Knights, who always tell truth, and liars, who always lie, live on an island. They have been distributed into two teams $A$ and $B$ for a game of tennis, and team $A$ had more members than team $B$. Two players from different teams started the game, whenever a player loses the game, he leaves it forever and he is replaces by a member of his team (that has never played before). The team, all of whose members left the game, loses. After the tournament, every member of team $A$ was asked: "Is it true that you have lost to a liar in some game?", and every member of team $B$ was asked: "Is it true that you have won at least two games, in which your opponent was a knight?". It turns out that every single answer was positive. Which team won?

Let \(n\) be a non-negative integer and define \(a_n = 2^n - n\). Determine all non-negative integers \(m\) such that \(s_m = a_0 + a_1 + \dots + a_m\) is a power of 2.

In a school there are $2007$ male and $2007$ female students. Each student joins not more than $100$ clubs in the school. It is known that any two students of opposite genders have joined at least one common club. Show that there is a club with at least $11$ male and $11$ female members.

Let $M$ be a set of $n \ge 4$ points in the plane, no three of which are collinear. Initially these points are connected with $n$ segments so that each point in $M$ is the endpoint of exactly two segments. Then, at each step, one may choose two segments $AB$ and $CD$ sharing a common interior point and replace them by the segments $AC$ and $BD$ if none of them is present at this moment. Prove that it is impossible to perform $n^3 /4$ or more such moves. [i]Proposed by Vladislav Volkov, Russia[/i]

In triangle $ABC$ $(b \geq c)$, point $E$ is the midpoint of shorter arc $BC$. If $D$ is the point such that $ED$ is the diameter of circumcircle $ABC$, prove that $\angle DEA = \frac{1}{2}(\beta-\gamma)$

Let $a,b,c$ be the lengths of the sides of a triangle, and $\alpha, \beta, \gamma$ respectively, the angles opposite these sides. Prove that if \[ a+b=\tan{\frac{\gamma}{2}}(a\tan{\alpha}+b\tan{\beta}) \] the triangle is isosceles.

Maxi chose $3$ digits, and by writing down all possible permutations of these digits, he obtained $6$ distinct $3$-digit numbers. If exactly one of those numbers is a perfect square and exactly three of them are prime, find Maxi’s $3$ digits. Give all of the possibilities for the $3$ digits.

None of the $n$ (not necessarily distinct) digits selected are equal to $0$ or $7$. It turns out that every $n$-digit number formed using these digits is divisible by $7$. Prove that $n$ is divisible by $6$.

Solve in positive integers the following equation: \[{1\over n^2}-{3\over 2n^3}={1\over m^2}\] [i]Proposed by A. Golovanov[/i]

Let $n$ be a positive integer. Define a chameleon to be any sequence of $3n$ letters, with exactly $n$ occurrences of each of the letters $a, b,$ and $c$. Define a swap to be the transposition of two adjacent letters in a chameleon. Prove that for any chameleon $X$ , there exists a chameleon $Y$ such that $X$ cannot be changed to $Y$ using fewer than $3n^2/2$ swaps.

Consider the infinite polynomial $G(x) = F_1x+F_2x^2 +F_3x^3 +...$ defined for $0 < x <\frac{\sqrt5 -1}{2}$ where Fk is the $k$th term of the Fibonacci sequence defined to be $F_k = F_{k-1} + F_{k-2}$ with $F_1 = 1$, $F_2 = 1$. Determine the value a such that $G(a) = 2$.

Prove that for all integers $n > 1$, $$\frac{1}{n}+\frac{1}{n + 1}+ ...+\frac{1}{n^2- 1}+\frac{1}{n^2} > 1$$

Let $\vartriangle ABC$ be a right triangle with legs $AB = 6$ and $AC = 8$. Let $I$ be the incenter of $\vartriangle ABC$ and $X$ be the other intersection of $AI$ with the circumcircle of $\vartriangle ABC$. Find $\overline{AI} \cdot \overline{IX}$.

Let a,b,c be three distinct positive numbers. Consider the quadratic polynomial $P (x) =\frac{c(x - a)(x - b)}{(c -a)(c -b)}+\frac{a(x - b)(x - c)}{(a - b)(a - c)}+\frac{b(x -c)(x - a)}{(b - c)(b - a)}+ 1$. The value of $P (2017)$ is (A): $2015$ (B): $2016$ (C): $2017$ (D): $2018$ (E): None of the above.

Find all $4$-tuple of real numbers $(x, y, z, w)$ that satisfy the following system of equations: $$x^2 + y^2 + z^2 + w^2 = 4$$ $$\frac{1}{x^2} +\frac{1}{y^2} +\frac{1}{z^2 }+\frac{1}{w^2} = 5 -\frac{1}{(xyzw)^2}$$

The quadrilateral ABCD is inscribed in a circle which has diameter BD. Points A’ and B’ are symmetric to A and B with respect to the line BD and AC respectively. If the lines A’C, BD intersect at P and AC, B’D intersect at Q, prove that PQ is perpendicular to AC.

Find all possible values of $n$ for which a rectangular board $9\times n$ can be partitioned into tiles of the shape: [img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvYi8wLzdjM2Y4ZmE0Zjg1YWZlZGEzNTQ1MmEyNTc3ZjJkNzBlMjExYmY1LnBuZw==&rn=U2NyZWVuIFNob3QgMjAyMS0wNC0yMiBhdCA1LjEzLjU3IEFNLnBuZw==[/img]

Let $N$ be the set of positive integers. Find all the functions $f: N\to N$ with $f (1) = 2$ and such that $max \{f(m)+f(n), m+n\}$ divides $min\{2m+2n,f (m+ n)+1\}$ for all $m, n$ positive integers

Let $ABC$ be a triangle, let the $A$-altitude meet $BC$ at $D$, let the $B$-altitude meet $AC$ at $E$, and let $T\neq A$ be the point on the circumcircle of $ABC$ such that $AT || BC$. Given that $D,E,T$ are collinear, if $BD=3$ and $AD=4$, then the area of $ABC$ can be written as $a+\sqrt{b}$, where $a$ and $b$ are positive integers. What is $a+b$? [i]2021 CCA Math Bonanza Individual Round #12[/i]

Let $E$ be an arbitrary point on the side $BC$ of the square $ABCD$. Prove that the inscribed circles of triangles $ABE$, $CDE$, $ADE$ have a common tangent.

Let \[f_1(x) = \frac{1}{x}\quad\text{and}\quad f_2(x) = 1 - x\] Let $H$ be the set of all compositions of the form $h_1 \circ h_2 \circ \ldots \circ h_k$, where each $h_i$ is either $f_1$ or $f_2$. For all $h$ in $H$, let $h^{(n)}$ denote $h$ composed with itself $n$ times. Find the greatest integer $N$ such that $\pi, h(\pi), \ldots, h^{(N)}(\pi)$ are all distinct for some $h$ in $H$.

The cells of some rectangular $M \times n$ array are colored, each by one of two colors, so that for any two columns the number of pairs of cells situated on a same row and bearing the same color is less than the number of pairs of cells situated on a same row and bearing different colors. i) Prove that if $M=2011$ then $n \leq 2012$ (a model for the extremal case $n=2012$ does indeed exist, but you are not asked to exhibit one). ii) Prove that if $M=2011=n$, each of the colors appears at most $1006\cdot 2011$ times, and at least $1005\cdot 2011$ times. iii) Prove that if however $M=2012$ then $n \leq 1007$. ([i]Dan Schwarz[/i])

On an infinite sheet of tiles, an infinite number of $1 \times 2$ tile rectangles are placed, their edges follow the lines of the tiles, and they do not touch each other, not even the corners. Is it true that the remaining checkered sheet can be completely covered with $1 \times 2$ checkered rectangles? [hide=original wording]Uz bezgalīgas rūtiņu lapas ir novietoti bezgaglīgi daudzi 1 x 2 rūtiņu taisnstūri, to malas iet pa rūtiņu līnijām, un tie nesaskaras cits ar citu pat ne ar stūriem. Vai tiesa, ka atlikušo rūtiņu lapu var pilnībā noklāt ar 1 x 2 rūtiņu tainstūriem? [/hide]

At a party with $n\geq 4$ people, if every $3$ people have exactly $1$ common friend, how many different values can $n$ take? $ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ \text{Infinitely many} \qquad\textbf{(E)}\ \text{None of the above} $

Give a family of curves $2(2\sin\theta-\cos\theta+3)x^2-(8\sin\theta+\cos\theta+1)=0$, where $\theta$ is a parameter. Find the maximum value of the length of the chord that $y=2x$ intersects the curve.