Find the smallest possible value of $x + y$ where $x, y \ge 1$ and $x$ and $y$ are integers that satisfy $x^2 - 29y^2 = 1$.

The sum of the first eighty positive odd integers subtracted from the sum of the first eighty positive even integers is $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 20 \qquad \textbf{(C)}\ 40 \qquad \textbf{(D)}\ 60 \qquad \textbf{(E)}\ 80$

A square is cut into three rectangles along two lines parallel to a side, as shown. If the perimeter of each of the three rectangles is 24, then the area of the original square is [asy] draw((0,0)--(9,0)--(9,9)--(0,9)--cycle); draw((3,0)--(3,9), dashed); draw((6,0)--(6,9), dashed);[/asy] $\text{(A)} \ 24 \qquad \text{(B)} \ 36 \qquad \text{(C)} \ 64 \qquad \text{(D)} \ 81 \qquad \text{(E)} \ 96$

Consider the convex quadrilateral $ABCD$. The point $P$ is in the interior of $ABCD$. The following ratio equalities hold: \[\angle PAD:\angle PBA:\angle DPA=1:2:3=\angle CBP:\angle BAP:\angle BPC\] Prove that the following three lines meet in a point: the internal bisectors of angles $\angle ADP$ and $\angle PCB$ and the perpendicular bisector of segment $AB$. [i]Proposed by Dominik Burek, Poland[/i]

Let $f:[0,\infty)\to(0,\infty)$ a continous function such that $\lim_{n\to\infty} \int^x_0 f(t)dt$ exists and it is finite. Prove that \[ \lim_{x\to\infty} \frac 1{\sqrt x} \int^x_0 \sqrt {f(t)} dt = 0. \] [i]Radu Miculescu[/i]

Let $ABCD$ be a quadrilateral inscribed in a circle $k$. Let the lines $AC\cap BD=O$, $AD\cap BC=P$, and $AB\cap CD=Q$. Line $QO$ intersects $k$ in points $M$ and $N$. Prove that $PM$ and $PN$ are tangent to $k$.

(a) Find the smallest number of lines drawn on the plane so that they produce exactly 2022 points of intersection. (Note: For 1 point of intersection, the minimum is 2; for 2 points, minimum is 3; for 3 points, minimum is 3; for 4 points, minimum is 4; for 5 points, the minimum is 4, etc.) (b) What happens if the lines produce exactly 2023 intersections?

a) A game master divides a group of $12$ players into two teams of six. The players do not know what the teams are, however the master gives each player a card containing the names of two other players: one of them is a teammate and the other is not, but the master does not tell the player which is which. Can the master write the names on the cards in such a way that the players can determine the teams? (All of the players can work together to do so.) b) On the next occasion, the game master writes the names of $3$ teammates and $1$ opposing player on each card (possibly in a mixed up order). Now he wants to write the names in such away that the players together cannot determine the two teams. Is it possible for him to achieve this? c) Can he write the names in such a way that the players together cannot determine the two teams, if now each card contains the names of $4$ teammates and $1$ opposing player (possibly in a mixed up order)?

Point $P$ is $\sqrt3$ units away from plane $A$. Let $Q$ be a region of $A$ such that every line through $P$ that intersects $A$ in $Q$ intersects $A$ at an angle between $30^o$ and $60^o$ . What is the largest possible area of $Q$?

A number plus $4$ is $2003$. What is the number?

Solve the following equation in real numbers: $\frac{(x^2-1)(|x|+1)}{x+sgnx}=[x+1].$

Suppose 100 people are gathered around at a park, each with an envelope with their name on it (all their names are distinct). Then, the envelopes are uniformly and randomly permuted between the people. If $N$ is the number of people who end up with their original envelope, find the expected value of $N^5$. [i]Proposed by Michael Duncan[/i]

Let $ ABCD$ be the inscribed quadrilateral with the circumcircle $ \omega$.Let $ \zeta$ be another circle that internally tangent to $ \omega$ and to the lines $ BC$ and $ AD$ at points $ M,N$ respectively.Let $ I_1,I_2$ be the incenters of the $ \triangle ABC$ and $ \triangle ABD$.Prove that $ M,I_1,I_2,N$ are collinear.

Let $f : \mathbb{Z} \to \mathbb{R}$ be a function satisfying $f(x) + f(y) + f(z) \ge 0$ for all integers $x, y, z$ with $x + y + z = 0$. Prove that \[ f(-2017) + f(-2016) + \cdots + f(2016) + f(2017) \ge 0. \]

For each positive integer $ n$, the mean of the first $ n$ terms of a sequence is $ n$. What is the $ 2008$th term of the sequence? $ \textbf{(A)}\ 2008 \qquad \textbf{(B)}\ 4015 \qquad \textbf{(C)}\ 4016 \qquad \textbf{(D)}\ 4,030,056 \qquad \textbf{(E)}\ 4,032,064$

Let $ABC$ be a triangle. Let $C^{\prime}$ and $A^{\prime}$ be the reflections of its vertices $C$ and $A$, respectively, in the altitude of triangle $ABC$ issuing from $B$. The perpendicular to the line $BA^{\prime}$ through the point $C^{\prime}$ intersects the line $BC$ at $U$; the perpendicular to the line $BC^{\prime}$ through the point $A^{\prime}$ intersects the line $BA$ at $V$. Prove that $UV \parallel CA$. Darij

Points $P,Q,R,S$ are taken on respective edges $AC$, $AB$, $BD$, and $CD$ of a tetrahedron $ABCD$ so that $PR$ and $QS$ intersect at point $N$ and $PS$ and $QR$ intersect at point $M$. The line $MN$ meets the plane $ABC$ at point $L$. Prove that the lines $AL$, $BP$, and $CQ$ are concurrent.

A necklace contains 2024 pearls, each one of them having one of the following colours: black, green and yellow. Each moment, we will switch each one of all pearls simultaneously to a new one following the following rules: i) If its two neighbours are of the same colour, then it'll be switched to that same colour. ii) If its two neighbours are of different colours, then it'll be switched to the third colour. a) Does there exist any necklace that can be transformed into a necklace that consists of only yellow pearls if initially half of the pearls are black and the other half is green? b) Does there exist a necklace that can be transformed into a necklace that consists of only yellow pearls if initially 998 pearls are black and the rest 1026 pearls are green?

Dorothy organized a party for the birthday of Duck Mom and she also prepared a cylindershaped cake. Since she was originally expecting to have $15$ guests, she divided the top of the cake into this many equal circular sectors, marking where the cuts need to be made. Just for fun Dorothy’s brother Donald split the top of the cake into $10$ equal circular sectors in such a way that some of the radii that he marked coincided with Dorothy’s original markings. Just before the arrival of the guests Douglas cut the cake according to all markings, and then he placed the cake into the fridge. This way they forgot about the cake and only got to eating it when only $6$ of them remained. Is it possible for them to divide the cake into $6$ equal parts without making any further cuts?

Let $ABCD$ be a cyclic quadrangle whose midpoints of diagonals $AC$ and $BD$ are $F$ and $G$, respectively. a) Prove the following implication: if the bisectors of angles at $B$ and $D$ of the quadrangle intersect at diagonal $AC$ then $\frac14 \cdot |AC| \cdot |BD| = | AG| \cdot |BF| \cdot |CG| \cdot |DF|$. b) Does the converse implication also always hold?

Prove the following inequality if we know that $a$ and $b$ are the legs of a right triangle , and $c$ is the length of the hypotenuse of this triangle: $$3a + 4b \le 5c.$$ When does equality holds?

Three runners, $X, Y$ and $Z$, participated in a race. $Z$ got held up at the start and began running last, while $Y$ was second from the start. During the race $Z$ exchanged positions with other contestants $6$ times, while $X$ did that $5$ times. It is known that $Y$ finished ahead of $X$. In what order did they finish?

Five friends sat in a movie theater in a row containing $5$ seats, numbered $1$ to $5$ from left to right. (The directions "left" and "right" are from the point of view of the people as they sit in the seats.) During the movie Ada went to the lobby to get some popcorn. When she returned, she found that Bea had moved two seats to the right, Ceci had moved one seat to the left, and Dee and Edie had switched seats, leaving an end seat for Ada. In which seat had Ada been sitting before she got up? $\textbf{(A) }1 \qquad \textbf{(B) } 2 \qquad \textbf{(C) } 3 \qquad \textbf{(D) } 4\qquad \textbf{(E) } 5$

Solve the system of equations: \[ x+y+z=a \] \[ x^2+y^2+z^2=b^2 \] \[ xy=z^2 \] where $a$ and $b$ are constants. Give the conditions that $a$ and $b$ must satisfy so that $x,y,z$ are distinct positive numbers.

A lattice point is a point in the plane with integer coordinates. How many lattice points are on the line segment whose endpoints are (3,17) and (48,281)? (Include both endpoints of the segment in your count.) $\textbf{(A)}\ 2 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 16 \qquad \textbf{(E)}\ 46$