Let $S$ be a circle with radius $2$, let $S_1$ be a circle,with radius $1$ and tangent, internally to $S$ in $B$ and let $S_2$ be a circle, with radius $1$ and tangent to $S_1$ in $A$, but $S_2$ isn't tangent to $S$. If $K$ is the point of intersection of the line $AB$ and the circle $S$, prove that $K$ is in the circle $S_2$.

Determine the unique pair of real numbers $(x,y)$ that satisfy the equation \[(4 x^2+ 6 x + 4)(4 y^2 - 12 y + 25 ) = 28 .\]

Let $O$ and $O'$ designate two dìagonally opposite vertices of a cube. Bisect those edges of the cube that contain neither of the points $O$ and $O'$. Prove that these midpoints of edges lie in a plane and form the vertices of a regular hexagon

In a country with $n$ towns, all the direct flights are of double destinations (back and forth). There are $r>2014$ rootes between different pairs of towns, that include no more than one intermediate stop (direction of each root matters). Find the minimum possible value of $n$ and the minimum possible $r$ for that value of $n$.

Some boys and girls are having a car wash to raise money for a class trip to China. Initially $ 40 \%$ of the group are girls. Shortly thereafter two girls leave and two boys arrive, and then $ 30 \%$ of the group are girls. How many girls were initially in the group? $ \textbf{(A)}\ 4 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 8 \qquad \textbf{(D)}\ 10 \qquad \textbf{(E)}\ 12$

Let $p>2$ be a prime. Define a sequence $(Q_{n}(x))$ of polynomials such that $Q_{0}(x)=1, Q_{1}(x)=x$ and $Q_{n+1}(x) =xQ_{n}(x) + nQ_{n-1}(x)$ for $n\geq 1.$ Prove that $Q_{p}(x)-x^p $ is divisible by $p$ for all integers $x.$

[b]p1.[/b] Let the series an be defined as $a_1 = 1$ and $a_n =\sum^{n-1}_{i=1} a_ia_{n-i}$ for all positive integers $n$. Evaluate $\sum^{\infty}_{i=1} \left(\frac14\right)^ia_i$. [b]p2.[/b] $a, b, c$ and $d$ are distinct real numbers such that $$a + \frac{1}{b}= b +\frac{1}{c}= c +\frac{1}{d}= d +\frac{1}{a}= x$$ Find |x|. [b]p3.[/b] Find all ordered tuples $(w, x, y, z)$ of complex numbers satisfying $$x + y + z + xy + yz + zx + xyz = -w$$ $$y + z + w + yz + zw + wy + yzw = -x$$ $$z + w + x + zw + wx + xz + zwx = -y$$ $$w + x + y + wx + xy + yw + wxy = -z$$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $n = 2188 = 3^7+1$ and let $A_0^{(0)}, A_1^{(0)}, ..., A_{n-1}^{(0)}$ be the vertices of a regular $n$-gon (in that order) with center $O$ . For $i = 1, 2, \dots, 7$ and $j=0,1,\dots,n-1$, let $A_j^{(i)}$ denote the centroid of the triangle \[ \triangle A_j^{(i-1)} A_{j+3^{7-i}}^{(i-1)} A_{j+2 \cdot 3^{7-i}}^{(i-1)}. \] Here the subscripts are taken modulo $n$. If \[ \frac{|OA_{2014}^{(7)}|}{|OA_{2014}^{(0)}|} = \frac{p}{q} \] for relatively prime positive integers $p$ and $q$, find $p+q$. [i]Proposed by Yang Liu[/i]

The numbers $\frac{1}{1}, \frac{1}{2}, ... , \frac{1}{2010}$ are written on a blackboard. A student chooses any two of the numbers, say $x$, $y$, erases them and then writes down $x + y + xy$. He continues to do this until only one number is left on the blackboard. What is this number?

$17.$ Let $a,$ $b,$ and $c.$ be such that $a+b+c=0$ and $$P=\frac{a^2}{2a^2+bc}+\frac{b^2}{2b^2+ca}+\frac{c^2}{2c^2+ab}$$ is defined. What is the value of $P ?$

For positive integers $x, y, $ $r_x(y)$ to represent the smallest positive integer $ r $ such that $ r \equiv y(\text{mod x})$ .For any positive integers $a, b, n ,$ Prove that $$\sum_{i=1}^{n} r_b(a i)\leq \frac{n(a+b)}{2}$$

There are 20 students in a high school class, and each student has exactly three close friends in the class. Five of the students have bought tickets to an upcoming concert. If any student sees that at least two of their close friends have bought tickets, then they will buy a ticket too. Is it possible that the entire class buys tickets to the concert? (Assume that friendship is mutual; if student $A$ is close friends with student $B$, then $B$ is close friends with $A$.)

Dexter's Laboratory has $2024$ robots, each with a program setup by Dexter. One day, his naughty sister Dee Dee intrudes and writes an integer in $\{1, 2, \dots, 113\}$ on each of the robot's forehead. Each robot detects the numbers on all other robots' foreheads, and guess its own number base on its program, individually and simultaneously. Find the largest positive integer $k$ such that Dexter can setup the programs so that, no matter how the numbers distribute, there are always at least $k$ robots who guess their numbers right. [i]Proposed by sn6dh[/i]

Let $ S$ be the smallest subset of the integers with the property that $ 0\in S$ and for any $ x\in S$, we have $ 3x\in S$ and $ 3x \plus{} 1\in S$. Determine the number of non-negative integers in $ S$ less than $ 2008$.

Is it possible to fit a regular polygon into a circle of radius one so that among the lengths of its diagonals there are 2023 different values whose product is equal to one? [i]Proposed by A. Kuznetsov[/i]

Solve the system of equations: $ \begin{matrix} x^2 + x - 1 = y \\ y^2 + y - 1 = z \\ z^2 + z - 1 = x. \end{matrix} $