ f\in\mathcal{C}\ .\] Determine the following two sets: [list][b]a)[/b] $\{V(f_a)\, |\, 0\le a\le 1\}$ , where $f_a(x)=0$ , if $0\le x\le a$ and $f_a(x)=x$ , if $a<x\le 1\, ;$ [b]b)[/b] $\{V(f)\, |\, f\in\mathcal{C}\}\ .$[/list] [/color]

A sequence of integers $x_1, x_2, \dots, x_k$ is called fibtastic if the difference between any two consecutive elements in the sequence is a Fibonacci number. The integers from $1$ to $2024$ are split into two groups, each written in increasing order. Group A is $a_1, a_2, \dots, a_m$ and Group B is $b_1, b_2, \dots, b_n.$ Find the largest integer $M$ such that we can guarantee that we can pick $M$ consecutive elements from either Group A or Group B which form a fibtastic sequence. As an illustrative example, if a group of numbers is $2, 4, 11, 12, 13, 16, 18, 27, 29, 30,$ the longest fibtastic sequence is $11, 12, 13, 16, 18,$ which has length $5.$

Let $ABC$ be an isosceles triangle with $AB = AC$. The points $D, E$ and $F$ are taken on the sides $BC, CA$ and $AB$, respectively, so that $\angle F DE = \angle ABC$ and $FE$ is not parallel to $BC$. Prove that the line $BC$ is tangent to the circumcircle of $\vartriangle DEF$ if and only if $D$ is the midpoint of the side $BC$.

The expression $\frac{x^2-3x+2}{x^2-5x+6}\div \frac{x^2-5x+4}{x^2-7x+12}$, when simplified is: $ \textbf{(A)}\ \frac{(x-1)(x-6)}{(x-3)(x-4)} \qquad\textbf{(B)}\ \frac{x+3}{x-3}\qquad\textbf{(C)}\ \frac{x+1}{x-1}\qquad\textbf{(D)}\ 1\qquad\textbf{(E)}\ 2$

There are 8 distinct points on a plane, where no three are collinear. An ant starts at one of the points, then walks in a straight line to each one of the other points, visiting each point exactly once and stopping at the final point. This creates a trail of 7 line segments. What is the maximum number of times the ant can cross its own path as it walks? [i]Proposed by Gabriel Wu[/i]

Let $n{}$ be a positive integer. Find the number of noncongruent triangles with integer sidelengths and a perimeter of $2n$.

Choose a point $(x,y)$ in the square bounded by $(0,0), (0,1), (1,0)$ and $(1,1)$. Your score is the minimal distance from your point to any other team's submitted point. Your answer must be in the form $(0.abcd, 0.efgh)$ where $a, b, c, d, e, f, g, h$ are decimal digits.

$AB$ and $CD$ are perpendicular diameters of a circle. $L$ is the tangent to the circle at $A$. $M$ is a variable point on the minor arc $AC$. The ray $BM, DM$ meet the line $L$ at $P$ and $Q$ respectively. Show that $AP\cdot AQ = AB\cdot PQ$. Show how to construct the point $M$ which gives$ BQ$ parallel to $DP$. If the lines $OP$ and $BQ$ meet at $N$ find the locus of $N$. The lines $BP$ and $BQ$ meet the tangent at $D$ at $P'$ and $Q'$ respectively. Find the relation between $P'$ and $Q$'. The lines $D$P and $DQ$ meet the line $BC$ at $P"$ and $Q"$ respectively. Find the relation between $P"$ and $Q"$.

[b]p1.[/b] Nikhil computes the sum of the first $10$ positive integers, starting from $1$. He then divides that sum by 5. What remainder does he get? [b]p2.[/b] In class, starting at $8:00$, Ava claps her hands once every $4$ minutes, while Ella claps her hands once every $6$ minutes. What is the smallest number of minutes after $8:00$ such that both Ava and Ella clap their hands at the same time? [b]p3.[/b] A triangle has side lengths $3$, $4$, and $5$. If all of the side lengths of the triangle are doubled, how many times larger is the area? [b]p4.[/b] There are $50$ students in a room. Every student is wearing either $0$, $1$, or $2$ shoes. An even number of the students are wearing exactly $1$ shoe. Of the remaining students, exactly half of them have $2$ shoes and half of them have $0$ shoes. How many shoes are worn in total by the $50$ students? [b]p5.[/b] What is the value of $-2 + 4 - 6 + 8 - ... + 8088$? [b]p6.[/b] Suppose Lauren has $2$ cats and $2$ dogs. If she chooses $2$ of the $4$ pets uniformly at random, what is the probability that the 2 chosen pets are either both cats or both dogs? [b]p7.[/b] Let triangle $\vartriangle ABC$ be equilateral with side length $6$. Points $E$ and $F$ lie on $BC$ such that $E$ is closer to $B$ than it is to $C$ and $F$ is closer to $C$ than it is to $B$. If $BE = EF = FC$, what is the area of triangle $\vartriangle AFE$? [b]p8.[/b] The two equations $x^2 + ax - 4 = 0$ and $x^2 - 4x + a = 0$ share exactly one common solution for $x$. Compute the value of $a$. [b]p9.[/b] At Shreymart, Shreyas sells apples at a price $c$. A customer who buys $n$ apples pays $nc$ dollars, rounded to the nearest integer, where we always round up if the cost ends in $.5$. For example, if the cost of the apples is $4.2$ dollars, a customer pays $4$ dollars. Similarly, if the cost of the apples is $4.5$ dollars, a customer pays $5$ dollars. If Justin buys $7$ apples for $3$ dollars and $4$ apples for $1$ dollar, how many dollars should he pay for $20$ apples? [b]p10.[/b] In triangle $\vartriangle ABC$, the angle trisector of $\angle BAC$ closer to $\overline{AC}$ than $\overline{AB}$ intersects $\overline{BC}$ at $D$. Given that triangle $\vartriangle ABD$ is equilateral with area $1$, compute the area of triangle $\vartriangle ABC$. [b]p11.[/b] Wanda lists out all the primes less than $100$ for which the last digit of that prime equals the last digit of that prime's square. For instance, $71$ is in Wanda's list because its square, $5041$, also has $1$ as its last digit. What is the product of the last digits of all the primes in Wanda's list? [b]p12.[/b] How many ways are there to arrange the letters of $SUSBUS$ such that $SUS$ appears as a contiguous substring? For example, $SUSBUS$ and $USSUSB$ are both valid arrangements, but $SUBSSU$ is not. [b]p13.[/b] Suppose that $x$ and $y$ are integers such that $x \ge 5$, $y \ge 3$, and $\sqrt{x - 5} +\sqrt{y - 3} = \sqrt{x + y}$. Compute the maximum possible value of $xy$. [b]p14.[/b] What is the largest integer $k$ divisible by $14$ such that $x^2-100x+k = 0$ has two distinct integer roots? [b]p15.[/b] What is the sum of the first $16$ positive integers whose digits consist of only $0$s and $1$s? [b]p16.[/b] Jonathan and Ajit are flipping two unfair coins. Jonathan's coin lands on heads with probability $\frac{1}{20}$ while Ajit's coin lands on heads with probability $\frac{1}{22}$ . Each year, they flip their coins at thesame time, independently of their previous flips. Compute the probability that Jonathan's coin lands on heads strictly before Ajit's coin does. [b]p17.[/b] A point is chosen uniformly at random in square $ABCD$. What is the probability that it is closer to one of the $4$ sides than to one of the $2$ diagonals? [b]p18.[/b] Two integers are coprime if they share no common positive factors other than $1$. For example, $3$ and $5$ are coprime because their only common factor is $1$. Compute the sum of all positive integers that are coprime to $198$ and less than $198$. [b]p19.[/b] Sumith lists out the positive integer factors of $12$ in a line, writing them out in increasing order as $1$, $2$, $3$, $4$, $6$, $12$. Luke, being the mischievious person he is, writes down a permutation of those factors and lists it right under Sumith's as $a_1$, $a_2$, $a_3$, $a_4$, $a_5$, $a_6$. Luke then calculates $$gcd(a_1, 2a_2, 3a_3, 4a_4, 6a_5, 12a_6).$$ Given that Luke's result is greater than $1$, how many possible permutations could he have written? [b]p20.[/b] Tetrahedron $ABCD$ is drawn such that $DA = DB = DC = 2$, $\angle ADB = \angle ADC = 90^o$, and $\angle BDC = 120^o$. Compute the radius of the sphere that passes through $A$, $B$, $C$, and $D$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Given a set $S$ of $2n-1$, $n\in \mathbb N$, different irrational numbers. Prove that there are $n$ different elements $x_1, x_2, \ldots, x_n\in S$ such that for all non-negative rational numbers $a_1, a_2, \ldots, a_n$ with $a_1+a_2+\ldots + a_n>0$ we have that $a_1x_1+a_2x_2+\cdots +a_nx_n$ is an irrational number.

Find $$\sum_{i=1}^{2016} i(i+1)(i+2) \pmod{2018}.$$

DK plays the following game in a simple graph: in each round, he does one of the two operations: ([i]i[/i]) Choose a vertex of odd degree and delete it. Before doing that, DK changes the relation between every two neighbors of the chosen vertex (that is, if they were connected by an edge, then remove this edge, and, if this edge did not exist, then put this edge on the graph). ([i]ii[/i]) Choose a vertex of even degree and change the relation between every two neighbors of it (note that the chosen vertex is not deleted). DK plays this game until there are no more edges on the graph. Show that the number of remaining vertices does not depend on the chosen operations.

Find the locus of excenters of right triangles with given hypotenuse.

The figure may be folded along the lines shown to form a number cube. Three number faces come together at each corner of the cube. What is the largest sum of three numbers whose faces come together at a corner? [asy] draw((0,0)--(0,1)--(1,1)--(1,2)--(2,2)--(2,1)--(4,1)--(4,0)--(2,0)--(2,-1)--(1,-1)--(1,0)--cycle); draw((1,0)--(1,1)--(2,1)--(2,0)--cycle); draw((3,1)--(3,0)); label("$1$",(1.5,1.25),N); label("$2$",(1.5,.25),N); label("$3$",(1.5,-.75),N); label("$4$",(2.5,.25),N); label("$5$",(3.5,.25),N); label("$6$",(.5,.25),N); [/asy] $\text{(A)}\ 11 \qquad \text{(B)}\ 12 \qquad \text{(C)}\ 13 \qquad \text{(D)}\ 14 \qquad \text{(E)}\ 15$

Find the maximum value of real number $A$ such that $$3x^2 + y^2 + 1 \geq A(x^2 + xy + x)$$ for all positive integers $x, y.$

[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].

Find the locus of points in the plane, the sum of whose distances from the sides of a regular polygon is five times the inradius of the pentagon.

Let $n$  be a positive integer. A row of $n+ 1$ squares is written from left to right, numbered $0, 1, 2, \cdots, n$ Two frogs, named Alphonse and Beryl, begin a race starting at square 0. For each second that passes, Alphonse and Beryl make a jump to the right according to the following rules: if there are at least eight squares to the right of Alphonse, then Alphonse jumps eight squares to the right. Otherwise, Alphonse jumps one square to the right. If there are at least seven squares to the right of Beryl, then Beryl jumps seven squares to the right. Otherwise, Beryl jumps one square to the right. Let A(n) and B(n) respectively denote the number of seconds for Alphonse and Beryl to reach square n. For example, A(40) = 5 and B(40) = 10. (a) Determine an integer n>200 for which $B(n) <A(n)$. (b) Determine the largest integer n for which$ B(n) \le A(n)$.