[color=darkblue]Points $ X$, $ Y$ and $ Z$ are situated on the sides $ (BC)$, $ (CA)$ and $ (AB)$ of the triangles $ ABC$, such that triangles $ XYZ$ and $ ABC$ are similiar. Prove that circumcircle of $ AYZ$ passes through a fixed point.[/color]

For an arbitrary real number $ x$, $ \lfloor x\rfloor$ denotes the greatest integer not exceeding $ x$. Prove that there is exactly one integer $ m$ which satisfy $ \displaystyle m\minus{}\left\lfloor \frac{m}{2005}\right\rfloor\equal{}2005$.

Plot points $A,B,C$ at coordinates $(0, 0)$, $(0, 1)$, and $(1,1)$ in the plane, respectively. Let $S$ denote the union of the two line segments $AB$ and $BC$. Let $X_1$ be the area swept out when Bobby rotates $S$ counterclockwise $45$ degrees about point $A$. Let $X_2$ be the area swept out when Calvin rotates $S$ clockwise $45$ degrees about point $A$. Find $\frac{X_1+X_2}{2}$ .

A polynomial is called Fermat polynomial if it can be written as the sum of squares of two polynomials with integer coefficients. Suppose that $f(x)$ is a Fermat polynomial such that $f(0)=1000$. Prove that $f(x)+2x$ is not a fermat polynomial

[b]p1.[/b] Sujay sees a shooting star go across the night sky, and took a picture of it. The shooting star consists of a star body, which is bounded by four quarter-circle arcs, and a triangular tail. Suppose $AB = 2$, $AC = 4$. Let the area of the shooting star be $X$. If $6X = a-b\pi$ for positive integers $a, b$, find $a + b$. [img]https://cdn.artofproblemsolving.com/attachments/0/f/f9c9ff23416565760df225c133330e795b9076.png[/img] [b]p2.[/b] Assuming that each distinct arrangement of the letters in $DISCUSSIONS$ is equally likely to occur, what is the probability that a random arrangement of the letters in $DISCUSSIONS$ has all the $S$’s together? [b]p3.[/b] Evaluate $$\frac{(1 + 2022)(1 + 2022^2)(1 + 2022^4) ... (1 + 2022^{2^{2022}})}{1 + 2022 + 2022^2 + ... + 2022^{2^{2023}-1}} .$$ [b]p4.[/b] Dr. Kraines has $27$ unit cubes, each of which has one side painted red while the other five are white. If he assembles his cubes into one $3 \times 3 \times 3$ cube by placing each unit cube in a random orientation, what is the probability that the entire surface of the cube will be white, with no red faces visible? If the answer is $2^a3^b5^c$ for integers $a$, $b$, $c$, find $|a + b + c|$. [b]p5.[/b] Let S be a subset of $\{1, 2, 3, ... , 1000, 1001\}$ such that no two elements of $S$ have a difference of $4$ or $7$. What is the largest number of elements $S$ can have? [b]p6.[/b] George writes the number $1$. At each iteration, he removes the number $x$ written and instead writes either $4x+1$ or $8x+1$. He does this until $x > 1000$, after which the game ends. What is the minimum possible value of the last number George writes? [b]p7.[/b] List all positive integer ordered pairs $(a, b)$ satisfying $a^4 + 4b^4 = 281 \cdot 61$. [b]p8.[/b] Karthik the farmer is trying to protect his crops from a wildfire. Karthik’s land is a $5 \times 6$ rectangle divided into $30$ smaller square plots. The $5$ plots on the left edge contain fire, the $5$ plots on the right edge contain blueberry trees, and the other $5 \times 4$ plots of land contain banana bushes. Fire will repeatedly spread to all squares with bushes or trees that share a side with a square with fire. How many ways can Karthik replace $5$ of his $20$ plots of banana bushes with firebreaks so that fire will not consume any of his prized blueberry trees? [b]p9.[/b] Find $a_0 \in R$ such that the sequence $\{a_n\}^{\infty}_{n=0}$ defined by $a_{n+1} = -3a_n + 2^n$ is strictly increasing. [b]p10.[/b] Jonathan is playing with his life savings. He lines up a penny, nickel, dime, quarter, and half-dollar from left to right. At each step, Jonathan takes the leftmost coin at position $1$ and uniformly chooses a position $2 \le k \le 5$. He then moves the coin to position $k$, shifting all coins at positions $2$ through $k$ leftward. What is the expected number of steps it takes for the half-dollar to leave and subsequently return to position $5$? PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

All vertices of a hexagon, whose sides may intersect at points other than the vertices, lie on a circle. (a) Draw a hexagon such that it has the largest possible number of points of self-intersection. (b) Prove that this number is indeed maximum. (NB Vassiliev)

Is it possible to draw in the plane the graph presented in the figure so that all the vertices are different points and all the edges are unit segments? (The segments can intersect at points different from vertices.)

A plane is cut into unit squares, which are then colored in $n$ colors. A polygon $P$ is created from $n$ unit squares that are connected by their sides. It is known that any cell polygon created by $P$ with translation, covers $n$ unit squares in different colors. Prove that the plane can be covered with copies of $P$ so that each cell is covered exactly once.

There are 7 pieces of paper in the hat. On the $ n $th piece of paper there is written the number $ 2^n-1 $ ($ n = 1, 2, \ldots, 7 $). We draw cards randomly until the sum exceeds 124. What is the most probable value of this sum?

Is it possible to have numbers $1, 2,..., 10$ put in a row in some order so that each of them, starting from the second, differs from the previous one by a whole number of percent?

$\vartriangle A_0B_0C_0$ has side lengths $A_0B_0 = 13$, $B_0C_0 = 14$, and $C_0A_0 = 15$. $\vartriangle A_1B_1C_1$ is inscribed in the incircle of $\vartriangle A_0B_0C_0$ such that it is similar to the first triangle. Beginning with $\vartriangle A_1B_1C_1$, the same steps are repeated to construct $\vartriangle A_2B_2C_2$, and so on infinitely many times. What is the value of $\sum_{i=0}^{\infty} A_iB_i$?

Alice, Bob, and Charlie are playing a game with $6$ cards numbered $1$ through $6.$ Each player is dealt $2$ cards uniformly at random. On each player’s turn, they play one of their cards, and the winner is the person who plays the median of the three cards played. Charlie goes last, so Alice and Bob decide to tell their cards to each other, trying to prevent him from winning whenever possible. Compute the probability that Charlie wins regardless.

Let $D$ be a point on side $[BC]$ of $\triangle ABC$ such that $|AB|=|AC|$, $|BD|=6$ and $|DC|=10$. If the incircles of $\triangle ABD$ and $\triangle ADC$ touch side $[AD]$ at $E$ and $F$, respectively, what is$|EF|$? $ \textbf{(A)}\ \dfrac{1}{\sqrt{2}} \qquad\textbf{(B)}\ \dfrac{2}{\sqrt{3}} \qquad\textbf{(C)}\ 1 \qquad\textbf{(D)}\ \dfrac{9}{8} \qquad\textbf{(E)}\ 2 $

Determine all values of $x$ in the interval $0 \leq x \leq 2\pi$ which satisfy the inequality \[ 2 \cos{x} \leq \sqrt{1+\sin{2x}}-\sqrt{1-\sin{2x}} \leq \sqrt{2}. \]

On the 8×8 chessboard there are two identical markers in the squares a1 and c3. Alice and Bob in turn make the following moves (the first move is Alice’s): a player picks any marker and moves it horizontally to the right or vertically upwards through any number of squares. The aim of each player is to get tothe square h8. Which player has a winning strategy no matter what does his opponent? (There may be only one marker on a square,the markers may not go through each other.) The 8x8 chessboard consists of columns lettered a to h from left to right and rows numbered 1-8 from bottom to top [img]https://cdn.artofproblemsolving.com/attachments/1/f/4c5548f606fda915e0a50a8cf886ff93e1f86d.png[/img]

Consider the sytem of equations \[ a_{11}x_1+a_{12}x_2+a_{13}x_3 = 0 \]\[a_{21}x_1+a_{22}x_2+a_{23}x_3 =0\]\[a_{31}x_1+a_{32}x_2+a_{33}x_3 = 0 \] with unknowns $x_1, x_2, x_3$. The coefficients satisfy the conditions: a) $a_{11}, a_{22}, a_{33}$ are positive numbers; b) the remaining coefficients are negative numbers; c) in each equation, the sum ofthe coefficients is positive. Prove that the given system has only the solution $x_1=x_2=x_3=0$.

A set $P$ of $2002$ persons is given. The family of subsets of $P$ containing exactly $1001$ persons has the property that the number of acquaintance pairs in each such subset is the same. (It is assumed that the acquaintance relation is symmetric). Find the best lower estimation of the acquaintance pairs in the set $P$.

In a round-robin tournament with $n$ players $P_1$, $P_2$, $\ldots$, $P_n$ (where $n > 1$), each player plays one game with each of the other players and the rules are such that no ties can occur. Let $w_r$ and $l_r$ be the number of games won and lost, respectively, by $P_r$. Show that \[ \sum_{r=1}^nw_r^2 = \sum_{r=1}^nl_r^2. \]

Consider an open interval of length $1/n$ on the real number line, where $n$ is a positive integer. Prove that the number of irreducible fractions $p/q$, with $1\le q\le n$, contained in the given interval is at most $(n+1)/2$.

Let $n$ be a positive integer. Also let $a_1, a_2, \dots, a_n$ and $b_1,b_2,\dots, b_n$ be real numbers such that $a_i+b_i>0$ for $i=1,2,\dots, n$. Prove that $$\sum_{i=1}^n \frac{a_ib_i-b_i^2}{a_i+b_i}\le\frac{\displaystyle \sum_{i=1}^n a_i\cdot \sum_{i=1}^n b_i - \left( \sum_{i=1}^n b_i\right) ^2}{\displaystyle\sum_{i=1}^n (a_i+b_i)}$$. (Proposed by Daniel Strzelecki, Nicolaus Copernicus University in Toruń, Poland)

Find the smallest nonnegative integer $n$ for which $\binom{2006}{n}$ is divisible by $7^3$.

Given a regular $n$-gon inscribed in a circle of radius 1, where $n > 3$. Define $G(n)$ as the average length of the diagonals of this $n$-gon. Prove that if $n \rightarrow \infty, G(n) \rightarrow \frac{4}{\pi}$.

The sum $2\frac17+3\frac12+5\frac{1}{19}$ is between $\text{(A)}\ 10\text{ and }10\frac12 \qquad \text{(B)}\ 10\frac12 \text{ and } 11 \qquad \text{(C)}\ 11\text{ and }11\frac12 \qquad \text{(D)}\ 11\frac12 \text{ and }12 \qquad \text{(E)}\ 12\text{ and }12\frac12$

Let $\omega$ be the circumcircle of acute triangle $ABC$ where $\angle A<\angle B$ and $M,N$ be the midpoints of minor arcs $BC,AC$ of $\omega$ respectively. The line $PC$ is parallel to $MN$, intersecting $\omega$ at $P$ (different from $C$). Let $I$ be the incentre of $ABC$ and let $PI$ intersect $\omega$ again at the point $T$. 1) Prove that $MP\cdot MT=NP\cdot NT$; 2) Let $Q$ be an arbitrary point on minor arc $AB$ and $I,J$ be the incentres of triangles $AQC,BCQ$. Prove that $Q,I,J,T$ are concyclic.

Determine all positive integers $n$ such that the following statement holds: If a convex polygon with with $2n$ sides $A_1 A_2 \ldots A_{2n}$ is inscribed in a circle and $n-1$ of its $n$ pairs of opposite sides are parallel, which means if the pairs of opposite sides \[(A_1 A_2, A_{n+1} A_{n+2}), (A_2 A_3, A_{n+2} A_{n+3}), \ldots , (A_{n-1} A_n, A_{2n-1} A_{2n})\] are parallel, then the sides \[ A_n A_{n+1}, A_{2n} A_1\] are parallel as well.