There are $n \ge 7$ points in the plane, no $3$ of which are collinear. At least $7$ pairs of points are joined by line segments. For every aforementioned line segment $s$, let $t(s)$ be the number of triangles for which the segment $s$ is a side. Prove that there exist different line segments $s_1, s_2, s_3,$ and $s_4$ such that \[t(s_1) = t(s_2) = t(s_3) = t(s_4)\] holds. Proposed by [i]Viktor Simjanoski[/i]

There are cities in country, and some cities are connected by roads. Not more than $100$ roads go from every city. Set of roads is called as ideal if all roads in set have not common ends, and we can not add one more road in set without breaking this rule. Every day minister destroy one ideal set of roads. Prove, that he need not more than $199$ days to destroy all roads in country.

Alice places down $n$ bishops on a $2015\times 2015$ chessboard such that no two bishops are attacking each other. (Bishops attack each other if they are on a diagonal.) [list=a] [*]Find, with proof, the maximum possible value of $n$. [*](A1 only) For this maximal $n$, find, with proof, the number of ways she could place her bishops on the chessboard. [/list]

Define the [i]mexth[/i] of \(k\) sets as the \(k\)th smallest positive integer that none of them contain, if it exists. Does there exist a family \(\mathcal F\) of sets of positive integers such that [list] [*]for any nonempty finite subset \(\mathcal G\) of \(\mathcal F\), the mexth of \(\mathcal G\) exists, and [*]for any positive integer \(n\), there is exactly one nonempty finite subset \(\mathcal G\) of \(\mathcal F\) such that \(n\) is the mexth of \(\mathcal G\). [/list] [i]Proposed by Espen Slettnes[/i]

A board $n \times n$ is divided into $n^2$ unit squares and a number is written in each unit square. Such a board is called [i] interesting[/i] if the following conditions hold: $\circ$ In all unit squares below the main diagonal, the number $0$ is written; $\circ$ Positive integers are written in all other unit squares. $\circ$ When we look at the sums in all $n$ rows, and the sums in all $n$ columns, those $2n$ numbers are actually the numbers $1,2,...,2n$ (not necessarily in that order). $a)$ Determine the largest number that can appear in a $6 \times 6$ [i]interesting[/i] board. $b)$ Prove that there is no [i]interesting[/i] board of dimensions $7\times 7$.

Let $ G$ be a simple graph with $ 2 \cdot n$ vertices and $ n^{2}+1$ edges. Show that this graph $ G$ contains a $ K_{4}-\text{one edge}$, that is, two triangles with a common edge.

A knight stands on an infinite chess board. Find all places it can reach in exactly $2n$ moves.

There are $ 25$ towns in a country. Find the smallest $ k$ for which one can set up two-direction flight routes connecting these towns so that the following conditions are satisfied: 1) from each town there are exactly $ k$ direct routes to $ k$ other towns; 2) if two towns are not connected by a direct route, then there is a town which has direct routes to these two towns.

$5$ points are given in the plane, any three non-collinear and any four non-concyclic. If three points determine a circle that has one of the remaining points inside it and the other one outside it, then the circle is said to be [i]good[/i]. Let the number of good circles be $n$; find all possible values of $n$.

Integers are placed on every cell of an infinite checkerboard. For each cell if it contains integer $a$ then the sum of the numbers in the cell under it and the cell right to it is $2a+1$. Prove that in every infinite diagonal row of direction [i] top-right down-left [/i] all numbers are different.

In a concert, 20 singers will perform. For each singer, there is a (possibly empty) set of other singers such that he wishes to perform later than all the singers from that set. Can it happen that there are exactly 2010 orders of the singers such that all their wishes are satisfied? [i]Proposed by Gerhard Wöginger, Austria[/i]

Every cell of a $20 \times 20$ table has to be coloured black or white (there are $2^{400}$ such colourings in total). Given any colouring $P$, we consider division of the table into rectangles with sides in the grid lines where no rectangle contains more than two black cells and where the number of rectangles containing at most one black cell is the least possible. We denote this smallest possible number of rectangles containing at most one black cell by $f(P)$. Determine the maximum value of $f(P)$ as $P$ ranges over all colourings.

Let $S$ be a square with the side length 20 and let $M$ be the set of points formed with the vertices of $S$ and another 1999 points lying inside $S$. Prove that there exists a triangle with vertices in $M$ and with area at most equal with $\frac 1{10}$. [i]Yugoslavia[/i]

Let \(n\) be a positive integer. The kingdom of Zoomtopia is a convex polygon with integer sides, perimeter \(6n\), and \(60^\circ\) rotational symmetry (that is, there is a point \(O\) such that a \(60^\circ\) rotation about \(O\) maps the polygon to itself). In light of the pandemic, the government of Zoomtopia would like to relocate its \(3n^2+3n+1\) citizens at \(3n^2+3n+1\) points in the kingdom so that every two citizens have a distance of at least \(1\) for proper social distancing. Prove that this is possible. (The kingdom is assumed to contain its boundary.) [i]Proposed by Ankan Bhattacharya, USA[/i]

A natural number $n$ is given and written in a row of $n$ numbers, each of which is equal to $0$ or $1$. Then $n - 1$ numbers are written below in a row - one number under each pair of adjacent numbers of the first row. At the same time, $0$ is written under a pair of identical numbers. and under a pair of different ones $1$. Then, under the second row, the third of $n- 2$ numbers is similarly written, etc., until we get a triangular table with $n$ rows. For a given $n$, find the largest possible number of units in such a table.

Peter has many squares of equal side. Some of the squares are black, some are white. Peter wants to assemble a big square, with side equal to $n$ sides of the small squares, so that the big square has no rectangle formed by the small squares such that all the squares in the vertices of the rectangle are of equal colour. How big a square is Peter able to assemble?

An $n \times n$ table is called good if one can paint its cells with three colors so that, for any two different rows and two different columns, the four cells at their intersections are not all of the same color. a)Show, that exists good $9 \times 9$ good table. b)Prove, that fif $n \times n$ table is good, then $n<11$

[b]p1.[/b] Len's Spanish class has four tests in the first term. Len scores $72$, $81$, and $78$ on the first three tests. If Len wants to have an 80 average for the term, what is the minimum score he needs on the last test? [b]p2.[/b] In $1824$, the Electoral College had $261$ members. Andrew Jackson won $99$ Electoral College votes and John Quincy Adams won $84$ votes. A plurality occurs when no candidate has more than $50\%$ of the votes. Should a plurality occur, the vote goes to the House of Representatives to break the tie. How many more votes would Jackson have needed so that a plurality would not have occurred? [b]p3.[/b] $\frac12 + \frac16 + \frac{1}{12} + \frac{1}{20} + \frac{1}{30}= 1 - \frac{1}{n}$. Find $n$. [b]p4.[/b] How many ways are there to sit Samuel, Esun, Johnny, and Prat in a row of $4$ chairs if Prat and Johnny refuse to sit on an end? [b]p5.[/b] Find an ordered quadruple $(w, x, y, z)$ that satisfies the following: $$3^w + 3^x + 3^y = 3^z$$ where $w + x + y + z = 2017$. [b]p6.[/b] In rectangle $ABCD$, $E$ is the midpoint of $CD$. If $AB = 6$ inches and $AE = 6$ inches, what is the length of $AC$? [b]p7.[/b] Call an integer interesting if the integer is divisible by the sum of its digits. For example, $27$ is divisible by $2 + 7 = 9$, so $27$ is interesting. How many $2$-digit interesting integers are there? [b]p8.[/b] Let $a\#b = \frac{a^3-b^3}{a-b}$ . If $a, b, c$ are the roots of the polynomial $x^3 + 2x^2 + 3x + 4$, what is the value of $a\#b + b\#c + c\#a$? [b]p9.[/b] Akshay and Gowri are examining a strange chessboard. Suppose $3$ distinct rooks are placed into the following chessboard. Find the number of ways that one can place these rooks so that they don't attack each other. Note that two rooks are considered attacking each other if they are in the same row or the same column. [img]https://cdn.artofproblemsolving.com/attachments/f/1/70f7d68c44a7a69eb13ce12291c0600d11027c.png[/img] [b]p10.[/b] The Earth is a very large sphere. Richard and Allen have a large spherical model of Earth, and they would like to (for some strange reason) cut the sphere up with planar cuts. If each cut intersects the sphere, and Allen holds the sphere together so it does not fall apart after each cut, what is the maximum number of pieces the sphere can be cut into after $6$ cuts? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Is it possible to paint $33$ squares on a $9\times 9$ game board, so that each row and each column of the board has a maximum of $4$ painted squares, but if we also paint any other square a row or column appears that has $5$ squares painted?

There are at least $3$ subway stations in a city. In this city, there exists a route that passes through more than $L$ subway stations, without revisiting. Subways run both ways, which means that if you can go from subway station A to B, you can also go from B to A. Prove that at least one of the two holds. $\text{(i)}$. There exists three subway stations $A$, $B$, $C$ such that there does not exist a route from $A$ to $B$ which doesn't pass through $C$. $\text{(ii)}$. There is a cycle passing through at least $\lfloor \sqrt{2L} \rfloor$ stations, without revisiting a same station more than once.

Given real numbers $ x_1 < x_2 < \ldots < x_n$ such that every real number occurs at most two times among the differences $ x_j \minus{} x_i$, $ 1\leq i < j \leq n$, prove that there exists at least $ \lfloor n/2\rfloor$ real numbers that occurs exactly one time among such differences.

Suppose a box contains $28$ balls: $1$ red, $2$ blue, $3$ yellow, $4$ orange, $5$ purple, $6$ green, and $7$ pink. One by one, each ball is removed uniformly at random and without replacement until all $28$ balls have been removed. Determine the probability that the most likely “scenario of exhaustion” occurs; that is, determine the probability that the first color to have all such balls removed from the box is red, that the second is blue, the third is yellow, the fourth is orange, the fifth is purple, the sixth is green, and the seventh is pink.

In a mathematical competition, in which $6$ problems were posed to the participants, every two of these problems were solved by more than $\frac 25$ of the contestants. Moreover, no contestant solved all the $6$ problems. Show that there are at least $2$ contestants who solved exactly $5$ problems each. [i]Radu Gologan and Dan Schwartz[/i]

The rook, standing on the surface of the checkered cube, beats the cells, located in the same row as well as on the continuations of this series through one or even several edges. (The picture shows an example for a $4 \times 4 \times 4$ cube,visible cells that some beat the rook, shaded gray.) What is the largest number do not beat each other rooks can be placed on the surface of the cube $50 \times 50 \times 50$?

Andrew and Barry play the following game: there are two heaps with $a$ and $b$ pebbles, respectively. In the first round Barry chooses a positive integer $k,$ and Andrew takes away $k$ pebbles from one of the two heaps (if $k$ is bigger than the number of pebbles in the heap, he takes away the complete heap). In the second round, the roles are reversed: Andrew chooses a positive integer and Barry takes away the pebbles from one of the two heaps. This goes on, in each round the two players are reversing the roles. The player that takes the last pebble loses the game. Which player has a winning strategy? [i]Submitted by András Imolay, Budapest[/i]