Let ${\cal C}_1$ and ${\cal C}_2$ be concentric circles, with ${\cal C}_2$ in the interior of ${\cal C}_1$. From a point $A$ on ${\cal C}_1$ one draws the tangent $AB$ to ${\cal C}_2$ ($B\in {\cal C}_2$). Let $C$ be the second point of intersection of $AB$ and ${\cal C}_1$, and let $D$ be the midpoint of $AB$. A line passing through $A$ intersects ${\cal C}_2$ at $E$ and $F$ in such a way that the perpendicular bisectors of $DE$ and $CF$ intersect at a point $M$ on $AB$. Find, with proof, the ratio $AM/MC$.

There is a $6 \times 6$ grid of lights. There is a switch at the top of each column and on the left of each row. A light will only turn on if the switches corresponding to both its column and its row are in the "on" position. Compute the number of different configurations of lights.

A rectangle $ D$ is partitioned in several ($ \ge2$) rectangles with sides parallel to those of $ D$. Given that any line parallel to one of the sides of $ D$, and having common points with the interior of $ D$, also has common interior points with the interior of at least one rectangle of the partition; prove that there is at least one rectangle of the partition having no common points with $ D$'s boundary. [i]Author: Kei Irie, Japan[/i]

Find the smallest possible value of $n$ such that $n+2$ people can stand inside or on the border of a regular $n$-gon with side length $6$ feet where each pair of people are at least $6$ feet apart. [i]Proposed by Jeff Lin[/i]

Given $n>4$, prove that every cyclic quadrilateral can be dissected into $n$ cyclic quadrilaterals.

A super ball rolling on the floor enters a half circular track (radius $R$). The ball rolls without slipping around the track and leaves (velocity $v$) traveling horizontally in the opposite direction. Afterwards, it bounces on the floor. How far (horizontally) from the end of the track will the ball bounce for the second time? The ball’s surface has a theoretically infinite coefficient of static friction. It is a perfect sphere of uniform density. All collisions with the ground are perfectly elastic and theoretically instantaneous. Variations could involve the initial velocity being given before the ball enters the track or state that the normal force between the ball and the track right before leaving is zero (centripetal acceleration). [i]Problem proposed by Brian Yue[/i]

Find all functions $ f :\mathbb{Z}\mapsto\mathbb{Z} $ such that following conditions holds: $a)$ $f(n) \cdot f(-n)=f(n^2)$ for all $n\in\mathbb{Z}$ $b)$ $f(m+n)=f(m)+f(n)+2mn$ for all $m,n\in\mathbb{Z}$

Let $n$ be a positive integer, and let $W = \ldots x_{-1}x_0x_1x_2 \ldots$ be an infinite periodic word, consisting of just letters $a$ and/or $b$. Suppose that the minimal period $N$ of $W$ is greater than $2^n$. A finite nonempty word $U$ is said to [i]appear[/i] in $W$ if there exist indices $k \leq \ell$ such that $U=x_k x_{k+1} \ldots x_{\ell}$. A finite word $U$ is called [i]ubiquitous[/i] if the four words $Ua$, $Ub$, $aU$, and $bU$ all appear in $W$. Prove that there are at least $n$ ubiquitous finite nonempty words. [i]Proposed by Grigory Chelnokov, Russia[/i]

Circles of diameter 1 inch and 3 inches have the same center. The smaller circle is painted red, and the portion outside the smaller circle and inside the larger circle is painted blue. What is the ratio of the blue-painted area to the red-painted area? $ \textbf{(A) } 2 \qquad \textbf{(B) } 3 \qquad \textbf{(C) } 6 \qquad \textbf{(D) } 8 \qquad \textbf{(E) } 9$

Consider a cross like in the figure but with size $2021$. Every square have a $+1$. Every minute we select a sub-cross of size $3$ and multiply their squares by $-1$. It´s posible achieve that all the squares of the cross with size $2021$ have a $-1$?

For a binary string $S$ (i.e. a string of 0 's and 1's) that contains at least one 0 , we produce a binary string $f(S)$ as follows: - If the substring 110 occurs in $S$, replace each instance of 110 with 01 to produce $f(S)$; - Otherwise, replace the leftmost occurrence of 0 in $S$ by 1 to produce $f(S)$. Given binary string $S$ of length $n$, we define the lifetime of $S$ to be the number of times $f$ can be applied to $S$ until the resulting string contains no more 0 's. For example, $$

For all positive real numbers $a,b,c$ with $a+b+c=3$, prove the inequality $$\frac{a^6}{c^2+2b^3} + \frac{b^6}{a^2+2c^3} + \frac{c^6}{b^2+2a^3} \geq 1.$$

Let $k_1$ and $k_2$ be two circles and let them cut each other at points $A$ and $B$. A line through $B$ is cutting $k_1$ and $k_2$ in $C$ and $D$ respectively, such that $C$ doesn't lie inside of $k_2$ and $D$ doesn't lie inside of $k_1$. Let $M$ be the intersection point of the tangent lines to $k_1$ and $k_2$ that are passing through $C$ and $D$, respectively. Let $P$ be the intersection of the lines $AM$ and $CD$. The tangent line to $k_1$ passing through $B$ intersects $AD$ in point $L$. The tangent line to $k_2$ passing through $B$ intersects $AC$ in point $K$. Let $KP \cap MD \equiv N$ and $LP \cap MC \equiv Q$. Prove that $MNPQ$ is a parallelogram.

Show that, for all integers $n$, $n^2+2n+12$ is not a multiple of 121.

Let $a_{1}=1$, $a_{2}=2$, $a_{3}$, $a_{4}$, $\cdots$ be the sequence of positive integers of the form $2^{\alpha}3^{\beta}$, where $\alpha$ and $\beta$ are nonnegative integers. Prove that every positive integer is expressible in the form \[a_{i_{1}}+a_{i_{2}}+\cdots+a_{i_{n}},\] where no summand is a multiple of any other.

Let $A$ be a closed subset of $\mathbb{R}^{n}$ and let $B$ be the set of all those points $b \in \mathbb{R}^{n}$ for which there exists exactly one point $a_{0}\in A $ such that $|a_{0}-b|= \inf_{a\in A}|a-b|$. Prove that $B$ is dense in $\mathbb{R}^{n}$; that is, the closure of $B$ is $\mathbb{R}^{n}$

Solve the system of equations: $$\begin{cases} x + [y] + \{z\}=3.9 \\ y + [z] + \{x\}= 3.5 \\ z + [x] + \{y\}= 2. \end{cases}$$

Find all real numbers $a,b$ and $c$ such that $a+bc=b+ca=c+ab$.

Let $a, b, c$ be positive real numbers in the interval $[0, 1]$ with $a+b, b+c, c+a \ge 1$. Prove that \[ 1 \le (1-a)^2 + (1-b)^2 + (1-c)^2 + \frac{2\sqrt{2} abc}{\sqrt{a^2+b^2+c^2}}. \]

Let $\Gamma$ be the circumcircle of acute triangle $ABC$. Points $D$ and $E$ are on segments $AB$ and $AC$ respectively such that $AD = AE$. The perpendicular bisectors of $BD$ and $CE$ intersect minor arcs $AB$ and $AC$ of $\Gamma$ at points $F$ and $G$ respectively. Prove that lines $DE$ and $FG$ are either parallel or they are the same line. [i]Proposed by Silouanos Brazitikos, Evangelos Psychas and Michael Sarantis, Greece[/i]

Show that $x^4 > x - \frac12$ for all real $x$.

Ankit, Box, and Clark are playing a game. First, Clark comes up with a prime number less than 100. Then he writes each digit of the prime number on a piece of paper (writing $0$ for the tens digit if he chose a single-digit prime), and gives one each to Ankit and Box, without telling them which digit is the tens digit, and which digit is the ones digit. The following exchange occurs: 1. Clark: There is only one prime number that can be made using those two digits. 2. Ankit: I don't know whether I'm the tens digit or the ones digit. 3. Box: I don't know whether I'm the tens digit or the ones digit. 4. Box: You don't know whether you're the tens digit or the ones digit. 5. Ankit: I don't know whether you're the tens digit or the ones digit. What was Clark's number?

On a circle, $n$ points are selected and the chords joining them in pairs are drawn. Assuming that no three of these chords are concurrent (except at the endpoints), how many points of intersection are there?

The beaver is chess piece that move to $2$ cells by horizontal or vertical. Every cell of $100 \times 100$ chessboard colored in some color,such that we can not get from one cell to another with same color with one move of beaver or knight. What minimal color do we need?

Determine the number of ordered pairs $(x,y)$ with $x$ and $y$ integers between $-5$ and $5,$ inclusive, such that $(x+y)(x+3y)=(x+2y)^2.$