Which of the following cannot be equal to $x^2 + \dfrac 1{4x}$ where $x$ is a positive real number? $ \textbf{a)}\ \sqrt 3 -1 \qquad\textbf{b)}\ 2\sqrt 2 - 2 \qquad\textbf{c)}\ \sqrt 5 - 1 \qquad\textbf{d)}\ 1 \qquad\textbf{e)}\ \text{None of above} $

Find the last three digits of the sum $2005^{11}$ + $2005^{12}$ +
...
+ $2005^{2006}$

Sets \(X\subset \mathbb{Z}^{+}\) and \(Y\subset \mathbb{Z}^{+}\) are called [i]comradely[/i], if every positive integer \(n\) can be written as \(n=xy\) for some \(x\in X\) and \(y\in Y\). Let \(X(n)\) and \(Y(n)\) denote the number of elements of \(X\) and \(Y\), respectively, among the first \(n\) positive integers. Let \(f\colon \mathbb{Z}^{+}\to \mathbb{R}^{+}\) be an arbitrary function that goes to infinity. Prove that one can find comradely sets \(X\) and \(Y\) such that \(\dfrac{X(n)}{n}\) and \(\dfrac{Y(n)}{n}\) goes to \(0\), and for all \(\varepsilon>0\) exists \(n \in \mathbb{Z}^+\) such that \[\frac{\min\big\{X(n), Y(n)\big\}}{f(n)}<\varepsilon. \]

Let $P(x)$ be a polynomial with complex coefficients such that $P(0)\neq 0$. Prove that there exists a multiple of $P(x)$ with real positive coefficients if and only if $P(x)$ has no real positive root.

On the bisector of angle $A$ of triangle $ABC$, points $D$ and $F$ are taken inside the triangle so that $\angle DBC = \angle FBA$. Prove that: a) $\angle DCB = \angle FCA$ b) a circle passing through $B$ and $F$ and tangent to the segment $BC$ is tangle to the circumscribed circle of the triangle $ABC$.

Find the smallest positive prime that divides $n^2 + 5n + 23$ for some integer $n$.

In each of the squares of a chessboard an arbitrary integer is written. A king starts to move on the board. Whenever the king moves to some square, the number in that square is increased by $1$. Is it always possible to make the numbers on the chessboard: (a) all even; (b) all divisible by $3$; (c) all equal?

Let $ABC $ be an acute triangle such that $AB\neq AC$ ,with circumcircle $ \Gamma$ and circumcenter $O$. Let $M$ be the midpoint of $BC$ and $D$ be a point on $ \Gamma$ such that $AD \perp BC$. let $T$ be a point such that $BDCT$ is a parallelogram and $Q$ a point on the same side of $BC$ as $A$ such that $\angle{BQM}=\angle{BCA}$ and $\angle{CQM}=\angle{CBA}$. Let the line $AO$ intersect $ \Gamma$ at $E$ $(E\neq A)$ and let the circumcircle of $\triangle ETQ$ intersect $ \Gamma$ at point $X\neq E$. Prove that the point $A,M$ and $X$ are collinear.

Define an $n$-digit pair cycle to be a number with $n^2 + 1$ digits between $1$ and $n$ with every possible pair of consecutive digits. For instance, $11221$ is a 2-digit pair cycle since it contains the consecutive digits $11$, $12$, $22$, and $21$. How many $3$-digit pair cycles exist?

Two circles, $\omega_1$ and $\omega_2$, are internally tangent at $A.$ Let $B$ be the point on $\omega_2$ opposite of $A.$ The radius of $\omega_1$ is $4$ times the radius of $\omega_2.$ Point $P$ is chosen on the circumference of $\omega_1$ such that the ratio $\tfrac{AP}{BP}=\tfrac{2\sqrt{3}}{\sqrt{7}}.$ Let $O$ denote the center of $\omega_2.$ Determine $\tfrac{OP}{AO}.$

Consider a group $G$ with at least $2$ elements and the property that each nontrivial element has infinite order. Let $H$ be a cyclic subgroup of $G$ such that the set $\{xH\mid x\in G\}$ has $2$ elements. \\ $\textbf{(a)}$ Prove that $G$ is cyclic. \\ $\textbf{(b)}$ Does the conclusion from $\textbf{(a)}$ stand true if $G$ contains nontrivial elements of finite order?

Prove that it is impossible to arrange the numbers $1,2,\ldots,1997$ around a circle in such a way that, if $x$ and $y$ are any two neighboring numbers, then $499\le|x-y|\le997$.

Let $n > 1$ be an integer and $X = \{1, 2, \cdots , n^2 \}$. If there exist $x, y$ such that $x^2\mid y$ in all subsets of $X$ with $k$ elements, find the least possible value of $k$.

Let $P(x)$ be a polynomial with real coefficients so that equation $P(m) + P(n) = 0$ has infinitely many pairs of integer solutions $(m,n)$. Prove that graph of $y = P(x)$ has a center of symmetry.

If $ 8\cdot2^x \equal{} 5^{y \plus{} 8}$, then when $ y \equal{} \minus{} 8,x \equal{}$ $ \textbf{(A)}\ \minus{} 4 \qquad\textbf{(B)}\ \minus{} 3 \qquad\textbf{(C)}\ 0 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ 8$

[u]Round 9[/u] [b]p25.[/b] Circle $\omega_1$ has radius $1$ and diameter $AB$. Let circle $\omega_2$ be a circle withm aximum radius such that it is tangent to $AB$ and internally tangent to $\omega_1$. A point $C$ is then chosen such that $\omega_2$ is the incircle of triangle $ABC$. Compute the area of $ABC$. [b]p26.[/b] Two particles lie at the origin of a Cartesian plane. Every second, the first particle moves from its initial position $(x, y)$ to one of either $(x +1, y +2)$ or $(x -1, y -2)$, each with probability $0.5$. Likewise, every second the second particle moves from it’s initial position $(x, y)$ to one of either $(x +2, y +3)$ or $(x -2, y -3)$, each with probability $0.5$. Let $d$ be the distance distance between the two particles after exactly one minute has elapsed. Find the expected value of $d^2$. [b]p27.[/b] Find the largest possible positive integer $n$ such that for all positive integers $k$ with $gcd (k,n) = 1$, $k^2 -1$ is a multiple of $n$. [u]Round 10[/u] [b]p28.[/b] Let $\vartriangle ABC$ be a triangle with side lengths $AB = 13$, $BC = 14$, $C A = 15$. Let $H$ be the orthcenter of $\vartriangle ABC$, $M$ be the midpoint of segment $BC$, and $F$ be the foot of altitude from $C$ to $AB$. Let $K$ be the point on line $BC$ such that $\angle MHK = 90^o$. Let $P$ be the intersection of $HK$ and $AB$. Let $Q$ be the intersection of circumcircle of $\vartriangle FPK$ and $BC$. Find the length of $QK$. [b]p29.[/b] Real numbers $(x, y, z)$ are chosen uniformly at random from the interval $[0,2\pi]$. Find the probability that $$\cos (x) \cdot \cos (y)+ \cos(y) \cdot \cos (z)+ \cos (z) \cdot \cos(x) + \sin (x) \cdot \sin (y)+ \sin (y) \cdot \sin (z)+ \sin (z) \cdot \sin (x)+1$$ is positive. [b]p30.[/b] Find the number of positive integers where each digit is either $1$, $3$, or $4$, and the sum of the digits is $22$. [u]Round 11[/u] [b]p31.[/b] In $\vartriangle ABC$, let $D$ be the point on ray $\overrightarrow{CB}$ such that $AB = BD$ and let $E$ be the point on ray $\overrightarrow{AC}$ such that $BC =CE$. Let $L$ be the intersection of $AD$ and circumcircle of $\vartriangle ABC$. The exterior angle bisector of $\angle C$ intersects $AD$ at $K$ and it follows that $AK = AB +BC +C A$. Given that points $B$, $E$, and $L$ are collinear, find $\angle C AB$. [b]p32.[/b] Let $a$ be the largest root of the equation $x^3 -3x^2 +1 0$. Find the remainder when $\lfloor a^{2019} \rfloor$ is divided by $17$. [b]p33.[/b] For all $x, y \in Q$, functions $f , g ,h : Q \to Q$ satisfy $f (x + g (y)) = g (h( f (x)))+ y$. If $f (6)=2$, $g\left( \frac12 \right) = 2$, and $h \left( \frac72 \right)= 2$, find all possible values of $f (2019)$. [u]Round 12[/u] [b]p34.[/b] An $n$-polyomino is formed by joining $n$ unit squares along their edges. A free polyomino is a polyomino considered up to congruence. That is, two free polyominos are the same if there is a combination of translations, rotations, and reflections that turns one into the other. Let $P(n)$ be the number of free $n$-polyominos. For example, $P(3) = 2$ and $P(4) = 5$. Estimate $P(20)+P(19)$. If your estimate is $E$ and the actual value is $A$, your score for this problem will be $$\max \, \left( 0, \left \lfloor 15-10 \cdot \left|\log_{10} \left( \frac{A}{E} \right) \right| \right \rfloor \right).$$ [b]p35.[/b] Estimate $$\sum^{2019}_{k=1} sin(k),$$ where $k$ is measured in radians. If your estimate is $E$ and the actual value is $A$, your score for this problem will be $\max \, (0,15-10 \cdot |E - A|)$ . [b]p36.[/b] For a positive integer $n$, let $r_{10}(n)$ be the number of $10$-tuples of (not necessarily positive) integers $(a_1,a_2,... ,a_9,a_{10})$ such that $$a^2_1 +a^2_2+ ...+a^2_9+a^2_{10}= n.$$ Estimate $r_{10}(20)+r_{10}(19)$. If your estimate is $E$ and the actual value is $A$, your score for this problem will be$$\max \, \left( 0, \left \lfloor 15-10 \cdot \left|\log_{10} \left( \frac{A}{E} \right) \right| \right \rfloor \right).$$ PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h3165997p28809441]here [/url] and 5-8 [url=https://artofproblemsolving.com/community/c3h3166012p28809547]here[/url].Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Define a sequence by $s_0 = 1$ and for $d \geq 1$, $s_d = s_{d-1} + X_d$, where $X_d$ is chosen uniformly at random from the set $\{1, 2, \dots, d\}$. What is the probability that the sequence $s_0, s_1, s_2, \dots$ contains infinitely many primes?

Find all pairs of integers $(x,y)$ satisfying the following equality $8x^2y^2 + x^2 + y^2 = 10xy$

$ABC$ is a triangle with an area of $1$ square meter. Given the point $D$ on $BC$, point $E$ on $CA$, point $F$ on $AB$, such that quadrilateral $AFDE$ is cyclic. Prove that the area of $DEF \le \frac{EF^2}{4 AD^2}$. [i](holmes)[/i]

At ARML, Santa is asked to give rubber duckies to $2013$ students, one for each student. The students are conveniently numbered $1,2,\cdots,2013$, and for any integers $1 \le m < n \le 2013$, students $m$ and $n$ are friends if and only if $0 \le n-2m \le 1$. Santa has only four different colors of duckies, but because he wants each student to feel special, he decides to give duckies of different colors to any two students who are either friends or who share a common friend. Let $N$ denote the number of ways in which he can select a color for each student. Find the remainder when $N$ is divided by $1000$. [i]Proposed by Lewis Chen[/i]

Triangle $ABC$ is inscribed in a unit circle $\omega$. Let $H$ be its orthocenter and $D$ be the foot of the perpendicular from $A$ to $BC$. Let $\triangle XY Z$ be the triangle formed by drawing the tangents to $\omega$ at $A, B, C$. If $\overline{AH} = \overline{HD}$ and the side lengths of $\triangle XY Z$ form an arithmetic sequence, the area of $\triangle ABC$ can be expressed in the form $\tfrac{p}{q}$ for relatively prime positive integers $p, q$. What is $p + q$?

A secant line splits a circle into two segments. Inside those segments, one draws two squares such that both squares has two corners on a secant line and two on the circumference. The ratio of the square's side lengths is $5:9$. Compute the ratio of the secant line versus circle radius.

Consider the numbers$$A= \frac{1}{4}\cdot \frac{3}{6}\cdot \frac{5}{8}\cdot ...\frac{595}{598}\cdot \frac{597}{600}$$and$$B= \frac{2}{5}\cdot \frac{4}{7}\cdot \frac{6}{9}\cdot ...\frac{596}{599}\cdot \frac{598}{601}$$. Prove that: (a) $A < B$, (b) $A < \frac{1}{5990}$

Seven cyclists follow one another, in a line, on a narrow way. By the end of the training, each cyclist complains about the style of driving of the one in front of him. They agree to rearrange themselves the next day, so that no cyclist would follow the same cyclist he follows the first day. How many such rearrangements are possible?

What is $\frac{2+4+6}{1+3+5}-\frac{1+3+5}{2+4+6}$? $\textbf{(A) }-1\qquad\textbf{(B) }\frac5{36}\qquad\textbf{(C) }\frac7{12}\qquad\textbf{(D) }\frac{49}{20}\qquad\textbf{(E) }\frac{43}3$