Betty Lou and Peggy Sue take turns flipping switches on a $100 \times 100$ grid. Initially, all switches are "off". Betty Lou always flips a horizontal row of switches on her turn; Peggy Sue always flips a vertical column of switches. When they finish, there is an odd number of switches turned "on'' in each row and column. Find the maximum number of switches that can be on, in total, when they finish.

The natural numbers $a$ and $b$ are such that $ \frac{a+1}{b}+ \frac{b+1}{a}$ is an integer. Show that the greatest common divisor of a and b is not greater than $\sqrt{a+b}$.

Let $a$ and $b$ be the two possible values of $\tan\theta$ given that \[\sin\theta + \cos\theta = \dfrac{193}{137}.\] If $a+b=m/n$, where $m$ and $n$ are relatively prime positive integers, compute $m+n$.

Let $a_0$, $a_1$, $a_2$, $\dots$ be an infinite sequence of real numbers such that $a_0 = \frac{4}{5}$ and \[ a_{n} = 2 a_{n-1}^2 - 1 \] for every positive integer $n$. Let $c$ be the smallest number such that for every positive integer $n$, the product of the first $n$ terms satisfies the inequality \[ a_0 a_1 \dots a_{n - 1} \le \frac{c}{2^n}. \] What is the value of $100c$, rounded to the nearest integer?

Alice and Bob play a game on a complete graph $G$ with $2014$ vertices. They take moves in turn with Alice beginning. At each move Alice directs one undirected edge of $G$. At each move Bob chooses a positive integer number $m,$ $1 \le m \le 1000$ and after that directs $m$ undirected edges of $G$. The game ends when all edges are directed. If there is some directed cycle in $G$ Alice wins. Determine whether Alice has a winning strategy.

Let $\triangle ABC$ be a triangle with $AB=5, BC=6, CA=7$. Suppose $P$ is a point inside $\triangle ABC$ such that $\triangle BPA\sim \triangle APC$. If $AP$ intersects $BC$ at $X$, find $\frac{BX}{CX}$. [i]Proposed by Nathan Ramesh

The values of y which will satisfy the equations $ 2x^2\plus{}6x\plus{}5y\plus{}1\equal{}0, 2x\plus{}y\plus{}3\equal{}0$ may be found by solving: $\textbf{(A)}\ y^2+14y-7=0 \qquad \textbf{(B)}\ y^2+8y+1=0 \qquad \textbf{(C)}\ y^2+10y-7=0 \qquad \textbf{(D)}\ y^2+y-12=0 \qquad \textbf{(E)}\ \text{None of these equations}$

Numbers $7^2$,$8^2,\ldots,2023^2$,$2024^2$ are written on the board. Is it possible to add to one of them $7$, to some other one $8$, $\ldots$, to the remaining $2024$ such that all numbers became prime [i]M. Zorka[/i]

There is convex $2009$-gon on the plane. [b]a)[/b] Find the greatest number of vertices of $2009$-gon such that no two forms the side of the polygon. [b]b)[/b] Find the greatest number of vertices of $2009$-gon such that among any three of them there is one that is not connected with other two by side.

Let $ABC$ be an acute triangle with $AC > AB > BC$. The perpendicular bisectors of $AC$ and $AB$ cut line $BC$ at $D$ and $E$ respectively. Let $P$ and $Q$ be points on lines $AC$ and $AB$ respectively, both different from $A$, such that $AB = BP$ and $AC = CQ$, and let $K$ be the intersection of lines $EP$ and $DQ$. Let $M$ be the midpoint of $BC$. Show that $\angle DKA = \angle EKM$.

Prove that for all integers $k>2$, there exists $k$ distinct positive integers $a_1, \dots, a_k$ such that $$\sum_{1 \le i<j \le k} \frac{1}{a_ia_j} =1.$$ [i]Proposed by Anant Mudgal[/i]

Let $n,k$ be arbitrary positive integers. We fill the entries of an $n\times k$ array with integers such that all the $n$ rows contain the integers $1,2,\dots,k$ in some order. Add up the numbers in all $k$ columns – let $S$ be the largest of these sums. What is the minimal value of $S$?

On Babba Island they use a two-letter alphabet, $a$ and $b$, and every (finite) sequence of letters is a word. For each set $P$ of six words of $4$ letters each, we denote $N_P$ to the set of all words that do not contain any of the words of $P$ as a syllable (subword). Prove that if $N_P$ is finite, then all its words are of length less than or equal to $10$, and find a set $P$ such that $N_P$ is finite and contains at least one word of length $10$.

The harmonic mean of two positive numbers is the reciprocal of the arithmetic mean of their reciprocals. For how many ordered pairs of positive integers $(x,y)$ with $x<y$ is the harmonic mean of $x$ and $y$ equal to $6^{20}.$

Can the equation $f(g(h(x))) = 0$, where $f$, $g$, $h$ are quadratic polynomials, have the solutions $1, 2, 3, 4, 5, 6, 7, 8$? [i]S. Tokarev[/i]

Find all pairs of non-negative integers $(x,y)$ such that \[\sqrt{x+y}-\sqrt{x}-\sqrt{y}+2=0.\]

Let $ABC$ be an acute-angled scalene triangle. Let $P$ be a point on the extension of $AB$ past $B$, and $Q$ a point on the extension of $AC$ past $C$ such that $BPQC$ is a cyclic quadrilateral. Let $N$ be the foot of the perpendicular from $A$ to $BC$. If $NP = NQ$ then prove that $N$ is also the centre of the circumcircle of $APQ$.

In an acute-angled isosceles triangle $ABC$, altitudes $CC_1$ and $BB_1$ intersect the line passing through the vertex $A$ and parallel to the line $BC$, at points $P$ and $Q$. Let $A_0$ be the midpoint of side $BC$, and $AA_1$ the altitude. Lines $A_0C_1$ and $A_0B_1$ intersect line $PQ$ at points $K$ and $L$. Prove that the circles circumscribed around triangles $PQA_1, KLA_0, A_1B_1C_1$ and a circle with a diameter $AA_1$ intersect at one point.

In the plane, three distinct non-collinear points $A,B,C$ are marked. In each step, Ege can do one of the following: [list] [*] For marked points $X,Y$, mark the reflection of $X$ across $Y$. [*]For distinct marked points $X,Y,Z,T$ which do not form a parallelogram, mark the center of spiral similarity which takes segment $XY$ to $ZT$. [*] For distinct marked points $X,Y,Z,T$, mark the intersection of lines $XY$ and $ZT$. [/list] No matter how the points $A,B,C$ are marked in the beginning, can Ege always mark, after finitely many moves, a) The circumcenter of $\triangle ABC$. b) The incenter of $\triangle ABC$. Proposed by [i]Deniz Can Karaçelebi[/i]

Given a positive integer $n$, let $a_1,a_2,..,a_n$ be a sequence of nonnegative integers. A sequence of one or more consecutive terms of $a_1,a_2,..,a_n$ is called $dragon$ if their aritmetic mean is larger than 1. If a sequence is a $dragon$, then its first term is the $head$ and the last term is the $tail$. Suppose $a_1,a_2,..,a_n$ is the $head$ or/and $tail$ of some $dragon$ sequence; determine the minimum value of $a_1+a_2+\cdots +a_n$ in terms of $n$.

Given $2009 \times 4018$ rectangular board. Frame is a rectangle $n \times n$ or $n \times(n + 2)$ for $ ( n \geq 3 )$ without all cells which don’t have any common points with boundary of rectangle. Rectangles $1\times1,1\times 2,1\times 3$ and $ 2\times 4$ are also frames. Two players by turn paint all cells of some frame that has no painted cells yet. Player that can't make such move loses. Who has a winning strategy?

Point $O$ is the center of the regular octagon $ABCDEFGH$, and $X$ is the midpoint of the side $\overline{AB}.$ What fraction of the area of the octagon is shaded? $\textbf{(A) }\frac{11}{32} \qquad\textbf{(B) }\frac{3}{8} \qquad\textbf{(C) }\frac{13}{32} \qquad\textbf{(D) }\frac{7}{16}\qquad \textbf{(E) }\frac{15}{32}$ [asy] pair A,B,C,D,E,F,G,H,O,X; A=dir(45); B=dir(90); C=dir(135); D=dir(180); E=dir(-135); F=dir(-90); G=dir(-45); H=dir(0); O=(0,0); X=midpoint(A--B); fill(X--B--C--D--E--O--cycle,rgb(0.75,0.75,0.75)); draw(A--B--C--D--E--F--G--H--cycle); dot("$A$",A,dir(45)); dot("$B$",B,dir(90)); dot("$C$",C,dir(135)); dot("$D$",D,dir(180)); dot("$E$",E,dir(-135)); dot("$F$",F,dir(-90)); dot("$G$",G,dir(-45)); dot("$H$",H,dir(0)); dot("$X$",X,dir(135/2)); dot("$O$",O,dir(0)); draw(E--O--X); [/asy]

Suppose that \[\operatorname{lcm}(1024,2016)=\operatorname{lcm}(1024,2016,x_1,x_2,\ldots,x_n),\] with $x_1$, $x_2$, $\cdots$, $x_n$ are distinct postive integers. Find the maximum value of $n$. [i]Proposed by Le Duc Minh[/i]

Alireza is currently standing at the point $(0, 0)$ in the $x-y$ plane. At any given time, Alireza can move from the point $(x, y)$ to the point $(x + 1, y)$ or the point $(x, y + 1)$. However, he cannot move to any point of the form $(x, y)$ where $y \equiv 2x\,\, (\mod \,\,5)$. Let $p_k$ be the number of paths Alireza can take starting from the point $(0, 0)$ to the point $(k + 1, 2k + 1)$. Evaluate the sum $$\sum^{\infty}_{k=1} \frac{p_k}{5^k}.$$.

Let $ W$ be a dense, open subset of the real line $ \mathbb{R}$. Show that the following two statements are equivalent: (1) Every function $ f : \mathbb{R} \rightarrow \mathbb{R}$ continuous at all points of $ \mathbb{R} \setminus W$ and nondecreasing on every open interval contained in $ W$ is nondecreasing on the whole $ \mathbb{R}$. (2) $ \mathbb{R} \setminus W$ is countable. [i]E. Gesztelyi[/i]