$2019$ people (all of whom are perfect logicians), labeled from $1$ to $2019$, partake in a paintball duel. First, they decide to stand in a circle, in order, so that Person $1$ has Person $2$ to his left and person $2019$ to his right. Then, starting with Person $1$ and moving to the left, every person who has not been eliminated takes a turn shooting. On their turn, each person can choose to either shoot one non-eliminated person of his or her choice (which eliminates that person from the game), or deliberately miss. The last person standing wins. If, at any point, play goes around the circle once with no one getting eliminated (that is, if all the people playing decide to miss), then automatic paint sprayers will turn on, and end the game with everyone losing. Each person will, on his or her turn, always pick a move that leads to a win if possible, and, if there is still a choice in what move to make, will prefer shooting over missing, and shooting a person closer to his or her left over shooting someone farther from their left. What is the number of the person who wins this game? Put “$0$” if no one wins.

One afternoon a bakery finds that it has $300$ cups of flour and $300$ cups of sugar on hand. Annie and Sam decide to use this to make and sell some batches of cookies and some cakes. Each batch of cookies will require $1$ cup of flour and $3$ cups of sugar. Each cake will require $2$ cups of flour and $1$ cup of sugar. Annie thinks that each batch of cookies should sell for $2$ dollars and each cake for $1$ dollar, but Sam thinks that each batch of cookies should sell for $1$ dollar and each cake should sell for $3$ dollars. Find the difference between the maximum dollars of income they can receive if they use Sam's selling plan and the maximum dollars of income they can receive if they use Annie's selling plan.

Find all function $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for any three real number $a,b,c$ , if $ a + f(b) + f(f(c)) = 0$ : $$ f(a)^3 + bf(b)^2 + c^2f(c) = 3abc $$. [i]Proposed by Amirhossein Zolfaghari [/i]

Suppose that $ I$ is incenter of triangle $ ABC$ and $ l'$ is a line tangent to the incircle. Let $ l$ be another line such that intersects $ AB,AC,BC$ respectively at $ C',B',A'$. We draw a tangent from $ A'$ to the incircle other than $ BC$, and this line intersects with $ l'$ at $ A_1$. $ B_1,C_1$ are similarly defined. Prove that $ AA_1,BB_1,CC_1$ are concurrent.

What is the maximum number of primes that divide both the numbers $n^3+2$ and $(n+1)^3+2$ where $n$ is a positive integer? $ \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 1 \qquad\textbf{(D)}\ 0 \qquad\textbf{(E)}\ \text{None of above} $

The $\textbf{symmetric difference}$, $\triangle$, of a pair of sets is the set of elements in exactly one set. For example, \[\{1,2,3\}\triangle\{2,3,4\}=\{1,4\}.\] There are fifteen nonempty subsets of $\{1,2,3,4\}$. Assign each subset to exactly one of the squares in the grid to the right so that the following conditions are satisfied. (i) If $A$ and $B$ are in squares connected by a solid line then $A\triangle B$ has exactly one element. (ii) If $A$ and $B$ are in squares connected by a dashed line then the largest element of $A$ is equal to the largest element of $B$. You do not need to prove that your configuration is the only one possible; you merely need to find a configuration that satisfies the constraints above. (Note: In any other USAMTS problem, you need to provide a full proof. Only in this problem is an answer without justification acceptable.) [asy] size(150); defaultpen(linewidth(0.8)); draw((0,1)--(0,3)--(3,3)^^(2,3)--(2,2)--(3,2)--(3,1)--(1,1)--(1,2)--(0,2)^^(2,1)--(2,0)--(0,0)); draw(origin--(0,1)^^(1,0)--(3,2)^^(1,1)--(0,2)^^(1,2)--(0,3)^^(1,3)--(2,2),linetype("4 4")); real r=1/4; path square=(r,r)--(r,-r)--(-r,-r)--(-r,r)--cycle; int limit; for(int i=0;i<=3;i=i+1) { if (i==0) limit=2; else limit=3; for(int j=0;j<=limit;j=j+1) filldraw(shift(j,i)*square,white); } [/asy]

Let $n$ be an integer greater than $3$. Show that it is possible to divide a square into $n^2 + 1$ or more disjointed rectangles and with sides parallel to those of the square so that any line parallel to one of the sides intersects at most the interior of $n$ rectangles. Note: We say that two rectangles are [i]disjointed [/i] if they do not intersect or only intersect at their perimeters.

Quadrilateral can be cut into two isosceles triangles in two different ways. a) Can this quadrilateral be nonconvex? b) If given quadrilateral is convex, is it necessarily a rhomb?

Let $\{a_0,a_1,\ldots\}$ and $\{b_0,b_1,\ldots\}$ be two infinite sequences of integers such that \[(a_{n}-a_{n-1})(a_n-a_{n-2}) +(b_n-b_{n-1})(b_n-b_{n-2})=0\] for all integers $n\geq 2$. Prove that there exists a positive integer $k$ such that \[a_{k+2011}=a_{k+2011^{2011}}.\]

In town $ A,$ there are $ n$ girls and $ n$ boys, and each girl knows each boy. In town $ B,$ there are $ n$ girls $ g_1, g_2, \ldots, g_n$ and $ 2n \minus{} 1$ boys $ b_1, b_2, \ldots, b_{2n\minus{}1}.$ The girl $ g_i,$ $ i \equal{} 1, 2, \ldots, n,$ knows the boys $ b_1, b_2, \ldots, b_{2i\minus{}1},$ and no others. For all $ r \equal{} 1, 2, \ldots, n,$ denote by $ A(r),B(r)$ the number of different ways in which $ r$ girls from town $ A,$ respectively town $ B,$ can dance with $ r$ boys from their own town, forming $ r$ pairs, each girl with a boy she knows. Prove that $ A(r) \equal{} B(r)$ for each $ r \equal{} 1, 2, \ldots, n.$

Two squares with side $12$ lie exactly on top of each other. One square is rotated around a corner point through an angle of $30$ degrees relative to the other square. Determine the area of the common piece of the two squares. [asy] unitsize (2 cm); pair A, B, C, D, Bp, Cp, Dp, P; A = (0,0); B = (-1,0); C = (-1,1); D = (0,1); Bp = rotate(-30)*(B); Cp = rotate(-30)*(C); Dp = rotate(-30)*(D); P = extension(C, D, Bp, Cp); fill(A--Bp--P--D--cycle, gray(0.8)); draw(A--B--C--D--cycle); draw(A--Bp--Cp--Dp--cycle); label("$30^\circ$", (-0.5,0.1), fontsize(10)); [/asy]

Let $n >1$ be a natural number and $T_{n}(x)=x^n + a_{n-1}x^{n-1} + a_{n-2}x^{n-2} + ... + a_1 x^1 + a_0$.\\ Assume that for each nonzero real number $t $ we have $T_{n}(t+\frac {1}{t})=t^n+\frac {1}{t^n} $.\\ Prove that for each $0\le i \le n-1 $ $gcd (a_i,n) >1$. [i]Proposed by Morteza Saghafian[/i]

Let $\mathbb{N}$ denote the set of positive integers, and let $S$ be a set. There exists a function $f :\mathbb{N} \rightarrow S$ such that if $x$ and $y$ are a pair of positive integers with their difference being a prime number, then $f(x) \neq f(y)$. Determine the minimum number of elements in $S$.

a) The numbers $1, 2, . . . , 49$ are arranged in a square table as follows: [img]https://cdn.artofproblemsolving.com/attachments/5/0/c2e350a6ad0ebb8c728affe0ebb70783baf913.png[/img] Among these numbers we select an arbitrary number and delete from the table the row and the column which contain this number. We do the same with the remaining table of $36$ numbers, etc., $7$ times. Find the sum of the numbers selected. b) The numbers $1, 2, . . . , k^2$ are arranged in a square table as follows: [img]https://cdn.artofproblemsolving.com/attachments/2/d/28d60518952c3acddc303e427483211c42cd4a.png[/img] Among these numbers we select an arbitrary number and delete from the table the row and the column which contain this number. We do the same with the remaining table of $(k - 1)^2$ numbers, etc., $k$ times. Find the sum of the numbers selected.

Let $ G$ be finite group and $ \mathcal{K}$ a conjugacy class of $ G$ that generates $ G$. Prove that the following two statements are equivalent: (1) There exists a positive integer $ m$ such that every element of $ G$ can be written as a product of $ m$ (not necessarily distinct) elements of $ \mathcal{K}$. (2) $ G$ is equal to its own commutator subgroup. [i]J. Denes[/i]

Is it possible to divide a circle by three chords, different from diameters, into several equal parts?

Let $z_1$ and $z_2$ be complex numbers. Prove that \[|z_1+z_2|+|z_1-z_2|\leqslant |z_1|+|z_2|+\max\{|z_1|,|z_2|\}.\][i]Vlad Cerbu and Sorin Rădulescu[/i]

If $x,y,z>0$, prove that $(3x+y)(3y+z)(3z+x) \ge 64xyz$. When we have equality;

a) Suppose that $n$ is an odd integer. Prove that $k(n-k)$ is divisible by $2$ for all positive integers $k$. b) Find an integer $k$ such that $k(100-k)$ is not divisible by $11$. c) Suppose that $p$ is an odd prime, and $n$ is an integer. Prove that there is an integer $k$ such that $k(n-k)$ is not divisible by $p$. d) Suppose that $p,q$ are two different odd primes, and $n$ is an integer. Prove that there is an integer $k$ such that $k(n-k)$ is not divisible by any of $p,q$.

Let $a_1,a_2,a_3,\dots$ be a sequence of positive real numbers such that $a_ka_{k+2}=a_{k+1}+1$ for all positive integers $k$. If $a_1$ and $a_2$ are positive integers, find the maximum possible value of $a_{2014}$.

Let $ABC$ be a triangle with an interior point $P$ such that $\angle APB = \angle APC = \alpha$ and $\alpha > 180^o-\angle BAC$. The circumcircle of triangle $APB$ cuts $AC$ at $E$, the circumcircle of triangle $APC$ cuts $AB$ at $F$. Let $Q$ be the point in the triangle $AEF$ such that $\angle AQE = \angle AQF =\alpha$. Let $D$ be the symmetric point of $Q$ wrt $EF$. Angle bisector of $\angle EDF$ cuts $AP$ at $T$. a) Prove that $\angle DET = \angle ABC, \angle DFT = \angle ACB$. b) Straight line $PA$ cuts straight lines $DE, DF$ at $M, N$ respectively. Denote $I, J$ the incenters of the triangles $PEM, PFN$, and $K$ the circumcenter of the triangle $DIJ$. Straight line $DT$ cut $(K)$ at $H$. Prove that $HK$ passes through the incenter of the triangle $DMN$.

The external angle bisector of the angle $A$ of an acute-angled triangle $ABC$ meets the circumcircle of $\vartriangle ABC$ at point $T$. The perpendicular from the orthocenter $H$ of $\vartriangle ABC$ to the line $TA$ meets the line $BC$ at point $P$. The line $TP$ meets the circumcircce of $\vartriangle ABC$ at point $D$. Prove that $AB^2+DC^2=AC^2+BD^2$ A. Voidelevich

Find all integer solutions of the equation $2x^6 + y^7 = 11$.

Let $n$ be a positive integer with $n \ge 3$. Show that \[n^{n^{n^{n}}}-n^{n^{n}}\] is divisible by $1989$.

A manufacturer built a machine which will address $500$ envelopes in $8$ minutes. He wishes to build another machine so that when both are operating together they will address $500$ envelopes in $2$ minutes. The equation used to find how many minutes $x$ it would require the second machine to address $500$ envelopes alone is: $\textbf{(A)}\ 8-x=2 \qquad \textbf{(B)}\ \dfrac{1}{8}+\dfrac{1}{x}=\dfrac{1}{2} \qquad \textbf{(C)}\ \dfrac{500}{8}+\dfrac{500}{x}=500 \qquad \textbf{(D)}\ \dfrac{x}{2}+\dfrac{x}{8}=1 \qquad\\ \textbf{(E)}\ \text{None of these answers}$