Suppose $2016$ points of the circumference of a circle are colored red and the remaining points are colored blue . Given any natural number $n\ge 3$, prove that there is a regular $n$-sided polygon all of whose vertices are blue.

John has a string of paper where $n$ real numbers $a_i \in [0, 1]$, for all $i \in \{1, \ldots, n\}$, are written in a row. Show that for any given $k < n$, he can cut the string of paper into non-empty $k$ pieces, between adjacent numbers, in such a way that the sum of the numbers on each piece does not differ from any other sum by more than $1$.

We have a square of side $1$ and a number $\ell$ such that $0 <\ell <\sqrt2$. Two players $A$ and $B$, in turn, draw in the square an open segment (without its two ends) of length $\ell $, starts A. Each segment after the first cannot have points in common with the previously drawn segments. He loses the player who cannot make his play. Determine if either player has a winning strategy.

Consider a cube and two of its vertices $A$ and $B$, which are the endpoints of a face diagonal. A [i]path [/i] is a sequence of cube angles, each step of one angle along a cube edge is walked to one of the three adjacent angles. Let $a$ be the number of paths of length $2012$ that starts at point $A$ and ends at $A$ and let b be the number of ways of length $2012$ that starts in $A$ and ends in $B$. Decide which of the two numbers $a$ and $b$ is the larger.

We have $2^m$ sheets of paper, with the number $1$ written on each of them. We perform the following operation. In every step we choose two distinct sheets; if the numbers on the two sheets are $a$ and $b$, then we erase these numbers and write the number $a + b$ on both sheets. Prove that after $m2^{m -1}$ steps, the sum of the numbers on all the sheets is at least $4^m$ . [i]Proposed by Abbas Mehrabian, Iran[/i]

There exist right triangles with integer side lengths such that the legs differ by $ 1$. For example, $3-4-5$ and $20-21-29$ are two such right triangles. What is the perimeter of the next smallest Pythagorean right triangle with legs differing by $ 1$?

All integers from $1$ to one million are written on a tape in some arbitrary order. Then the tape is cut into pieces containing two consecutive digits each. Prove that these pieces contain all two-digit integers for sure, regardless of the initial order of integers.[i](4 points)[/i] [i]Alexey Tolpygo[/i]

In a football league, there are $n\geq 6$ teams. Each team has a homecourt jersey and a road jersey with different color. When two teams play, the home team always wear homecourt jersey and the road team wear their homecourt jersey if the color is different from the home team's homecourt jersey, or otherwise the road team shall wear their road jersey. It is required that in any two games with 4 different teams, the 4 teams' jerseys have at least 3 different color. Find the least number of color that the $n$ teams' $2n$ jerseys may use.

Writing down the first $4$ rows of the Pascal triangle in the usual way and then adding up the numbers in vertical columns, we obtain $7$ numbers as shown above. If we repeat this procedure with the first $1024$ rows of the Pascal triangle, how many of the $2047$ numbers thus obtained will be odd? [img]https://cdn.artofproblemsolving.com/attachments/8/a/4dc4a815d8b002c9f36a6da7ad6e1c11a848e9.png[/img]

Given a positive integer $n$, determine the maximum number of lattice points in the plane a square of side length $n +\frac{1}{2n+1}$ may cover.

Thirty nine nonzero numbers are written in a row. The sum of any two neighbouring numbers is positive, while the sum of all the numbers is negative. Is the product of all these numbers negative or positive? (4 points) Boris Frenkin

All the integers from $1$ to $100$ are arranged in a $10 \times 10$ table as shown below. Prove that if some ten numbers are removed from the table, the remaining $90$ numbers contain 10 numbers in Arithmetic Progression. $1 \,\,\,\,2\,\, \,\,3 \,\,\,\,... \,\,10$ $11 \,\,12 \,\,13 \,\,... \,\,20$ $\,\,.\,\,\,\,.\,\,\,.$ $\,\,.\,\,\,\,.\,\,\,\,.$ $91 \,\,92 \,\,93\,\, ... \,\,100$

Martu wants to build a set of cards with the following properties: • Each card has a positive integer on it. • The number on each card is equal to one of $5$ possible numbers. • If any two cards are taken and added together, it is always possible to find two other cards in the set such that the sum is the same. Determine the fewest number of cards Martu's set can have and give an example for that number.

Let $a_k$ be the number of nonnegative integers $ n $ with the properties: a) $n\in[0, 10^k)$ has exactly $ k $ digits, such that he zeroes on the first positions of $ n $ are included in the decimal writting. b) the digits of $ n $ can be permutated such that the new number is divisible by $11.$ Show that $a_{2m}=10a_{2m-1}$ for every $m\in\mathbb{N}.$

In the coordinate plane consider the set $ S$ of all points with integer coordinates. For a positive integer $ k$, two distinct points $A$, $ B\in S$ will be called $ k$-[i]friends[/i] if there is a point $ C\in S$ such that the area of the triangle $ ABC$ is equal to $ k$. A set $ T\subset S$ will be called $ k$-[i]clique[/i] if every two points in $ T$ are $ k$-friends. Find the least positive integer $ k$ for which there exits a $ k$-clique with more than 200 elements. [i]Proposed by Jorge Tipe, Peru[/i]

Define the unit $N$-hypercube to be the set of points $[0, 1]^N \subset R^N$ . For example, the unit $0$-hypercube is a point, and the unit $3$-hypercube is the unit cube. Define a $k$-face of the unit $N$-hypercube to be a copy of the $k$-hypercube in the exterior of the $N$-hypercube. More formally, a $k$-face of the unit $N$-hypercube is a set of the form $$\prod_{i=1}^{N} S_i$$ where $S_i$ is either $\{0\}$, $\{1\}$, or $[0, 1]$ for each $1 \le i \le N$, and there are exactly $k$ indices $i$ such that $S_i = [0, 1]$. The expected value of the dimension of a random face of the unit $ 8$-hypercube (where the dimension of a face can be any value between $0$ and $N$) can be written in the form $p/q$ where $p$ and $q$ are relatively prime positive integers. Find $p + q$.

Twelve candidates for mayor participate in a TV talk show. At some point a candidate said: "One lie has been told." Another said: "Now two lies have been told". "Now three lies," said a third. This continued until the twelfth said: "Now twelve lies have been told". At this point the moderator ended the discussion. It turned out that at least one of the candidates correctly stated the number of lies told before he made the claim. How many lies were actually told by the candidates?

Lucy starts by writing $s$ integer-valued $2022$-tuples on a blackboard. After doing that, she can take any two (not necessarily distinct) tuples $\mathbf{v}=(v_1,\ldots,v_{2022})$ and $\mathbf{w}=(w_1,\ldots,w_{2022})$ that she has already written, and apply one of the following operations to obtain a new tuple: \begin{align*} \mathbf{v}+\mathbf{w}&=(v_1+w_1,\ldots,v_{2022}+w_{2022}) \\ \mathbf{v} \lor \mathbf{w}&=(\max(v_1,w_1),\ldots,\max(v_{2022},w_{2022})) \end{align*} and then write this tuple on the blackboard. It turns out that, in this way, Lucy can write any integer-valued $2022$-tuple on the blackboard after finitely many steps. What is the smallest possible number $s$ of tuples that she initially wrote?

Let $n \ge 2$ be a natural number. For an $n$-element subset $F$ of $\{1, . . . , 2n\}$ we define $m(F)$ as the minimum of all $lcm \,\, (x, y)$ , where $x$ and $y$ are two distinct elements of $F$. Find the maximum value of $m(F)$.

For positive integers $n$, let $C(n)$ be the number of representation of $n$ as a sum of nonincreasing powers of $2$, where no power can be used more than three times. For example, $C(8)=5$ since the representations of $8$ are: $$8,4+4,4+2+2,4+2+1+1,\text{ and }2+2+2+1+1.$$Prove or disprove that there is a polynomial $P(x)$ such that $C(n)=\lfloor P(n)\rfloor$ for all positive integers $n$.

A square $ (n \minus{} 1) \times (n \minus{} 1)$ is divided into $ (n \minus{} 1)^2$ unit squares in the usual manner. Each of the $ n^2$ vertices of these squares is to be coloured red or blue. Find the number of different colourings such that each unit square has exactly two red vertices. (Two colouring schemse are regarded as different if at least one vertex is coloured differently in the two schemes.)

We are given $N$ lines ($N > 1$ ) in a plane, no two of which are parallel and no three of which have a point in common. Prove that it is possible to assign, to each region of the plane determined by these lines, a non-zero integer of absolute value not exceeding $N$ , such that the sum of the integers o n either side of any of the given lines is equal to $0$ . (S . Fomin, Leningrad)

On the table are $300$ coins. Petya, Vasya and Tolya play the next game. They go in turn in the following order: Petya, Vasya, Tolya, Petya. Vasya, Tolya, etc. In one move, Petya can take $1, 2, 3$, or $4$ coins from the table, Vasya, $1$ or $2$ coins, and Tolya, too, $1$ or $2$ coins. Can Vasya and Tolya agree so that, as if Petya were playing, one of them two will take the last coin off the table?

Initially, the numbers $1,1,-1$ written on the board.At every step,Mikail chooses the two numbers $a,b$ and substitutes them with $2a+c$ and $\frac{b-c}{2}$ where $c$ is the unchosen number on the board. Prove that at least $1$ negative number must be remained on the board at any step.

Recall that an icosahedron is a $3$-dimensional solid characterized by its $20$ congruent faces, each of which is an equilateral triangle. Determine the number of rigid rotations that preserve the symmetry of the icosahedron. (Each vertex moves to the location of another vertex.)