Find all polynomials with real coefficients such that one can find an integer valued series $a_0, a_1, \dots$ satisfying $\lfloor P(x) \rfloor = a_{ \lfloor x^2 \rfloor}$ for all $x$ real numbers.

A game is played on a $n \times n$ chessboard. In the beginning Bars the cat occupies any cell according to his choice. The $d$ sparrows land on certain cells according to their choice (several sparrows may land in the same cell). Bars and the sparrows play in turns. In each turn of Bars, he moves to a cell adjacent by a side or a vertex (like a king in chess). In each turn of the sparrows, precisely one of the sparrows flies from its current cell to any other cell of his choice. The goal of Bars is to get to a cell containing a sparrow. Can Bars achieve his goal a) if $d=\lfloor \frac{3\cdot n^2}{25}\rfloor$, assuming $n$ is large enough? b) if $d=\lfloor \frac{3\cdot n^2}{19}\rfloor$, assuming $n$ is large enough? c) if $d=\lfloor \frac{3\cdot n^2}{14}\rfloor$, assuming $n$ is large enough?

How many ways are there to paint each of the integers $2, 3, \dots, 9$ either red, green, or blue so that each number has a different color from each of its proper divisors? $\textbf{(A)}\ 144\qquad\textbf{(B)}\ 216\qquad\textbf{(C)}\ 256\qquad\textbf{(D)}\ 384\qquad\textbf{(E)}\ 432$

Andrea and Bruno play a game on a table with $11$ rows and $9$ columns. First Andrea divides the table in $33$ zones. Each zone is formed by $3$ contiguous cells aligned vertically or horizontally, as shown in the figure. [code] ._ |_| |_| _ _ _ |_| |_|_|_| [/code] Then, Bruno writes one of the numbers $0, 1, 2, 3, 4, 5$ in each cell in such a way that the sum of the numbers in each zone is equal to $5$. Bruno wins if the sum of the numbers written in each of the $9$ columns of the table is a prime number. Otherwise, Andrea wins. Show that Bruno always has a winning strategy.

Find all continuous functions $f: \mathbb R \to \mathbb R$ such that \[f(x^2-y^2)=f(x)^2 + f(y)^2, \quad \forall x,y \in \mathbb R.\]

Prove that for natural numbers $p$ and $q$, there exists a natural number $x$ such that $$(\sqrt{p}+\sqrt{p-1})^q=\sqrt{x}+\sqrt{x-1}$$ (As an example, if $p = 3, q = 2$, then $x$ can be taken to be $25$.)

In a town there are $n$ avenues running from south to north. They are numbered $1$ through $n$ (from west to east). There are $n$ streets running from west to east – they are also numbered $1$ through $n$ (from south to north). If you drive through the junction of the $k$th avenue and the $\ell$th street, you have to pay $k\ell$ kroner. How much do you at least have to pay for driving from the junction of the $1$st avenue and the $1$st street to the junction of the nth avenue and the $n$th street? (You also pay for the starting and finishing junctions.)

How many integers between $1000$ and $2000$ have all three of the numbers $15$, $20$, and $25$ as factors? $\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 5$

Is there an infinite sequence of real numbers $a_1,a_2,a_3,\dots$ such that \[a_1^m+a_2^m+a_3^m+\cdots=m\] for every positive integer $m?$

Consider $n$ students with numbers $1, 2, \ldots, n$ standing in the order $1, 2, \ldots, n.$ Upon a command, any of the students either remains on his place or switches his place with another student. (Actually, if student $A$ switches his place with student $B,$ then $B$ cannot switch his place with any other student $C$ any more until the next command comes.) Is it possible to arrange the students in the order $n,1, 2, \ldots, n-1$ after two commands ?

For any angle α with $0 < \alpha < 180^{\circ}$, we call a closed convex planar set an $\alpha$-set if it is bounded by two circular arcs (or an arc and a line segment) whose angle of intersection is $\alpha$. Given a (closed) triangle $T$ , find the greatest $\alpha$ such that any two points in $T$ are contained in an $\alpha$-set $S \subset T .$

There are $20$ cities in the County of Newton. Their populations are shown in the bar chart below. The average population of all the cities is indicated by the horizontal dashed line. Which of the following is closest to the total population of all $20$ cities? [asy] // made by SirCalcsALot size(300); pen shortdashed=linetype(new real[] {6,6}); // axis draw((0,0)--(0,9300), linewidth(1.25)); draw((0,0)--(11550,0), linewidth(1.25)); for (int i = 2000; i < 9000; i = i + 2000) { draw((0,i)--(11550,i), linewidth(0.5)+1.5*grey); label(string(i), (0,i), W); } for (int i = 500; i < 9300; i=i+500) { draw((0,i)--(150,i),linewidth(1.25)); if (i % 2000 == 0) { draw((0,i)--(250,i),linewidth(1.25)); } } int[] data = {8750, 3800, 5000, 2900, 6400, 7500, 4100, 1400, 2600, 1470, 2600, 7100, 4070, 7500, 7000, 8100, 1900, 1600, 5850, 5750}; int data_length = 20; int r = 550; for (int i = 0; i < data_length; ++i) { fill(((i+1)*r,0)--((i+1)*r, data[i])--((i+1)*r,0)--((i+1)*r, data[i])--((i+1)*r,0)--((i+1)*r, data[i])--((i+2)*r-100, data[i])--((i+2)*r-100,0)--cycle, 1.5*grey); draw(((i+1)*r,0)--((i+1)*r, data[i])--((i+1)*r,0)--((i+1)*r, data[i])--((i+1)*r,0)--((i+1)*r, data[i])--((i+2)*r-100, data[i])--((i+2)*r-100,0)); } draw((0,4750)--(11450,4750),shortdashed); label("Cities", (11450*0.5,0), S); label(rotate(90)*"Population", (0,9000*0.5), 10*W); [/asy] $\textbf{(A) }65{,}000 \qquad \textbf{(B) }75{,}000 \qquad \textbf{(C) }85{,}000 \qquad \textbf{(D) }95{,}000 \qquad \textbf{(E) }105{,}000$

Let $O_1O_2O_3$ be an acute angled triangle.Let $\omega_1$, $\omega_2$, $\omega_3$ be the circles with centres $O_1$, $O_2$, $O_3$ respectively such that any of two are tangent to each other. Circumcircle of $O_1O_2O_3$ intersects $\omega_1$ at $A_1$ and $B_1$, $\omega_2$ at $A_2$ and $B_2$, $\omega_3$ at $A_3$ and $B_3$ respectively. Prove that the incenter of triangle which can be constructed by lines $A_1B_1$, $A_2B_2$, $A_3B_3$ and the incenter of $O_1O_2O_3$ are coincide.

Let $a, b$, and $c$ be positive real numbers not greater than $2$. Prove the inequality $\frac{abc}{a + b + c} \le \frac43$

Find all positive integers $n$ for which the number $1!+2!+3!+\cdots+n!$ is a perfect power of an integer.

John has cut out these two polygons made out of unit squares. He joins them to each other to form a larger polygon (but they can't overlap). Find the smallest possible perimeter this larger polygon can have. He can rotate and reflect the cut out polygons.