Let $P(x)$ be a polynomial with integer coefficients that has at least one rational root. Let $n$ be a positive integer. Alan and Allan are playing a game. First, Alan writes down $n$ integers at $n$ different locations on a board. Then Allan may make moves of the following kind: choose a position that has integer $a$ written, then choose a different position that has integer $b$ written, then at the first position erase $a$ and in its place write $a+P(b)$. After any nonnegative number of moves, Allan may choose to end the game. Once Allan ends the game, his score is the number of times the mode (most common element) of the integers on the board appears. Find, in terms of $P(x)$ and $n$, the maximum score Allan can guarantee. [i]Henrick Rabinovitz[/i]

An $n \times m$ matrix is nice if it contains every integer from $1$ to $mn$ exactly once and $1$ is the only entry which is the smallest both in its row and in its column. Prove that the number of $n \times m$ nice matrices is $(nm)!n!m!/(n+m-1)!$.

Let a point $P$ lie inside a triangle $ABC$. The rays starting at $P$ and crossing the sides $BC$, $AC$, $AB$ under the right angle meet the circumcircle of $ABC$ at $A_{1}$, $B_{1}$, $C_{1}$ respectively. It is known that lines $AA_{1}$, $BB_{1}$, $CC_{1}$ concur at point $Q$. Prove that all such lines $PQ$ concur.

$A$ and $B$ lie on a circle. $P$ lies on the minor arc $AB$. $Q$ and $R$ (distinct from $P$) also lie on the circle, so that $P$ and $Q$ are equidistant from $A$, and $P$ and $R$ are equidistant from $B$. Show that the intersection of $AR$ and $BQ$ is the reflection of $P$ in $AB$.

In a convex quadrilateral $ABCD$, the lines $AB$ and $CD$ meet at point $E$ and the lines $AD$ and $BC$ meet at point $F$. Let $P$ be the intersection point of diagonals $AC$ and $BD$. Suppose that $\omega_1$ is a circle passing through $D$ and tangent to $AC$ at $P$. Also suppose that $\omega_2$ is a circle passing through $C$ and tangent to $BD$ at $P$. Let $X$ be the intersection point of $\omega_1$ and $AD$, and $Y$ be the intersection point of $\omega_2$ and $BC$. Suppose that the circles $\omega_1$ and $\omega_2$ intersect each other in $Q$ for the second time. Prove that the perpendicular from $P$ to the line $EF$ passes through the circumcenter of triangle $XQY$ . Proposed by Iman Maghsoudi

In a $100\times25$ rectangle table, fill in a positive real number in each blank. Let the number in the $i$th line, the $j$th column be $x_{i,j}(i=1,2,\cdots,100,j=1,2,\cdots,25)$ (shown in Fig.1 ). Then, we rearrange the numbers in each column: $x'_{1,j}\geq x'_{2,j}\geq\cdots\geq x'_{100,j}(j=1,2,\cdots,25)$ (shown in Fig.2 ). Find the minumum value of $k$, satisfying: As long as $\sum_{j=1}^{25}x_{i,j}\leq1$ for numbers in Fig.1 ($i=1,2,\cdots,100$), then $\sum_{j=1}^{25}x'_{i,j}\leq1$ for $i\geq k$ in Fig.2. $$\textbf{Fig.1}\\ \begin{tabular}{|c|c|c|c|} \hline $x_{1,1}$&$x_{1,2}$&$\cdots$&$x_{1,25}$\\ \hline $x_{2,1}$&$x_{2,2}$&$\cdots$&$x_{2,25}$\\ \hline $\cdots$&$\cdots$&$\cdots$&$\cdots$\\ \hline $x_{100,1}$&$x_{100,2}$&$\cdots$&$x_{100,25}$\\ \hline \end{tabular} \qquad\textbf{Fig.2}\\ \begin{tabular}{|c|c|c|c|} \hline $x'_{1,1}$&$x'_{1,2}$&$\cdots$&$x'_{1,25}$\\ \hline $x'_{2,1}$&$x'_{2,2}$&$\cdots$&$x'_{2,25}$\\ \hline $\cdots$&$\cdots$&$\cdots$&$\cdots$\\ \hline $x'_{100,1}$&$x'_{100,2}$&$\cdots$&$x'_{100,25}$\\ \hline \end{tabular}$$

Solve the equation $x^3 - [x] = 3$.

The circuit diagram drawn (see figure ) contains a battery $B$, a lamp $L$ and five switches $S_1$ to $S_5$. The probability that switch $S_3$ is closed (makes contact) is $\frac23$, for the other four switches that probability is $\frac12$ (the probabilities are mutually independent). Calculate the probability that the light is on. [asy] unitsize (2 cm); draw((-1,1)--(-0.5,1)); draw((-0.25,1)--(1,1)--(1,0.25)); draw((1,-0.25)--(1,-1)--(0.05,-1)); draw((-0.05,-1)--(-1,-1)--(-1,0.25)); draw((-1,0.5)--(-1,1)); draw((-1,1)--(-0.5,0.5)); draw((-0.25,0.25)--(0,0)); draw((-1,0)--(-0.75,0)); draw((-0.5,0)--(0,0)); draw((0,1)--(0,0.75)); draw((0,0.5)--(0,0)); draw((-0.25,1)--(-0.5,1.25)); draw((-1,0.25)--(-1.25,0.5)); draw((-0.5,0.5)--(-0.25,0.5)); draw((0,0.75)--(0.25,0.5)); draw((-0.75,0)--(-0.5,-0.25)); draw(Circle((1,0),0.25)); draw(((1,0) + 0.25*dir(45))--((1,0) + 0.25*dir(225))); draw(((1,0) + 0.25*dir(135))--((1,0) + 0.25*dir(315))); draw((0.05,-0.9)--(0.05,-1.1)); draw((-0.05,-0.8)--(-0.05,-1.2)); label("$L$", (1.25,0), E); label("$B$", (-0.1,-1.1), SW); label("$S_1$", (-0.5,1.25), NE); label("$S_2$", (-1.25,0.5), SW); label("$S_3$", (-0.5,0.5), SW); label("$S_4$", (0.25,0.5), NE); label("$S_5$", (-0.5,-0.25), SW); [/asy]

We consider the inscriptible pentagon $ABCDE$ in which $AB=BC=CD$ and the centroid of the pentagon coincides with the circumcenter. Prove that the pentagon $ABCDE$ is regular. [i]The centroid of a pentagon is the point in the plane of the pentagon whose position vector is equal to the average of the position vectors of the vertices.[/i]

The symbol $|a|$ means $+a$ if $a$ is greater than or equal to zero, and $-a$ if $a$ is less than or equal to zero; the symbol $<$ means "less than"; the symbol $>$ means "greater than." The set of values $x$ satisfying the inequality $|3-x|<4$ consists of all $x$ such that: $ \textbf{(A)}\ x^2<49 \qquad\textbf{(B)}\ x^2>1 \qquad\textbf{(C)}\ 1<x^2<49\qquad\textbf{(D)}\ -1<x<7\qquad\textbf{(E)}\ -7<x<1 $

The numbers from$ 1$ to $2016$ are divided into three (disjoint) subsets $A, B$ and $C$, each one contains exactly $672$ numbers. Prove that you can find three numbers, each from a different subset, such that the sum of two of them is equal to the third. [hide=original wording]Skaitļi no 1 līdz 2016 ir sadalīti trīs (nešķeļošās) apakškopās A, B un C, katranotām satur tieši 672 skaitļus. Pierādīt, ka var atrast trīs tādus skaitļus, katru no citas apakškopas, ka divu no tiem summa ir vienāda ar trešo. [/hide]

Find all integers that can be written in the form $\frac{(x+y+z)^2}{xyz}$, where $x,y,z$ are positive integers.

Find all functions $f : (0, +\infty) \to (0, +\infty)$ which are increasing on $[1, +\infty)$ and for all positive reals $a, b, c$ they fulfill the following relation $f(ab)f(bc)f(ca)=f(a^2b^2c^2)+f(a^2)+f(b^2)+f(c^2)$.

The twenty-first century began on January $1$, $2001$ and runs through December $31$, $2100$. Note that March $1$, $2014$ fell on Saturday, so there were five Mondays in March $2014$. In how many years of the twenty-first century does March have five Mondays?

Show that $n!=a^{n-1}+b^{n-1}+c^{n-1}$ has only finitely many solutions in positive integers. [i]Proposed by Dorlir Ahmeti, Albania[/i]

Across all $x \in \mathbb{R}$, find the maximum value of the expression $$\sin x + \sin 3x + \sin 5x.$$

Consider the sequence formed by the first digits of the powers of $5$:$$1,5,2,1,6,...$$ Prove any segment in this sequence, when written in reversed order, will be encountered in the sequence of the first digits of the powers of $2:$ $$1,2,4,8,1,3,6,1...$$

Given the nine-sided regular polygon $ A_1 A_2 A_3 A_4 A_5 A_6 A_7 A_8 A_9$, how many distinct equilateral triangles in the plane of the polygon have at least two vertices in the set $ \{A_1,A_2,...A_9\}$? $ \textbf{(A)} \ 30 \qquad \textbf{(B)} \ 36 \qquad \textbf{(C)} \ 63 \qquad \textbf{(D)} \ 66 \qquad \textbf{(E)} \ 72$

Let $ABC$ be a triangle with $\angle BAC = 75^o$ and $\angle ABC = 45^o$. If $BC =\sqrt3 + 1$, what is the perimeter of $\vartriangle ABC$?

How many different numbers with $6$ digits and multiples of $45$ can be written by adding one digit to the left and one to the right of $2008$?

In trapezoid $ABCD$, $AD$ is parallel to $BC$. Knowing that $AB=AD+BC$, prove that the bisector of $\angle A$ also bisects $CD$.

Let $ m$ and $ n$ be positive integers such that $ x^2 \minus{} mx \plus{}n \equal{} 0$ has real roots $ \alpha$ and $ \beta$. Prove that $ \alpha$ and $ \beta$ are integers [b]if and only if[/b] $ [m\alpha] \plus{} [m\beta]$ is the square of an integer. (Here $ [x]$ denotes the largest integer not exceeding $ x$)

[u]Round 1[/u] [b]p1.[/b] A circle is inscribed inside a square such that the cube of the radius of the circle is numerically equal to the perimeter of the square. What is the area of the circle? [b]p2.[/b] If the coefficient of $z^ky^k$ is $252$ in the expression $(z + y)^{2k}$, find $k$. [b]p3.[/b] Let $f(x) = \frac{4x^4-2x^3-x^2-3x-2}{x^4-x^3+x^2-x-1}$ be a function defined on the real numbers where the denominator is not zero. The graph of $f$ has a horizontal asymptote. Compute the sum of the x-coordinates of the points where the graph of $f$ intersects this horizontal asymptote. If the graph of f does not intersect the asymptote, write $0$. [u]Round 2 [/u] [b]p4.[/b] How many $5$-digit numbers have strictly increasing digits? For example, $23789$ has strictly increasing digits, but $23889$ and $23869$ do not. [b]p5.[/b] Let $$y =\frac{1}{1 +\frac{1}{9 +\frac{1}{5 +\frac{1}{9 +\frac{1}{5 +...}}}}}$$ If $y$ can be represented as $\frac{a\sqrt{b} + c}{d}$ , where $b$ is not divisible by any squares, and the greatest common divisor of $a$ and $d$ is $1$, find the sum $a + b + c + d$. [b]p6.[/b] “Counting” is defined as listing positive integers, each one greater than the previous, up to (and including) an integer $n$. In terms of $n$, write the number of ways to count to $n$. [u]Round 3 [/u] [b]p7.[/b] Suppose $p$, $q$, $2p^2 + q^2$, and $p^2 + q^2$ are all prime numbers. Find the sum of all possible values of $p$. [b]p8.[/b] Let $r(d)$ be a function that reverses the digits of the $2$-digit integer $d$. What is the smallest $2$-digit positive integer $N$ such that for some $2$-digit positive integer $n$ and $2$-digit positive integer $r(n)$, $N$ is divisible by $n$ and $r(n)$, but not by $11$? [b]p9.[/b] What is the period of the function $y = (\sin(3\theta) + 6)^2 - 10(sin(3\theta) + 7) + 13$? [u]Round 4 [/u] [b]p10.[/b] Three numbers $a, b, c$ are given by $a = 2^2 (\sum_{i=0}^2 2^i)$, $b = 2^4(\sum_{i=0}^4 2^i)$, and $c = 2^6(\sum_{i=0}^6 2^i)$ . $u, v, w$ are the sum of the divisors of a, b, c respectively, yet excluding the original number itself. What is the value of $a + b + c -u - v - w$? [b]p11.[/b] Compute $\sqrt{6- \sqrt{11}} - \sqrt{6+ \sqrt{11}}$. [b]p12.[/b] Let $a_0, a_1,..., a_n$ be such that $a_n\ne 0$ and $$(1 + x + x^3)^{341}(1 + 2x + x^2 + 2x^3 + 2x^4 + x^6)^{342} =\sum_{i=0}^n a_ix^i.$$ Find the number of odd numbers in the sequence $a_0, a_1,..., a_n$. PS. You should use hide for answers. Rounds 5-7 have been posted [url=https://artofproblemsolving.com/community/c4h2781343p24424617]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A mustache is created by taking the set of points $(x, y)$ in the $xy$-coordinate plane that satisfy $4 + 4 \cos(\pi x/24) \le y \le 6 + 6\cos(\pi x/24)$ and $-24 \le x \le 24$. What is the area of the mustache?

In the acute-angled triangle $ABC$ all angles are greater than $45^o$. Let $AM$ and $BN$ be the heights of this triangle and let $X$ and $Y$ be the points on $MA$ and $NB$, respecively, such that $|MX| =|MB|$ and $|NY| =|NA|$. Prove that $MN$ and $XY$ are parallel.