We say that a set $S$ of integers is [i]rootiful[/i] if, for any positive integer $n$ and any $a_0, a_1, \cdots, a_n \in S$, all integer roots of the polynomial $a_0+a_1x+\cdots+a_nx^n$ are also in $S$. Find all rootiful sets of integers that contain all numbers of the form $2^a - 2^b$ for positive integers $a$ and $b$.

Define a sequence recursively by $F_0 = 0$, $F_1 = 1$, and $F_n = $ the remainder when $F_{n-1} + F_{n-2}$ is divided by $3$, for all $n \ge 2$. Thus the sequence starts $0,1,1,2,0,2 \ldots$. What is $F_{2017} + F_{2018} + F_{2019} + F_{2020} + F_{2021} + F_{2022} + F_{2023} + F_{2024}$? $\textbf{(A)}\ 6\qquad\textbf{(B)}\ 7\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 9\qquad\textbf{(E)}\ 10$

On a $999\times 999$ board a [i]limp rook[/i] can move in the following way: From any square it can move to any of its adjacent squares, i.e. a square having a common side with it, and every move must be a turn, i.e. the directions of any two consecutive moves must be perpendicular. A [i]non-intersecting route[/i] of the limp rook consists of a sequence of pairwise different squares that the limp rook can visit in that order by an admissible sequence of moves. Such a non-intersecting route is called [i]cyclic[/i], if the limp rook can, after reaching the last square of the route, move directly to the first square of the route and start over. How many squares does the longest possible cyclic, non-intersecting route of a limp rook visit? [i]Proposed by Nikolay Beluhov, Bulgaria[/i]

In triangle $ABC$, sides $CB$ and $CA$ are equal to $a$ and $b$, respectively. The bisector of the angle $ACB$ intersects the side $AB$ at the point $K$ and the circumscribed circle of the triangle at point $M$. The circumscribed circle of the triangle $AMK$ intersects for second time the straight line $CA$ at the point $P$. Find the length of the segment $AP$.

Suppose that $a,b$ are real numbers, function $f(x) = ax+b$ satisfies $\mid f(x) \mid \leq 1$ for any $x \in [0,1]$. Find the range of values of $S= (a+1)(b+1).$

The infinite integer plane $Z\times Z = Z^2$ consists of all number pairs $(x, y)$, where $x$ and $y$ are integers. Let $a$ and $b$ be non-negative integers. We call any move from a point $(x, y)$ to any of the points $(x\pm a, y \pm b)$ or $(x \pm b, y \pm a) $ a $(a, b)$-knight move. Determine all numbers $a$ and $b$, for which it is possible to reach all points of the integer plane from an arbitrary starting point using only $(a, b)$-knight moves.

Isabella is making sushi. She slices a piece of salmon into the shape of a solid triangular prism. The prism is $2$ cm thick, and its triangular faces have side lengths $7$ cm, $ 24$cm, and $25$ cm. Find the volume of this piece of salmon in cm$^3$. [i]Proposed by Isabella Li[/i]

Each of two similar triangles was cut into two triangles so that one of the resulting parts of one triangle is similar to one of the parts of the other triangle. Is it true that the remaining parts are also similar? (D. Shnol)

Each point of a plane is colored in one of three colors: red, black and blue. Prove that there exists a rectangle in this plane whose vertices all have the same color.

A quadrilateral $ABCD$ with perpendicular diagonals is inscribed into a circle $\omega$. Two arcs $\alpha$ and $\beta$ with diameters AB and $CD$ lie outside $\omega$. Consider two crescents formed by the circle $\omega$ and the arcs $\alpha$ and $\beta$ (see Figure). Prove that the maximal radii of the circles inscribed into these crescents are equal. (F.Nilov)

Let $S$ be the set of all rational numbers expressible in the form \[\frac{(a_1^2+a_1-1)(a_2^2+a_2-1)\ldots (a_n^2+a_n-1)}{(b_1^2+b_1-1)(b_2^2+b_2-1)\ldots (b_n^2+b_n-1)}\] for some positive integers $n, a_1, a_2 ,\ldots, a_n, b_1, b_2, \ldots, b_n$. Prove that there is an infinite number of primes in $S$.

For any positive integer $n$ and $0 \leqslant i \leqslant n$, denote $C_n^i \equiv c(n,i)\pmod{2}$, where $c(n,i) \in \left\{ {0,1} \right\}$. Define \[f(n,q) = \sum\limits_{i = 0}^n {c(n,i){q^i}}\] where $m,n,q$ are positive integers and $q + 1 \ne {2^\alpha }$ for any $\alpha \in \mathbb N$. Prove that if $f(m,q)\left| {f(n,q)} \right.$, then $f(m,r)\left| {f(n,r)} \right.$ for any positive integer $r$.

On the side $ AB$ of a cyclic quadrilateral $ ABCD$ there is a point $ X$ such that diagonal $ BD$ bisects $ CX$ and diagonal $ AC$ bisects $ DX$. What is the minimum possible value of $ AB\over CD$? [i]Proposed by S. Berlov[/i]

The set of all real solutions of the inequality \[ |x \minus{} 1| \plus{} |x \plus{} 2| < 3\] is $ \textbf{(A)}\ x \in ( \minus{} 3,2) \qquad \textbf{(B)}\ x \in ( \minus{} 1,2) \qquad \textbf{(C)}\ x \in ( \minus{} 2,1) \qquad$ $ \textbf{(D)}\ x \in \left( \minus{} \frac32,\frac72\right) \qquad \textbf{(E)}\ \O \text{ (empty})$ Note: I updated the notation on this problem.

Point $O$ is the center of the circumcircle of the acute triangle $ABC$. The line $AO$ intersects the side $BC$ at point $D$ so that $OD = BD = 1/3 BC$ . Find the angles of the triangle $ABC$. Justify the answer.

Kevin likes drawing. He takes a large piece of paper and draws on it every rectangle with positive integer side lengths and perimeter at most 2017, with no two rectangles overlapping. Compute the total area of the paper that is covered by a rectangle.

Find the number of rectangles who have the following properties: a) Have for vertices, points $(x,y)$ of plane $Oxy$ with $x,y$ non negative integers and $ x \le 8$ , $y\le 8$ b) Have sides parallel to axes c) Have area $E$, with $30<E\le 40$

Find natural $a, b, c, d$ that satisfy the system $$\begin{cases} ab+cd=34 \\ ac-bd=19 \end{cases}$$

Find all continuous and nondecreasing functions $ f:[0,\infty)\longrightarrow\mathbb{R} $ that satisfy the inequality: $$ \int_0^{x+y} f(t) dt\le \int_0^x f(t) dt +\int_0^y f(t) dt,\quad\forall x,y\in [0,\infty) . $$

Point $A$ lies outside circle $o$ of center $O$. From point $A$ draw two lines tangent to a circle $o$ in points $B$ and $C$. A tangent to a circle $o$ cuts segments $AB$ and $AC$ in points $E$ and $F$, respectively. Lines $OE$ and $OF$ cut segment $BC$ in points $P$ and $Q$, respectively. Prove that from line segments $BP$, $PQ$, $QC$ can construct triangle similar to triangle $AEF$.

Let $ABC$ be a triangle, with $A'$, $B'$, and $C'$ the points of tangency of the incircle with $BC$, $CA$, and $AB$ respectively. Let $X$ be the intersection of the excircle with respect to $A$ with $AB$, and $M$ the midpoint of $BC$. Let $D$ be the intersection of $XM$ with $B'C'$. Show that $\angle C'A'D' = 90^o$.

Prove that for all integers $n > 2$, \[ 3| \sum\limits_{i=0}^{[n/3]} (-1)^i C _n ^{3i} \]

Let $m$ be the area and let $n$ be the perimeter of a regular octagon. The ratio $\frac{m^2}{n}$ can be expressed as $p \tan (q \pi)$ where $p$ is a positive integer. Find $pq$.

Two equal parallel chords are drawn $ 8$ inches apart in a circle of radius $ 8$ inches. The area of that part of the circle that lies between the chords is: $ \textbf{(A)}\ 21\frac {1}{3}\pi \minus{} 32\sqrt {3} \qquad\textbf{(B)}\ 32\sqrt {3} \plus{} 21\frac {1}{3}\pi \qquad\textbf{(C)}\ 32\sqrt {3} \plus{} 42\frac {2}{3}\pi$ $ \textbf{(D)}\ 16\sqrt {3} \plus{} 42\frac {2}{3}\pi \qquad\textbf{(E)}\ 42\frac {2}{3}\pi$

Find the natural numbers $ n $ which have the property that $$ 2011=\left| \mathbb{Q}\cap\bigcup_{k=1}^n\left\{ x\in\mathbb{R} | 1+k^2x^2=2k\left( x-\lfloor x\rfloor \right) \right\} \right| . $$ [i]Marian Teler[/i]