Denote by $Q$ and $N^*$ the set of all rational and positive integer numbers, respectively. Suppose that $\frac{ax + b}{cx + d} \in Q$ for every $x \in N^*$: Prove that there exist integers $A,B,C,D$ such that $\frac{ax + b}{cx + d}= \frac{Ax + B}{Cx+D}$ for all $x \in N^* $

Acute triangle $\triangle ABC$ with $AB<AC$, circumcircle $\Gamma$ and circumcenter $O$ is given. Midpoint of side $AB$ is $D$. Point $E$ is chosen on side $AC$ so that $BE=CE$. Circumcircle of triangle $BDE$ intersects $\Gamma$ at point $F$ (different from point $B$). Point $K$ is chosen on line $AO$ satisfying $BK \perp AO$ (points $A$ and $K$ lie in different half-planes with respect to line $BE$). Prove that the intersection of lines $DF$ and $CK$ lies on $\Gamma$.

Quadrilateral $ABCD$ with perpendicular diagonals $AC$ and $BD$ is inscribed in a circle. Altitude $DE$ in triangle $ABD$ intersects diagonal $AC$ in $F$. Prove that $FB = BC$

Let $ABC$ be a triangle with $\angle A=60^{\circ}$. Point $D, E$ in lines $\overrightarrow{AB}, \overrightarrow{AC}$ respectively satisfies $DB=BC=CE$. ($A,B,D$ lies on this order, and $A,C,E$ likewise) Circle with diameter $BC$ and circle with diameter $DE$ meets at two points $X, Y$. Prove that $\angle XAY\ge 60^{\circ}$

Let $n\geq 4$ be a positive integer.Out of $n$ people,each of two individuals play table tennis game(every game has a winner).Find the minimum value of $n$,such that for any possible outcome of the game,there always exist an ordered four people group $(a_{1},a_{2},a_{3},a_{4})$,such that the person $a_{i}$ wins against $a_{j}$ for any $1\leq i<j\leq 4$

Let $\mathcal B$ be a family of open balls in $\mathbb R^n$ and $c<\lambda\left(\bigcup\mathcal B\right)$ where $\lambda$ is the $n$-dimensional Lebesgue measure. Show that there exists a finite family of pairwise disjoint balls $\{U_i\}^k_{i=1}\subseteq\mathcal B$ such that $$\sum_{j=1}^k\lambda(U_j)>\frac c{3^n}.$$

Determine all functions $f: \mathbb{R}\rightarrow\mathbb{R}$ so that $\forall x: x\cdot f(\frac x2) - f(\frac2x) = 1$

Determine all triplets $(x, y, z)$ of positive integers with $x \le y \le z$ that satisfy $\left(1+\frac1x \right)\left(1+\frac1y \right)\left(1+\frac1z \right) = 3$

Let $\overline{AB}$ be a line segment of length 2, $C_1$ be the circle with diameter $\overline{AB}$, $C_0$ be the circle with radius 2 externally tangent to $C_1$ at $A$, and $C_2$ be the circle with radius 3 externally tangent to $C_1$ at $B$. Duck $D_1$ is located at point $B$, Duck $D_2$ is located on $C_2$, 270 degrees counterclockwise from $B$, and Duck $D_0$ is located on $C_0$, 270 degrees counterclockwise from $A$. At the same time, the ducks all start running counterclockwise around their corresponding circles, with each duck taking the same amount of time to complete a full lap around its circle. When the 3 ducks are first collinear, the distance between $D_0$ and $D_2$ can be expressed as $p\sqrt{q}$. Find $p+q$. [i]2022 CCA Math Bonanza Individual Round #10[/i]

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].