Regular polygons with $5, 6, 7, $ and $8$ sides are inscribed in the same circle. No two of the polygons share a vertex, and no three of their sides intersect at a common point. At how many points inside the circle do two of their sides intersect? $\textbf{(A)}\ 52 \qquad\textbf{(B)}\ 56 \qquad\textbf{(C)}\ 60 \qquad\textbf{(D)}\ 64 \qquad\textbf{(E)}\ 68$

Circles $\omega_1$ and $\omega_2$ intersect at points $A$ and $B$. At point $A$ to $\omega_1$ and $\omega_2$ the tangents $\ell_1$ and $\ell_2$ are drawn respectively. The points $T_1$ and $T_2$ are chosen respectively on the circles $\omega_1$ and $\omega_2$ so that the angular measures of the arcs $T_1A$ and $AT_2$ are equal (the measure of the circular arc is calculated clockwise). The tangent $t_1$ at the point $ T_1$ to the circle $\omega_1$ intersects $\ell_2$ at the point $M_1$. Similarly, the tangent $t_2$ at the point $T_2$ to the circle $\omega_2$ intersects $\ell_1$ at point $M_2$. Prove that the midpoints of the segments $M_1M_2$ are on the same a straight line that does not depend on the position of points $T_1$, $T_2$.

We say that three positive integers $a$, $b$ and $c$ form a [i]family[/i] if the following two conditions are satisfied: [list] [*]$a + b + c = 900$. [*]There exists an integer $n$, with $n \geqslant 2$, such that $$\frac{a}{n-1}=\frac{b}{n}=\frac{c}{n+1}.$$ [/list] Determine the number of such families.

A mathematics teacher writes a quadratic equation on the blackboard of the form $$x^2+mx \star n = 0$$, with $m$ and $n$ integers. The sign of $n$ is blurred. Even so, Claudia solves it and obtains integer solutions, one of which is $2011$. Find all possible values of $m$ and $n$.

Determine all integers $m\ge2$ for which there exists an integer $n\ge1$ with $\gcd(m,n)=d$ and $\gcd(m,4n+1)=1$. [i]Proposed by Gerhard Woeginger, Austria[/i]

A certain town is represented as an infinite plane, which is divided by straight lines into squares. The lines are streets, while the squares are blocks. Along a certain street there stands a policeman on each $100$th intersection . Somewhere in the town there is a bandit , whose position and speed are unknown, but he can move only along the streets. The aim of the police is to see the bandit . Does there exist an algorithm available to the police to enable them to achieve their aim? (A. Andjans, Riga)

If $F(n+1)=\frac{2F(n)+1}{2}$ for $n=1,2,\ldots$, and $F(1)=2$, then $F(101)$ equals: $\text{(A)} \ 49 \qquad \text{(B)} \ 50 \qquad \text{(C)} \ 51 \qquad \text{(D)} \ 52 \qquad \text{(E)} \ 53$

Suppose that $S$ is a finite set of points in the plane such that the area of triangle $\triangle ABC$ is at most $1$ whenever $A,B,$ and $C$ are in $S.$ Show that there exists a triangle of area $4$ that (together with its interior) covers the set $S.$

Determine for what values of $x$ the expressions $2x + 2$,$x + 4$, $x + 2$ can represent the sidelengths of a right triangle.

Exactly one number from the set $\{ -1,0,1 \}$ is written in each unit cell of a $2005 \times 2005$ table, so that the sum of all the entries is $0$. Prove that there exist two rows and two columns of the table, such that the sum of the four numbers written at the intersections of these rows and columns is equal to $0$.

Let $S$ be a finite set, and let $\mathcal{A}$ be the set of all functions from $S$ to $S$. Let $f$ be an element of $\mathcal{A}$, and let $T=f(S)$ be the image of $S$ under $f$. Suppose that $f\circ g\circ f\ne g\circ f\circ g$ for every $g$ in $\mathcal{A}$ with $g\ne f$. Show that $f(T)=T$.

Let $n$ be a given positive integer. Sisyphus performs a sequence of turns on a board consisting of $n + 1$ squares in a row, numbered $0$ to $n$ from left to right. Initially, $n$ stones are put into square $0$, and the other squares are empty. At every turn, Sisyphus chooses any nonempty square, say with $k$ stones, takes one of these stones and moves it to the right by at most $k$ squares (the stone should say within the board). Sisyphus' aim is to move all $n$ stones to square $n$. Prove that Sisyphus cannot reach the aim in less than \[ \left \lceil \frac{n}{1} \right \rceil + \left \lceil \frac{n}{2} \right \rceil + \left \lceil \frac{n}{3} \right \rceil + \dots + \left \lceil \frac{n}{n} \right \rceil \] turns. (As usual, $\lceil x \rceil$ stands for the least integer not smaller than $x$. )

We call a set $X$ of real numbers [i]three-averaging[/i] if for every two distinct elements $a$ and $b$ of $X$, there exists an element $c$ in $X$ (different from both $a$ and $b$) such that the number $(a + b + c)/3$ also belongs to $X$. For instance, the set $\{ 0, 1008, 2016 \}$ is three-averaging. What is the least possible number of elements in a three-averaging set with more than 3 elements?

Two circles $(O)$ and $(O')$ intersect at $A$ and $B$. Take two points $P,Q$ on $(O)$ and $(O')$, respectively, such that $AP=AQ$. The line $PQ$ intersects $(O)$ and $(O')$ respectively at $M,N$. Let $E,F$ respectively be the centers of the two arcs $BP$ and $BQ$ (which don't contains $A$). Prove that $MNEF$ is a cyclic quadrilateral.

In a quadrilateral $ABCD$ we have $\angle A = \angle C = 90^o$. Let $E$ be a point in the interior of $ABCD$. Let $M$ be the midpoint of $BE$. Prove that $\angle ADB = \angle EDC$ if and only if $|MA| = |MC|$.

In a ballroom, seven gentlemen $A, B, C, D, E, F$ and $G$ sit directly across from seven queens $a, b, c, d, e, f$ and $g$ in random order. When the gentlemen rise and walks across the dance floor to bow to each of their ladies, someone notices that at least two of the men travel equally long distances. Will it always be like this? The figure shows an example. In the example, $|Bb| =|Ee|$ and $|Dd|=|Cc|$. [img]https://cdn.artofproblemsolving.com/attachments/8/3/1e18a30b1e9acc90b24210fc7991b58062a69f.png[/img]

Let $n$ be a positive integer, and $a_1,...,a_n, b_1,..., b_n$ be $2n$ positive real numbers such that $a_1 + ... + a_n = b_1 + ... + b_n = 1$. Find the minimal value of $ \frac {a_1^2} {a_1 + b_1} + \frac {a_2^2} {a_2 + b_2} + ...+ \frac {a_n^2} {a_n + b_n}$.

What is the side length, in meters, of a square with area $49 \text{ m}^2$? $\textbf{(A) }3\qquad\textbf{(B) }4\qquad\textbf{(C) }5\qquad\textbf{(D) }6\qquad\textbf{(E) }7$

Let $a,b,c$ be given positive integers. Prove that there exists some positive integer $N$ such that \[ a\mid Nbc+b+c,\ b\mid Nca+c+a,\ c\mid Nab+a+b \] if and only if, denoting $d=\gcd(a,b,c)$ and $a=dx$, $b=dy$, $c=dz$, the positive integers $x,y,z$ are pairwise coprime, and also $\gcd(d,xyz) \mid x+y+z$. (Dan Schwarz)

Given an isosceles triangle $ABC$ with a right angle at $C,$ construct the center $M$ and radius $r$ of a circle cutting on segments $AB, BC, CA$ the segments $DE, FG,$ and $HK,$ respectively, such that $\angle DME + \angle FMG + \angle HMK = 180^\circ$ and $DE : FG : HK = AB : BC : CA.$

$30$ people sit around a table, some of which are Yale students. Each person is asked if the person to their right is a Yale student. Yale students will always answer correctly, but non-Yale students will answer randomly. Find the smallest possible number of Yale students such that, after hearing everyone’s answers and knowing the number of Yale students, it is possible to identify for certain at least one Yale student.

Let $N$ be a positive integer. There are $N$ tasks, numbered $1, 2, 3, \ldots, N$, to be completed. Each task takes one minute to complete and the tasks must be completed subjected to the following conditions: [list] [*] Any number of tasks can be performed at the same time. [*] For any positive integer $k$, task $k$ begins immediately after all tasks whose numbers are divisors of $k$, not including $k$ itself, are completed. [*] Task 1 is the first task to begin, and it begins by itself. [/list] Suppose $N = 2017$. How many minutes does it take for all of the tasks to complete? Which tasks are the last ones to complete?

Let $a_j (j = 1, 2, \ldots, n)$ be entirely arbitrary numbers except that no one is equal to unity. Prove \[ a_1 + \sum^n_{i=2} a_i \prod^{i-1}_{j=1} (1 - a_j) = 1 - \prod^n_{j=1} (1 - a_j).\]

The acuteangled triangle $ABC$ with circumcenter $O$ is given. The midpoints of the sides $BC, CA$ and $AB$ are $D, E$ and $F$ respectively. An arbitrary point $M$ on the side $BC$, different of $D$, is choosen. The straight lines $AM$ and $EF$ intersects at the point $N$ and the straight line $ON$ cut again the circumscribed circle of the triangle $ODM$ at the point $P$. Prove that the reflection of the point $M$ with respect to the midpoint of the segment $DP$ belongs on the nine points circle of the triangle $ABC$.

The positive integer formed after writing $k$ consecutive positive integers from smallest to largest is called a $k-\text{continuous}$ number. For example $99100101$ is a $3-\text{continuous}$ number. Prove that: for $\forall N$, $k\in\mathbb Z^+$, there must be a $k-\text{continuous}$ number that can be divisible by $N$.