Given is a directed graph with $28$ vertices, such that there do not exist vertices $u, v$, such that $u \rightarrow v$ and $v \rightarrow u$. Every $16$ vertices form a directed cycle. Prove that among any $17$ vertices, we can choose $15$ which form a directed cycle.

In Oddland there are stamps with values of $1$ cent, $3$ cents, $5$ cents, etc., each for odd number there is exactly one stamp type. Oddland Post dictates: For two different values on a letter must be the number of stamps of the lower one value must be at least as large as the number of tokens of the higher value. In Squareland, on the other hand, there are stamps with values of $1$ cent, $4$ cents, $9$ cents, etc. there is exactly one stamp type for each square number. Brands can be found in Squareland can be combined as required without further regulations. Prove for every positive integer $n$: there are the same number in the two countries possibilities to send a letter with stamps worth a total of $n$ cents. It makes no difference if you have the same stamps on arrange a letter differently. (Stephan Wagner)

Given are $n$ pairwise intersecting convex $k$-gons on the plane. Any of them can be transferred to any other by a homothety with a positive coefficient. Prove that there is a point in a plane belonging to at least $1 +\frac{n-1}{2k}$ of these $k$-gons.

Consider all sets $A$ of one hundred different natural numbers with the property that any three elements $a,b,c \in A$ (not necessarily different) are the sides of a non-obtuse triangle. Denote by $S(A)$ the sum of the perimeters of all such triangles. Compute the smallest possible value of $S(A)$.

During the mathematics Olympiad, students solved three problems. Each task was evaluated with an integer number of points from $0$ to $7$. There is at most one problem for each pair of students, for which they got after the same number of points. Determine the maximum number of students could participate in the Olympics.

Cynthia loves Pokemon and she wants to catch them all. In Victory Road, there are a total of $80$ Pokemon. Cynthia wants to catch as many of them as possible. However, she cannot catch any two Pokemon that are enemies with each other. After exploring around for a while, she makes the following two observations: 1. Every Pokemon in Victory Road is enemies with exactly two other Pokemon. 2. Due to her inability to catch Pokemon that are enemies with one another, the maximum number of the Pokemon she can catch is equal to $n$. What is the sum of all possible values of $n$?

Let $p$ be a prime number and let $X$ be a finite set containing at least $p$ elements. A collection of pairwise mutually disjoint $p$-element subsets of $X$ is called a $p$-family. (In particular, the empty collection is a $p$-family.) Let $A$(respectively, $B$) denote the number of $p$-families having an even (respectively, odd) number of $p$-element subsets of $X$. Prove that $A$ and $B$ differ by a multiple of $p$.

Through a point in the interior of a sphere we put three pairwise perpendicular planes. Those planes dissect the surface of the sphere in eight curvilinear triangles. Alternately the triangles are coloured black and wide to make the sphere surface look like a checkerboard. Prove that exactly half of the sphere's surface is coloured black.

Numbers from $1$ to $1000$ are arranged around a circle. Prove that it is possible to form $500$ non-intersecting line segments, each joining two such numbers, and so that in each case the difference between the numbers at each end (in absolute value) is not greater than $749$. (AA Razborov, Moscow)

$60$ points lie on the plane, such that no three points are collinear. Prove that one can divide the points into $20$ groups, with $3$ points in each group, such that the triangles ( $20$ in total) consist of three points in a group have a non-empty intersection.

[b]p1.[/b] A rectangle $ABCD$ is cut from a piece of paper and folded along a straight line so that the diagonally opposite vertices $A$ and $C$ coincide. Find the length of the resulting crease in terms of the length ($\ell$) and width ($w$) of the rectangle. (Justify your answer.) [b]p2.[/b] The residents of Andromeda use only bills of denominations $\$3 $and $\$5$ . All payments are made exactly, with no change given. What whole-dollar payments are not possible? (Justify your answer.) [b]p3.[/b] A set consists of $21$ objects with (positive) weights $w_1, w_2, w_3, ..., w_{21}$ . Whenever any subset of $10$ objects is selected, then there is a subset consisting of either $10$ or $11$ of the remaining objects such that the two subsets have equal fotal weights. Find all possible weights for the objects. (Justify your answer.) [b]p4.[/b] Let $P(x) = x^3 + x^2 - 1$ and $Q(x) = x^3 - x - 1$ . Given that $r$ and $s$ are two distinct solutions of $P(x) = 0$ , prove that $rs$ is a solution of $Q(x) = 0$ [b]p5.[/b] Given: $\vartriangle ABC$ with points $A_1$ and $A_2$ on $BC$ , $B_1$ and $B_2$ on $CA$, and $C_1$ and $C_2$ on $AB$. $A_1 , A_2, B_1 , B_2$ are on a circle, $B_1 , B_2, C_1 , C_2$ are on a circle, and $C_1 , C_2, A_1 , A_2$ are on a circle. The centers of these circles lie in the interior of the triangle. Prove: All six points $A_1$ , $A_2$, $B_1$, $B_2$, $C_1$, $C_2$ are on a circle. [img]https://cdn.artofproblemsolving.com/attachments/7/2/2b99ddf4f258232c910c062e4190d8617af6fa.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Each point on a circle is colored with one of $10$ colors. Is it true that for any coloring there are $4$ points of the same color that are vertices of a quadrilateral with two parallel sides (an isosceles trapezoid or a rectangle)?

The numbers $1, 2, . . . , 10$ are randomly arranged in a circle. Let $p$ be the probability that for every positive integer $k < 10$, there exists an integer $k' > k$ such that there is at most one number between $k$ and $k'$ in the circle. If $p$ can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a$ and $b$, compute $100a + b$.

There are six empty purses on the table. How many ways are there to put 12 two-euro coins in purses in such a way that at most one purse remains empty?

The nonnegative integers $2000$, $17$ and $n$ are written on the blackboard. Alice and Bob play the following game: Alice begins, then they play in turns. A move consists in replacing one of the three numbers by the absolute difference of the other two. No moves are allowed, where all three numbers remain unchanged. The player who is in turn and cannot make an allowed move loses the game. [list] [*] Prove that the game will end for every number $n$. [*] Who wins the game in the case $n = 2017$? [/list] [i]Proposed by Richard Henner[/i]

How many ways are there to line up $19$ girls (all of different heights) in a row so that no girl has a shorter girl both in front of and behind her?

What is the maximum number of knights that can be placed on a chessboard of size \(8 \times 8\) such that any knight, after making 1 or 2 arbitrary moves, does not land on a square occupied by another knight? [i]Proposed by Bogdan Rublov[/i]

Let $n$ be a natural number. There are $n$ blue points , $n$ red points and one green point on the circle . Prove that it is possible to draw $n$ lengths whose ends are in the given points, so that a maximum of one segment emerges from each point, no more than two segments intersect and the endpoints of none of the segments are blue and red points. [hide=original wording]Нека je ? природан број. На кружници се налази ? плавих, ? црвених и једна зелена тачка. Доказати да је могуће повући ? дужи чији су крајеви у датим тачкама, тако да из сваке тачке излази максимално једна дуж, никоје две дужи се не сијеку и крајње тачке ниједне од дужи нису плава и црвена тачка.[/hide]

Forty-nine students solve a set of 3 problems. The score for each problem is a whole number of points from 0 to 7. Prove that there exist two students $ A$ and $ B$ such that, for each problem, $ A$ will score at least as many points as $ B.$

A Mathlon is a competition where there are $M$ athletic events. $A, B$ and $C$ were the only participants of a Mathlon. In each event, $p_1$ points were given to the first place, $p_2$ points to the second place and $p_3$ points to third place, with $p_1> p_2> p_3> 0$ where $p_1$, $p_2$ and $p_3$ are integer numbers. The final result was $22$ points for $A$, $9$ for $B$, and $9$ for $C$. $B$ won the $100$ meter dash. Determine $M$ and who was the second in high jump.

Let $k$ be a positive integer. At the European Chess Cup every pair of players played a game in which somebody won (there were no draws). For any $k$ players there was a player against whom they all lost, and the number of players was the least possible for such $k$. Is it possible that at the Closing Ceremony all the participants were seated at the round table in such a way that every participant was seated next to both a person he won against and a person he lost against. [i]Proposed by Matija Bucić.[/i]

Determine whether there exist $100$ disks $D_2,D_3,\ldots ,D_{101}$ in the plane such that the following conditions hold for all pairs $(a,b)$ of indices satisfying $2\le a< b\le 101$: [list] [*] If $a|b$ then $D_a$ is contained in $D_b$. [*] If $\gcd (a,b)=1$ then $D_a$ and $D_b$ are disjoint. [/list] (A disk $D(O,r)$ is a set of points in the plane whose distance to a given point $O$ is at most a given positive real number $r$.)

A graph $G=(V,E)$ is given. If at least $n$ colors are required to paints its vertices so that between any two same colored vertices no edge is connected, then call this graph ''$n-$colored''. Prove that for any $n \in \mathbb{N}$, there is a $n-$colored graph without triangles.

Inside a square with side $60$, $121$ points are drawn. Prove them are three points that are vertices of a triangle of area not exceeding $30$.

Let $m$ and $n$ be positive integers. Mr. Fat has a set $S$ containing every rectangular tile with integer side lengths and area of a power of $2$. Mr. Fat also has a rectangle $R$ with dimensions $2^m \times 2^n$ and a $1 \times 1$ square removed from one of the corners. Mr. Fat wants to choose $m + n$ rectangles from $S$, with respective areas $2^0, 2^1, \ldots, 2^{m + n - 1}$, and then tile $R$ with the chosen rectangles. Prove that this can be done in at most $(m + n)!$ ways. [i]Palmer Mebane.[/i]