Given is a square board measuring $1 995 \times 1 995$. These cells are painted with black and white paints in a checkerboard pattern, so that the corner cells are black. A spider sitting on one of the black cells can crawl to the cell on the same side as the one it occupies in one step. Prove that a spider can always reach a fly sitting motionless in another black cell by visiting all the cells of the board once.

There are $n{}$ passengers in the queue to board a $n{}$-seat plane. The first one in the queue is an absent-minded old lady who, after boarding the plane, sits down at a randomly selected place. Each subsequent passenger sits in his seat if it is free, and in a random seat otherwise. How many passengers will be out of their seats on average? [i]Proposed by A. Zaslavsky[/i]

Let $a$ and $b$ be positive integers, and let $A$ and $B$ be finite sets of integers satisfying (i) $A$ and $B$ are disjoint; (ii) if an integer $i$ belongs to either to $A$ or to $B$, then either $i+a$ belongs to $A$ or $i-b$ belongs to $B$. Prove that $a\left\lvert A \right\rvert = b \left\lvert B \right\rvert$. (Here $\left\lvert X \right\rvert$ denotes the number of elements in the set $X$.)

At a meeting, there are \( N \) people who do not know each other. Prove that it is possible to introduce them in such a way that no three of them have the same number of acquaintances.

Let $n\ge 2$ be a positive integer. Fill up a $n\times n$ table with the numbers $1,2,...,n^2$ exactly once each. Two cells are termed adjacent if they have a common edge. It is known that for any two adjacent cells, the numbers they contain differ by at most $n$. Show that there exist a $2\times 2$ square of adjacent cells such that the diagonally opposite pairs sum to the same number.

Let $ABC$ be an equilateral triangle with the side of $20$ units. Amir divides this triangle into $400$ smaller equilateral triangles with the sides of $1$ unit. Reza then picks $4$ of the vertices of these smaller triangles. The vertices lie inside the triangle $ABC$ and form a parallelogram with sides parallel to the sides of the triangle $ABC.$ There are exactly $46$ smaller triangles that have at least one point in common with the sides of this parallelogram. Find all possible values for the area of this parallelogram. [asy] unitsize(150); defaultpen(linewidth(0.7)); int n = 20; /* # of vertical lines, including BC */ pair A = (0,0), B = dir(-30), C = dir(30); draw(A--B--C--cycle,linewidth(1)); dot(A,UnFill(0)); dot(B,UnFill(0)); dot(C,UnFill(0)); label("$A$",A,W); label("$C$",C,NE); label("$B$",B,SE); for(int i = 1; i < n; ++i) { draw((i*A+(n-i)*B)/n--(i*A+(n-i)*C)/n); draw((i*B+(n-i)*A)/n--(i*B+(n-i)*C)/n); draw((i*C+(n-i)*A)/n--(i*C+(n-i)*B)/n); }[/asy] [Thanks azjps for drawing the diagram.] [hide="Note"][i]Note:[/i] Vid changed to Amir, and Eva change to Reza![/hide]

Eight card players are seated around a table. One remarks that at some moment, any player and his two neighbours have altogether an odd number of winning cards. Show that any player has at that moment at least one winning card.

$7501$ points in a $96 \times 96$ square is marked. We call the $4 \times 4$ square without its central $2 \times 2$ square a [i]frame[/i]. Prove that there exist a frame with sides parallel to the $96 \times 96$ square (not necessarily from the grid lines) that contains at least $10$ marked points. [i]Proposed by Negar Babashah, Shima Amirbeygie[/i] [b]Rated 5[/b]

Let $ S \equal{} \{x_1, x_2, \ldots, x_{k \plus{} l}\}$ be a $ (k \plus{} l)$-element set of real numbers contained in the interval $ [0, 1]$; $ k$ and $ l$ are positive integers. A $ k$-element subset $ A\subset S$ is called [i]nice[/i] if \[ \left |\frac {1}{k}\sum_{x_i\in A} x_i \minus{} \frac {1}{l}\sum_{x_j\in S\setminus A} x_j\right |\le \frac {k \plus{} l}{2kl}\] Prove that the number of nice subsets is at least $ \dfrac{2}{k \plus{} l}\dbinom{k \plus{} l}{k}$. [i]Proposed by Andrey Badzyan, Russia[/i]

Prove that every polygon that has a center of symmetry can be dissected into a square such that it is divided into finitely many polygonal pieces, and all the pieces can only be translated. (In other words, the original polygon can be divided into polygons $A_1,A_2,\dotsc ,A_n$, a square can be divided into polygons a $B_1,B_2,\dotsc ,B_n$ such that for $1\leqslant i\leqslant n$ polygon $B_i$ is a translated copy of polygon $A_i$.)

Let $A$ and $B$ be two $n \times n$ square arrays. The cells of $A$ are labelled by the numbers from $1$ to $n^2$ from left to right starting from the top row, whereas the cells of $B$ are labelled by the numbers from $1$ to $n^2$ along rising north-easterly diagonals starting with the upper left-hand corner. Stack the array $B$ on top of the array $A$. If two overlapping cells have the same number, they are coloured red. Determine those $n$ for which there is at least one red cell other than the cells at top left corner, bottom right corner and the centre (when $n$ is odd). Below shows the arrays for $n=4$. [img]https://cdn.artofproblemsolving.com/attachments/8/e/cc8a435cb28420ccf91340023d440e39f0e849.png[/img]

At a party, there are $2011$ people with a glass of fruit juice each sitting around a circular table. Once a second, they clink glasses obeying the following two rules: (a) They do not clink glasses crosswise. (b) At each point of time, everyone can clink glasses with at most one other person. How many seconds pass at least until everyone clinked glasses with everybody else? [i](Swiss Mathematical Olympiad 2011, Final round, problem 1)[/i]

Let $ABC$ be an equilateral triangle. From the vertex $A$ we draw a ray towards the interior of the triangle such that the ray reaches one of the sides of the triangle. When the ray reaches a side, it then bounces off following the law of reflection, that is, if it arrives with a directed angle $\alpha$, it leaves with a directed angle $180^{\circ}-\alpha$. After $n$ bounces, the ray returns to $A$ without ever landing on any of the other two vertices. Find all possible values of $n$.

Tom Sawyer must whitewash a circular fence consisting of $N$ planks. He whitewashes the fence going clockwise and following the rule: He whitewashes the first plank, skips two planks, whitewashes one, skips three, and so on. Some planks may be whitewashed several times. Tom believes that all planks will be whitewashed sooner or later, but aunt Polly is sure that some planks will remain unwhitewashed forever. Prove that Tom is right if $N$ is a power of two, otherwise aunt Polly is right.

Two persons are dividing a piece of cheese. The first person cuts it into two pieces, then the second person cuts one of these pieces into two, then again the first person cuts one of the pieces into two, and so until they have 5 pieces. After that the first person chooses one of the pieces, then the second person chooses one of remaining pieces and so on until all pieces are taken. For each of the players, what is the maximal amount of cheese he can get for certain, regardless of the other's actions?

[b]p1.[/b] How many ordered triples of integers $(a, b, c)$ where $1 \le a, b, c \le 10$ are such that for every natural number, the equation $(a + n)x^2 + (b + 2n)x + c + n = 0$ has at least one real root? [b]p2.[/b] Find the smallest integer $n$ such that we can cut a $n \times n$ grid into $5$ rectangles with distinct side lengths in $\{1, 2, 3..., 10\}$. Every value is used exactly once. [b]p3.[/b] A plane is flying at constant altitude along a circle of radius $12$ miles with center at a point $A$.The speed of the aircraft is v. At some moment in time, a missile is fired at the aircraft from the point $A$, which has speed v and is guided so that its velocity vector always points towards the aircraft. How far does the missile travel before colliding with the aircraft? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In an exam with k questions, n students are taking part. A student fails the exam if he answers correctly less than half of all questions. Call a question easy if more than half of all students answer it correctly. For which pairs (k, n) of positive integers is it possible that (a) all students fail the exam although all questions are easy; (b) no student fails the exam although no question is easy?

On a chessboard ($8\times 8$ squares with sides of length $1$) two diagonally opposite corner squares are taken away. Can the board now be covered with nonoverlapping rectangles with sides of lengths $1$ and $2$?

Suppose that $\{D_n\}_{n\ge 1}$ is an finite sequence of disks in the plane whose total area is less than $1$. Prove that it is possible to rearrange the disks so that they are disjoint from each other and all contained inside a disk of area $4$.

You have a $9 \times 9$ board with white squares. A tile can be moved from one square to another neighbor (tiles that share one side). If we paint some squares of black, we say that such coloration is [i]good [/i] if there is a white square where we can place a chip that moving through white squares can return to the initial square having passed through at least $3$ boxes, without passing the same square twice. Find the highest possible value of $k$ such that any form of painting $k$ squares of black are a [i]good [/i] coloring.

On the Island of Camelot live $13$ grey, $15$ brown and $17$ crimson chameleons . If two chameleons of different colours meet , they both simultaneously change colour to the third colour (e .g . if a grey and brown chameleon meet each other they both change to crimson) . Is it possible that they will eventually all be the same colour? (V . G . Ilichev)

Let $n$ be an integer greater than 2. A positive integer is said to be [i]attainable [/i]if it is 1 or can be obtained from 1 by a sequence of operations with the following properties: 1.) The first operation is either addition or multiplication. 2.) Thereafter, additions and multiplications are used alternately. 3.) In each addition, one can choose independently whether to add 2 or $n$ 4.) In each multiplication, one can choose independently whether to multiply by 2 or by $n$. A positive integer which cannot be so obtained is said to be [i]unattainable[/i]. [b]a.)[/b] Prove that if $n\geq 9$, there are infinitely many unattainable positive integers. [b]b.)[/b] Prove that if $n=3$, all positive integers except 7 are attainable.

There are several candles of the same size on the Chapel of Bones. On the first day a candle is lit for a hour. On the second day two candles are lit for a hour, on the third day three candles are lit for a hour, and successively, until the last day, when all the candles are lit for a hour. On the end of that day, all the candles were completely consumed. Find all the possibilities for the number of candles.

Consider a convex polygon having $n$ vertices, $n\geq 4$. We arbitrarily decompose the polygon into triangles having all the vertices among the vertices of the polygon, such that no two of the triangles have interior points in common. We paint in black the triangles that have two sides that are also sides of the polygon, in red if only one side of the triangle is also a side of the polygon and in white those triangles that have no sides that are sides of the polygon. Prove that there are two more black triangles that white ones.

Let $ABCDEF$ regular hexagon of side length $1$ and $O$ is its center. In addition to the sides of the hexagon, line segments from $O$ to the every vertex are drawn, making as total of $12$ unit segments. Find the number paths of length $2003$ along these segments that star at $O$ and terminate at $O$.