Circles $\omega$ and $\Omega$ meet at points $A$ and $B$. Let $M$ be the midpoint of the arc $AB$ of circle $\omega$ ($M$ lies inside $\Omega$). A chord $MP$ of circle $\omega$ intersects $\Omega$ at $Q$ ($Q$ lies inside $\omega$). Let $\ell_P$ be the tangent line to $\omega$ at $P$, and let $\ell_Q$ be the tangent line to $\Omega$ at $Q$. Prove that the circumcircle of the triangle formed by the lines $\ell_P$, $\ell_Q$ and $AB$ is tangent to $\Omega$. [i]Ilya Bogdanov, Russia and Medeubek Kungozhin, Kazakhstan[/i]

There is an empty table with $2^{100}$ rows and $100$ columns. Alice and Eva take turns filling the empty cells of the first row of the table, Alice plays first. In each move, Alice chooses an empty cell and puts a cross in it; Eva in each move chooses an empty cell and puts a zero. When no empty cells remain in the first row, the players move on to the second row, and so on (in each new row Alice plays first). The game ends when all the rows are filled. Alice wants to make as many different rows in the table as possible, while Eva wants to make as few as possible. How many different rows will be there in the table if both follow their best strategies? Proposed by Denis Afrizonov

Sofia summed all the page numbers from a book starting at $1$ and getting $2013$. Pablo saw how she did this and realized Sofia skipped a page. How many pages does the book have, and what page did Sofia skip?

Let $A=\{1,3,3^2,\ldots, 3^{2014}\}$. We obtain a partition of $A$ if $A$ is written as a disjoint union of nonempty subsets. [list=a] [*]Prove that there is no partition of $A$ such that the product of elements in each subset is a square. [*]Prove that there exists a partition of $A$ such that the sum of elements in each subset is a square.[/list]

An infinite sequence of numbers $a, b, c, d,...$ is obtained by term-by-term addition of two geometric progressions. Can this sequence begin with the following numbers:. a) $1,1,3,5$ ? b) $1,2,3,5$ ? c) $1,2,3, 4$ ?

Let $ A $ be a ring in which $ 1\neq 0. $ If $ a,b\in A, $ then the following affirmations are equivalent: $ \text{(i)}\quad aba=a\wedge ba^2b=1 $ $ \text{(ii)}\quad ab=ba=1 $ $ \text{(iii)}\quad \exists !b\in A\quad aba=a $

Let $a$ and $b$ be two-digit positive integers. Find the greatest possible value of $a+b$, given that the greatest common factor of $a$ and $b$ is $6$. [i]Proposed by Jacob Xu[/i] [hide=Solution][i]Solution[/i]. $\boxed{186}$ We can write our two numbers as $6x$ and $6y$. Notice that $x$ and $y$ must be relatively prime. Since $6x$ and $6y$ are two digit numbers, we just need to check values of $x$ and $y$ from $2$ through $16$ such that $x$ and $y$ are relatively prime. We maximize the sum when $x = 15$ and $y = 16$, since consecutive numbers are always relatively prime. So the sum is $6 \cdot (15+16) = \boxed{186}$.[/hide]

Let $ABCD$ be a quadrilateral such that $\angle ABC = \angle CDA = 90^o$, and $BC = 7$. Let $E$ and $F$ be on $BD$ such that $AE$ and $CF$ are perpendicular to BD. Suppose that $BE = 3$. Determine the product of the smallest and largest possible lengths of $DF$.

The sequence of integers $a_1, a_2, ,,$ is defined as follows: $$a_n=\begin{cases} 0\,\,\,\, if\,\,\,\, n\,\,\,\, has\,\,\,\, an\,\,\,\, even\,\,\,\, number\,\,\,\, of\,\,\,\, divisors\,\,\,\, greater\,\,\,\, than\,\,\,\, 2014 \\ 1 \,\,\,\, if \,\,\,\, n \,\,\,\, has \,\,\,\, an \,\,\,\, odd \,\,\,\, number \,\,\,\, of \,\,\,\, divisors \,\,\,\, greater \,\,\,\, than \,\,\,\, 2014\end{cases}$$ Show that the sequence $a_n$ never becomes periodic.

Is it possible to draw a closed $20$-link polyline on the plane and number its links with the numbers $1, 2, 3, \ldots, 20$ in the order of traversal so that for each natural $i = 1, 2, 3, \ldots, 10$ the links numbered $i$ and $10+i$ intersect each other and do not intersect the other links? [i] I. Efremov[/i]

We call a graph with n vertices $k-flowing-chromatic$ if: 1. we can place a chess on each vertex and any two neighboring (connected by an edge) chesses have different colors. 2. we can choose a hamilton cycle $v_1,v_2,\cdots , v_n$, and move the chess on $v_i$ to $v_{i+1}$ with $i=1,2,\cdots ,n$ and $v_{n+1}=v_1$, such that any two neighboring chess also have different colors. 3. after some action of step 2 we can make all the chess reach each of the n vertices. Let T(G) denote the least number k such that G is k-flowing-chromatic. If such k does not exist, denote T(G)=0. denote $\chi (G)$ the chromatic number of G. Find all the positive number m such that there is a graph G with $\chi (G)\le m$ and $T(G)\ge 2^m$ without a cycle of length small than 2017.

A sequence of polynomials $P_m(x, y, z), m = 0, 1, 2, \cdots$, in $x, y$, and $z$ is defined by $P_0(x, y, z) = 1$ and by \[P_m(x, y, z) = (x + z)(y + z)P_{m-1}(x, y, z + 1) - z^2P_{m-1}(x, y, z)\] for $m > 0$. Prove that each $P_m(x, y, z)$ is symmetric, in other words, is unaltered by any permutation of $x, y, z.$

Find the number of integers $n$ with $1 \le n \le 2017$ so that $(n-2)(n-0)(n-1)(n-7)$ is an integer multiple of $1001$.

Xenia and Yagve take turns in playing the following game: A coin is placed on the first box in a row of nine cells. At each turn the player may choose to move the coin forward one step, move the coin forward four steps, or move coin back two steps. For a move to be allowed, the coin must land on one of them of nine cells. The winner is one who gets to move the coin to the last ninth cell. Who wins, given that Xenia makes the first move, and both players play optimally?

Given points $a$ and $b$ in the plane, let $a\oplus b$ be the unique point $c$ such that $abc$ is an equilateral triangle with $a,b,c$ in the clockwise orientation. Solve $(x\oplus (0,0))\oplus(1,1)=(1,-1)$ for $x$.

A square with side length 2 and a circle share the same center. The total area of the regions that are inside the circle and outside the square is equal to the total area of the regions that are outside the circle and inside the square. What is the radius of the circle? [asy]defaultpen(linewidth(0.8));pair a=(4,4), b=(0,0), c=(0,4), d=(4,0), o=(2,2); draw(a--d--b--c--cycle); draw(circle(o, 2.5));[/asy] $ \textbf{(A)}\ \frac{2}{\sqrt{\pi}} \qquad \textbf{(B)}\ \frac{1\plus{}\sqrt{2}}{2} \qquad \textbf{(C)}\ \frac{3}{2} \qquad \textbf{(D)}\ \sqrt{3} \qquad \textbf{(E)}\ \sqrt{\pi}$

Let $l$ denote the length of the smallest diagonal of all rectangles inscribed in a triangle $T$ . (By inscribed, we mean that all four vertices of the rectangle lie on the boundary of $T$ .) Determine the maximum value of $\frac{l^2}{S(T)}$ taken over all triangles ($S(T )$ denotes the area of triangle $T$ ).

For real numbers $a_1, a_2, \ldots, a_{200}$, we consider the value $S = a_1a_2 + a_2a_3 + \ldots + a_{199}a_{200} + a_{200}a_1$. In one operation, you can change the sign of any number (that is, change $a_i$ to $-a_i$), and then calculate the value of $S$ for the new numbers again. What is the smallest number of operations needed to always be able to make $S$ nonnegative? [i]Proposed by Oleksii Masalitin[/i]

Let $k$ and $n$ be two given distinct positive integers greater than $1$. There are finitely many (not necessarily distinct) integers written on the blackboard. Kázmér is allowed to erase $k$ consecutive elements of an arithmetic sequence with a difference not divisible by $k$. Similarly, Nándor is allowed to erase $n$ consecutive elements of an arithmetic sequence with a difference that is not divisible by $n$. The initial numbers on the blackboard have the property that both Kázmér and Nándor can erase all of them (independently from each other) in a finite number of steps. Prove that the difference of biggest and the smallest number on the blackboard is at least $\varphi(n)+\varphi(k)$. [i]Proposed by Boldizsár Varga, Budapest[/i]

There are $1988$ towns and $4000$ roads in a certain country (each road connects two towns) . Prove that there is a closed path passing through no more than $20$ towns. (A. Razborov , Moscow)

Find all pairs \((x, y)\) of real numbers that satisfy the system \[ (x + 1)(x^2 + 1) = y^3 + 1 \] \[ (y + 1)(y^2 + 1) = x^3 + 1 \]

Given $ 12$ points in a plane no three of which are collinear, the number of lines they determine is: $ \textbf{(A)}\ 24 \qquad\textbf{(B)}\ 54 \qquad\textbf{(C)}\ 120 \qquad\textbf{(D)}\ 66 \qquad\textbf{(E)}\ \text{none of these}$

In triangle $ABC$ with centroid $G$, let $M \in (AB)$ and $N \in (AC)$ be points on two of its sides. Prove that points $M, G, N$ are collinear if and only if $\frac{MB}{MA}+\frac{NC}{NA}=1$.

Suppose $k>3$ is a divisor of $2^p+1$, where $p$ is prime. Prove that $k\ge2p+1$.

[b]O[/b]n a blackboard a stranger writes the values of $s_7(n)^2$ for $n=0,1,...,7^{20}-1$, where $s_7(n)$ denotes the sum of digits of $n$ in base $7$. Compute the average value of all the numbers on the board.