Let $ a_{0},a_{1},\ldots,a_{n}$ be integers, one of which is nonzero, and all of the numbers are not less than $ \minus{} 1$. Prove that if \[ a_{0} \plus{} 2a_{1} \plus{} 2^{2}a_{2} \plus{} \cdots \plus{} 2^{n}a_{n} \equal{} 0,\] then $ a_{0} \plus{} a_{1} \plus{} \cdots \plus{} a_{n} > 0$.

The pentagons $P_1P_2P_3P_4P_5$ and$I_1I_2I_3I_4I_5$ are cyclic, where $I_i$ is the incentre of the triangle $P_{i-1}P_iP_{i+1}$ (reckoned cyclically, that is $P_0=P_5$ and $P_6=P_1$). Show that the lines $P_1I_1, P_2I_2, P_3I_3, P_4I_4$ and $P_5I_5$ meet in a single point.

Consider the sequence $(3^{2^n} + 1)_{n\geq 1}$. i) Prove there exist infinitely many primes, none dividing any term of the sequence. ii) Prove there exist infinitely many primes, each dividing some term of the sequence. [i](Dan Schwarz)[/i]

The sum of $n$ terms of an arithmetic progression is $153$, and the common difference is $2$. If the first interm is an integer, and $n>1$, then the number of possible values for $n$ is: $ \textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 6 $

Chim Tu has a large rectangular table. On it, there are finitely many pieces of paper with nonoverlapping interiors, each one in the shape of a convex polygon. At each step, Chim Tu is allowed to slide one piece of paper in a straight line such that its interior does not touch any other piece of paper during the slide. Can Chim Tu always slide all the pieces of paper off the table in finitely many steps?

Find all real quadruples $(a,b,c,d)$ satisfying the system of equations $$ \left\{ \begin{array}{ll} ab+cd = 6 \\ ac + bd = 3 \\ ad + bc = 2 \\ a + b + c + d = 6. \end{array} \right. $$

For how many values of $ a$ is it true that the line $ y \equal{} x \plus{} a$ passes through the vertex of the parabola $ y \equal{} x^2 \plus{} a^2$? $ \textbf{(A)}\ 0\qquad \textbf{(B)}\ 1\qquad \textbf{(C)}\ 2\qquad \textbf{(D)}\ 10\qquad \textbf{(E)}\ \text{infinitely many}$

In triangle $ABC$ points $P$ and $Q$ lies on the external bisector of $\angle A$ such that $B$ and $P$ lies on the same side of $AC$. Perpendicular from $P$ to $AB$ and $Q$ to $AC$ intersect at $X$. Points $P'$ and $Q'$ lies on $PB$ and $QC$ such that $PX=P'X$ and $QX=Q'X$. Point $T$ is the midpoint of arc $BC$ (does not contain $A$) of the circumcircle of $ABC$. Prove that $P',Q'$ and $T$ are collinear if and only if $\angle PBA+\angle QCA=90^{\circ}$.

Let $a_0,a_1,a_2,...$ be an infinite sequence of positive integers such that $a_0 = 1$ and $a_i^2 > a_{i-1}a_{i+1}$ for all $i > 0$. (a) Prove that $a_i < a_1^i$ for all $i > 1$. (b) Prove that $a_i > i$ for all $i$.

Let $ f$ be a polynomial with integer coefficients. Assume that there exists integers $ a$ and $ b$ such that $ f(a)\equal{}41$ and $ f(b)\equal{}49$. Prove that there exists an integer $ c$ such that $ 2009$ divides $ f(c)$. [i]Nanang Susyanto, Jogjakarta[/i]

In the diagram below, $BP$ bisects $\angle ABC$, $CP$ bisects $\angle BCA$, and $PQ$ is perpendicular to $BC$. If $BQ\cdot QC=2PQ^2$, prove that $AB+AC=3BC$. [img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvOC8zL2IwZjNjMDAxNWEwMTc1ZGNjMTkwZmZlZmJlMGRlOGRhYjk4NzczLnBuZw==&rn=VlRSTUMgMjAwNi5wbmc=[/img]

[b](a)[/b] Prove that every positive integer $n$ can be written uniquely in the form \[n=\sum_{j=1}^{2k+1}(-1)^{j-1}2^{m_j},\] where $k\geq 0$ and $0\le m_1<m_2\cdots <m_{2k+1}$ are integers. This number $k$ is called [i]weight[/i] of $n$. [b](b)[/b] Find (in closed form) the difference between the number of positive integers at most $2^{2017}$ with even weight and the number of positive integers at most $2^{2017}$ with odd weight.

Let $A$, $M$, and $C$ be nonnegative integers such that $A+M+C=10$. What is the maximum value of $A\cdot M\cdot C+A\cdot M+M\cdot C+C\cdot A$? $\text{(A)}\ 49 \qquad\text{(B)}\ 59 \qquad\text{(C)}\ 69 \qquad\text{(D)}\ 79\qquad\text{(E)}\ 89$

Let $A,B$ be matrices of dimension $2010\times2010$ which commute and have real entries, such that $A^{2010}=B^{2010}=I$, where $I$ is the identity matrix. Prove that if $\operatorname{tr}(AB)=2010$, then $\operatorname{tr}(A)=\operatorname{tr}(B)$.

Let $ABC$ be a triangle. Let $\Omega$ be the brocard point. Prove that $\left(\frac{A\Omega}{BC}\right)^2+\left(\frac{B\Omega}{AC}\right)^2+\left(\frac{C\Omega}{AB}\right)^2\ge 1$

Each of the $20$ balls is tossed independently and at random into one of the $5$ bins. Let $p$ be the probability that some bin ends up with $3$ balls, another with $5$ balls, and the other three with $4$ balls each. Let $q$ be the probability that every bin ends up with $4$ balls. What is $\frac{p}{q}$? $\textbf{(A)}\ 1 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 8 \qquad\textbf{(D)}\ 12 \qquad\textbf{(E)}\ 16$

Let $(m,n,N)$ be a triple of positive integers. Bruce and Duncan play a game on an m\times n array, where the entries are all initially zeroes. The game has the following rules. $\bullet$ The players alternate turns, with Bruce going first. $\bullet$ On Bruce's turn, he picks a row and either adds $1$ to all of the entries in the row or subtracts $1$ from all the entries in the row. $\bullet$ On Duncan's turn, he picks a column and either adds $1$ to all of the entries in the column or subtracts $1$ from all of the entries in the column. $\bullet$ Bruce wins if at some point there is an entry $x$ with $|x|\ge N$. Find all triples $(m, n,N)$ such that no matter how Duncan plays, Bruce has a winning strategy.

Cyclic quadrilateral $ABCD$ with circumcenter $O$ is given. Point $P$ is the intersection of diagonals $AC$ and $BD$. Let $M$ and $N$ be the midpoint of the sides $AD$ and $BC$, respectively. Suppose that $\omega_1$, $\omega_2$ and $\omega_3$ be the circumcircle of triangles $ADP$, $BCP$ and $OMN$, respectively. The intersection point of $\omega_1$ and $\omega_3$, which is not on the arc $APD$ of $\omega_1$, is $E$ and the intersection point of $\omega_2$ and $\omega_3$, which is not on the arc $BPC$ of $\omega_2$, is $F$. Prove that $OF=OE$. Proposed by Seyed Amirparsa Hosseini Nayeri

The sequence $ \{x_n\}$ is defined by \[ \left\{ \begin{array}{l}x_1 \equal{} \frac{1}{2} \\x_n \equal{} \frac{{\sqrt {x_{n \minus{} 1} ^2 \plus{} 4x_{n \minus{} 1} } \plus{} x_{n \minus{} 1} }}{2} \\\end{array} \right.\] Prove that the sequence $ \{y_n\}$, where $ y_n\equal{}\sum_{i\equal{}1}^{n}\frac{1}{{{x}_{i}}^{2}}$, has a finite limit and find that limit.

Integers $1,2, ...,2n$ are arbitrarily assigned to boxes labeled with numbers $1, 2,..., 2n$. Now, we add the number assigned to the box to the number on the box label. Show that two such sums give the same remainder modulo $2n$.

[b]8.[/b] Among all point lattices on the plane intersecting every closed convex region of unit width, which on's fundamental parallelogram has the largest area? ([b]G.36[/b]) [L. Fejes-Tóth]

Let $ABC$ be a triangle with $AB=20$ and $AC=18$. $E$ is on segment $AC$ and $F$ is on segment $AB$ such that $AE=AF=8$. Let $BE$ and $CF$ intersect at $G$. Given that $AEGF$ is cyclic, then $BC=m\sqrt{n}$ for positive integers $m$ and $n$ such that $n$ is not divisible by the square of any prime. Compute $100m+n$. [i] Proposed by James Lin[/i]

Let $ABC$ be an isosceles triangle with $AC=BC$, whose incentre is $I$. Let $P$ be a point on the circumcircle of the triangle $AIB$ lying inside the triangle $ABC$. The lines through $P$ parallel to $CA$ and $CB$ meet $AB$ at $D$ and $E$, respectively. The line through $P$ parallel to $AB$ meets $CA$ and $CB$ at $F$ and $G$, respectively. Prove that the lines $DF$ and $EG$ intersect on the circumcircle of the triangle $ABC$. [i]Proposed by Hojoo Lee[/i]

In acute triangle $ABC,$ the feet of the altitudes are $A_1,B_1,$ and $C_1$ (with the usual notations on sides $BC,CA,$ and $AB$ respectively). The circumcircles of triangles $AB_1C_1$ and $BC_1A_1$ intersect at the circumcircle of triangle $ABC$ ar points $P\neq A$ and $Q\neq B,$ respectively. Prove that lines $AQ, BP$ and the Euler line of triangle $ABC$ are either concurrent or parallel to each other. [i]Proposed by Géza Kós, Budapest[/i]

Consider the rectangle in the coordinate plane with corners $(0, 0)$, $(16, 0)$, $(16, 4)$, and $(0, 4)$. For a constant $x_0 \in [0, 16]$, the curves \[ \{(x, y) : y = \sqrt{x} \,\text{ and }\, 0 \leq x \leq 16\} \quad \text{and} \quad \{(x_0, y) : 0 \leq y \leq 4\} \] partition this rectangle into four 2D regions. Over all choices of $x_0$, determine the smallest possible sum of the areas of the bottom-left and top-right 2D regions in this partition. (The bottom-left region is $\{(x, y) : 0 \leq x < x_0 \,\text{ and }\, 0 \leq y < \sqrt{x}\}$, and the top-right region is $\{(x, y) : x_0 < x \leq 16 \,\text{ and }\, \sqrt{x} < y \leq 4\}$.)