For what natural number $n> 100$ can $n$ pairwise distinct numbers be arranged on a circle such that each number is either greater than $100$ numbers following it clockwise or less than all of them? and would any property be violated when deleting any of those numbers?

a) Fold a $10 \times 10$ square from a $1 \times 118$ rectangular strip. b) Fold a $10 \times 10$ square from a $1 \times (100+9\sqrt3)$ rectangular strip (approximately $1\times 115.58$). The strip can be bent, but not torn.

In the coordinate plane, consider the set of all segments of integer lengths whose endpoints have integer coordinates. Prove that no two of these segments form an angle of $45^o$. Are there such segments in coordinate space?

Let $P_i (x_i, y_i)$ (with $i = 1, 2, 3, 4, 5$) be five points with integer coordinates, no three collinear. Show that among all triangles with vertices at these points, at least three have integer areas.

Let $N$ be a positive integer, and consider an $N \times N$ grid. A [i]right-down path[/i] is a sequence of grid cells such that each cell is either one cell to the right of or one cell below the previous cell in the sequence. A [i]right-up path[/i] is a sequence of grid cells such that each cell is either one cell to the right of or one cell above the previous cell in the sequence. Prove that the cells of the $N \times N$ grid cannot be partitioned into less than $N$ right-down or right-up paths. For example, the following partition of the $5 \times 5$ grid uses $5$ paths. [asy] size(4cm); draw((5,-1)--(0,-1)--(0,-2)--(5,-2)--(5,-3)--(0,-3)--(0,-4)--(5,-4),gray+linewidth(0.5)+miterjoin); draw((1,-5)--(1,0)--(2,0)--(2,-5)--(3,-5)--(3,0)--(4,0)--(4,-5),gray+linewidth(0.5)+miterjoin); draw((0,0)--(5,0)--(5,-5)--(0,-5)--cycle,black+linewidth(2.5)+miterjoin); draw((0,-1)--(3,-1)--(3,-2)--(1,-2)--(1,-4)--(4,-4)--(4,-3)--(2,-3)--(2,-2),black+linewidth(2.5)+miterjoin); draw((3,0)--(3,-1),black+linewidth(2.5)+miterjoin); draw((1,-4)--(1,-5),black+linewidth(2.5)+miterjoin); draw((4,-3)--(4,-1)--(5,-1),black+linewidth(2.5)+miterjoin); [/asy] [i]Proposed by Zixiang Zhou, Canada[/i]

Let $f:\{1,2,\dots,2019\}\to\{-1,1\}$ be a function, such that for every $k\in\{1,2,\dots,2019\}$, there exists an $\ell\in\{1,2,\dots,2019\}$ such that $$ \sum_{i\in\mathbb{Z}:(\ell-i)(i-k)\geqslant 0} f(i)\leqslant 0. $$ Determine the maximum possible value of $$ \sum_{i\in\mathbb{Z}:1\leqslant i\leqslant 2019} f(i). $$

[b]p1.[/b] A function $f$ defined on the set of positive numbers satisfies the equality $$f(xy) = f(x) + f(y), x, y > 0.$$ Find $f(2007)$ if $f\left( \frac{1}{2007} \right) = 1$. [b]p2.[/b] The plane is painted in two colors. Show that there is an isosceles right triangle with all vertices of the same color. [b]p3.[/b] Show that the number of ways to cut a $2n \times 2n$ square into $1\times 2$ dominoes is divisible by $2$. [b]p4.[/b] Two mirrors form an angle. A beam of light falls on one mirror. Prove that the beam is reflected only finitely many times (even if the angle between mirrors is very small). [b]p5.[/b] A sequence is given by the recurrence relation $a_{n+1} = (s(a_n))^2 +1$, where $s(x)$ is the sum of the digits of the positive integer $x$. Prove that starting from some moment the sequence is periodic. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $P$ be the set of all subsets of ${1, 2, ... , n}$. Show that there are $1^n + 2^n + ... + m^n$ functions $f : P \longmapsto {1, 2, ... , m}$ such that $f(A \cap B) = min( f(A), f(B))$ for all $A, B.$

$1019$ stones are placed into two non-empty boxes. Each second Alex chooses a box with an even amount of stones and shifts half of these stones into another box. Prove that for each $k$, $1\le k\le1018$, at some moment there will be a box with exactly $k$ stones. [i](O. Izhboldin)[/i]

Two walkers are at the same altitude in a range of mountains. The path joining them is piecewise linear with all its vertices above the two walkers. Can they each walk along the path until they have changed places, so that at all times their altitudes are equal?

A pile of n pebbles is placed in a vertical column. This configuration is modified according to the following rules. A pebble can be moved if it is at the top of a column which contains at least two more pebbles than the column immediately to its right. (If there are no pebbles to the right, think of this as a column with 0 pebbles.) At each stage, choose a pebble from among those that can be moved (if there are any) and place it at the top of the column to its right. If no pebbles can be moved, the configuration is called a final configuration. For each n, show that, no matter what choices are made at each stage, the final configuration obtained is unique. Describe that configuration in terms of n.

For every positive integer $n$, denote by $D_n$ the number of permutations $(x_1, \dots, x_n)$ of $(1,2,\dots, n)$ such that $x_j\neq j$ for every $1\le j\le n$. For $1\le k\le \frac{n}{2}$, denote by $\Delta (n,k)$ the number of permutations $(x_1,\dots, x_n)$ of $(1,2,\dots, n)$ such that $x_i=k+i$ for every $1\le i\le k$ and $x_j\neq j$ for every $1\le j\le n$. Prove that $$\Delta (n,k)=\sum_{i=0}^{k=1} \binom{k-1}{i} \frac{D_{(n+1)-(k+i)}}{n-(k+i)}$$ (Proposed by Combinatorics; Ferdowsi University of Mashhad, Iran; Mirzavaziri)

Let $S=\{1,2,\cdots, n\}$ and let $T$ be the set of all ordered triples of subsets of $S$, say $(A_1, A_2, A_3)$, such that $A_1\cup A_2\cup A_3=S$. Determine, in terms of $n$, \[ \sum_{(A_1,A_2,A_3)\in T}|A_1\cap A_2\cap A_3|\]

Consider the set $A = \{1, 2, ..., n\}$, where $n \in N, n \ge 6$. Show that $A$ is the union of three pairwise disjoint sets, with the same cardinality and the same sum of their elements, if and only if $n$ is a multiple of $3$.

Let $P_1, P_2, \ldots, P_n$ be distinct $2$-element subsets of $\{1, 2, \ldots, n\}$. Suppose that for every $1 \le i < j \le n$, if $P_i \cap P_j \neq \emptyset$, then there is some $k$ such that $P_k = \{i, j\}$. Prove that if $a \in P_i$ for some $i$, then $a \in P_j$ for exactly one value of $j$ not equal to $i$.

$2013$ smurfs are sitting at a large round table. Each of them has two tickets. on each card represents a number from $\{1, 2, . . ., 2013\}$ such that each of the numbers from this set occurs exactly twice. Every smurf takes the card every minute with the smaller of the two numbers, it smurfs on to its left neighbor and receives a card from his right neighbor. Show that there will come a time when a smurf has two cards with the same number.

[b]p1.[/b] Solve the equation $\log_x (x + 2) = 2$. [b]p2.[/b] Solve the inequality: $0.5^{|x|} > 0.5^{x^2}$. [b]p3.[/b] The integers from 1 to 2015 are written on the blackboard. Two randomly chosen numbers are erased and replaced by their difference giving a sequence with one less number. This process is repeated until there is only one number remaining. Is the remaining number even or odd? Justify your answer. [b]p4.[/b] Four circles are constructed with the sides of a convex quadrilateral as the diameters. Does there exist a point inside the quadrilateral that is not inside the circles? Justify your answer. [b]p5.[/b] Prove that for any finite sequence of digits there exists an integer the square of which begins with that sequence. [b]p6.[/b] The distance from the point $P$ to two vertices $A$ and $B$ of an equilateral triangle are $|P A| = 2$ and $|P B| = 3$. Find the greatest possible value of $|P C|$. PS. You should use hide for answers.

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.

There are $m$ identical rectangular chocolate bars and $n$ people. Each chocolate bar may be cut into two (possibly unequal) pieces at most once. For which $m$ and $n$ is it possible to split the chocolate evenly among all the people? [i]Selected from the Kvant Magazine (D. Bugaenko and N. Konstantinov)[/i]

Let $M=\{1,2,3,\ldots, 10000\}.$ Prove that there are $16$ subsets of $M$ such that for every $a \in M,$ there exist $8$ of those subsets that intersection of the sets is exactly $\{a\}.$

Given a simple graph with $ 4038 $ vertices. Assume we arbitrarily choose $ 2019 $ vertices as a group (the other $ 2019 $ is another group, of course), there are always $ k $ edges that connect two groups. Find all possible value of $ k $.

Let $k$,$n$ be positive integers, $k,n>1$, $k<n$ and a $n \times n$ grid of unit squares is given. Ana and Maya take turns in coloring the grid in the following way: in each turn, a unit square is colored black in such a way that no two black cells have a common side or vertex. Find the smallest positive integer $n$ , such that they can obtain a configuration in which each row and column contains exactly $k$ black cells. Draw one example.

Let $n$ and $k$ be two integers with $n>k\geqslant 1$. There are $2n+1$ students standing in a circle. Each student $S$ has $2k$ [i]neighbors[/i] - namely, the $k$ students closest to $S$ on the left, and the $k$ students closest to $S$ on the right. Suppose that $n+1$ of the students are girls, and the other $n$ are boys. Prove that there is a girl with at least $k$ girls among her neighbors. [i]Proposed by Gurgen Asatryan, Armenia[/i]

$m,n,p$ are positive integers which do not simultaneously equal to zero. $3$D Cartesian space is divided into unit cubes by planes each perpendicular to one of $3$ axes and cutting corresponding axis at integer coordinates. Each unit cube is filled with an integer from $1$ to $60$. A filling of integers is called [i]Dien Bien[/i] if, for each rectangular box of size $\{2m+1,2n+1,2p+1\}$, the number in the unit cube which has common centre with the rectangular box is the average of the $8$ numbers of the $8$ unit cubes at the $8$ corners of that rectangular box. How many [i]Dien Bien[/i] fillings are there? Two fillings are the same if one filling can be transformed to the other filling via a translation. [hide]translation from [url=http://artofproblemsolving.com/community/c6h592875p3515526]here[/url][/hide]

Let $\triangle P_1P_2P_3$ be an equilateral triangle. For each $n\ge 4$, [i]Mingmingsan[/i] can set $P_n$ as the circumcenter or orthocenter of $\triangle P_{n-3}P_{n-2}P_{n-1}$. Find all positive integer $n$ such that [i]Mingmingsan[/i] has a strategy to make $P_n$ equals to the circumcenter of $\triangle P_1P_2P_3$. [i]Proposed by Li4 and Untro368.[/i]