Bob plotted $2003$ green points on the plane, so all triangles with three green vertices have area less than $1$. Prove that the $2003$ green points are contained in a triangle $T$ of area less than $4$.

Do there exist positive integers $k$ and $n$ such that for any finite graph $G$ with diameter $k+1$ there exists a set $S$ of at most $n$ vertices such that for any $v\in V(G)\setminus S$, there exists a vertex $u\in S$ of distance at most $k$ from $v$? [i]David Yang.[/i]

Let $ n$ be an even positive integer. Let $ A_1, A_2, \ldots, A_{n \plus{} 1}$ be sets having $ n$ elements each such that any two of them have exactly one element in common while every element of their union belongs to at least two of the given sets. For which $ n$ can one assign to every element of the union one of the numbers 0 and 1 in such a manner that each of the sets has exactly $ \frac {n}{2}$ zeros?

For any integer $n\geq 2$, let $N(n)$ be the maxima number of triples $(a_i, b_i, c_i)$, $i=1, \ldots, N(n)$, consisting of nonnegative integers $a_i$, $b_i$ and $c_i$ such that the following two conditions are satisfied: [list][*] $a_i+b_i+c_i=n$ for all $i=1, \ldots, N(n)$, [*] If $i\neq j$ then $a_i\neq a_j$, $b_i\neq b_j$ and $c_i\neq c_j$[/list] Determine $N(n)$ for all $n\geq 2$. [i]Proposed by Dan Schwarz, Romania[/i]

Nine weights are placed in a scale with the respective values $1kg,2kg,...,9kg$. In how many ways can we place six weights in the left side and three weights in the right side such that the right side is heavier than the left one?

Andriy and Olesya take turns (Andriy starts) in a $2 \times 1$ rectangle, drawing horizontal segments of length $2$ or vertical segments of length $1$, as shown in the figure below. [img]https://i.ibb.co/qWqWxgh/Kyiv-MO-2021-Round-1-7-2.png[/img] After each move, the value $P$ is calculated - the total perimeter of all small rectangles that are formed (i.e., those inside which no other segment passes). The winner is the one after whose move $P$ is divisible by $2021$ for the first time. Who has a winning strategy? [i]Proposed by Bogdan Rublov[/i]

For $n$ an odd positive integer, the unit squares of an $n\times n$ chessboard are coloured alternately black and white, with the four corners coloured black. A it tromino is an $L$-shape formed by three connected unit squares. For which values of $n$ is it possible to cover all the black squares with non-overlapping trominos? When it is possible, what is the minimum number of trominos needed?

Let $n$ be a positive integer. For a set of points in the plane $M$, we call $2$ distinct points $A,B \in M$ [i]connected[/i] if the line $AB$ contains exactly $n+1$ points from $M$. Find the minimum value of a positive integer $m$ such that there exists a set $M$ of $m$ points in the plane with the property that any point $A \in M$ is connected with exactly $2n$ other points from $M$.

Consider a sequence of words each consisting of two letters, $A$ and $B$ . The first word is "$A$" , while the second word is "$B$" . The $k$-th word is obtained from the ($k -2$)-nd by writing after it the ($k -1$)th one. (So the first few elements of the sequence are "$A$" , "$B$" ,"$AB$" , "$BAB$" , "$ABBAB$" . ) Does there exist in this sequence a "periodical" word, i.e. a word of the form $P P P ... P$ , where $P$ is a word , repeated at least once? (Remark: For instance, the word $BABBBABB$ is of the form $PP$ , in which $P$ is repeated exactly once . ) (A. Andjans, Riga)

Let us call the [i]distance [/i] between the numbers $\overline{a_1a_2a_3a_4a_5}$ and $\overline{b_1b_2b_3b_4b_5}$ the maximum $i$ for which $a_i \ne b_i$. All five-digit numbers are written out one after another in some order. What is the minimum possible sum of distances between adjacent numbers?

Let $n \geq 1$ be an integer. What is the maximum number of disjoint pairs of elements of the set $\{ 1,2,\ldots , n \}$ such that the sums of the different pairs are different integers not exceeding $n$?

The following figure is given: [img]https://cdn.artofproblemsolving.com/attachments/9/b/97a30e248fcd6f098a900c89721a2e1b3b3f0e.png[/img] Determine the number of paths from the starting square $A$ to the target square $Z$, where a path consists of steps from a square to its top or right neighbor square . (W. Janous, WRG Ursulinen, Innsbruck)

Show that if $994$ integers are chosen from $1, 2,\cdots , 1992$ and one of the chosen integers is less than $64$, then there exist two among the chosen integers such that one of them is a factor of the other.

[url=https://artofproblemsolving.com/community/c893771h1861957p12597232]8.1[/url] The point $E$ lies on the base $[AD]$ of the trapezoid $ABCD$. The perimeters of the triangles $ABE, BCE$ and $CDE$ are equal. Prove that $|BC| = |AD|/2$ [b]8.2[/b] In a convex pentagon, the lengths of all sides are equal. Find the point on the longest diagonal from which all sides are visible underneath angles not exceeding a right angle. [url=https://artofproblemsolving.com/community/c893771h1862007p12597620]8.3[/url] Every city in the certain state is connected by airlines with no more than with three other ones, but one can get from every city to every other city changing a plane once only or directly. What is the maximal possible number of the cities? [url=https://artofproblemsolving.com/community/c893771h1861966p12597273]8.4*/7.4*[/url] (asterisk problems in separate posts) [url=https://artofproblemsolving.com/community/c893771h1862002p12597605]8.5[/url] Four different three-digit numbers starting with the same digit have the property that their sum is divisible by three of them without a remainder. Find these numbers. [url=https://artofproblemsolving.com/community/c893771h1861967p12597280]8.6[/url] Given a finite sequence of zeros and ones, which has two properties: a) if in some arbitrary place in the sequence we select five digits in a row and also select five digits in any other place in a row, then these fives will be different (they may overlap); b) if you add any digit to the right of the sequence, then property (a) will no longer hold true. Prove that the first four digits of our sequence coincide with the last four. PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988085_1969_leningrad_math_olympiad]here[/url].

Find $k \in \mathbb{N}$ such that [b]a.)[/b] For any $n \in \mathbb{N}$, there does not exist $j \in \mathbb{Z}$ which satisfies the conditions $0 \leq j \leq n - k + 1$ and $\left( \begin{array}{c} n\\ j\end{array} \right), \left( \begin{array}{c} n\\ j + 1\end{array} \right), \ldots, \left( \begin{array}{c} n\\ j + k - 1\end{array} \right)$ forms an arithmetic progression. [b]b.)[/b] There exists $n \in \mathbb{N}$ such that there exists $j$ which satisfies $0 \leq j \leq n - k + 2$, and $\left( \begin{array}{c} n\\ j\end{array} \right), \left( \begin{array}{c} n\\ j + 1\end{array} \right), \ldots , \left( \begin{array}{c} n\\ j + k - 2\end{array} \right)$ forms an arithmetic progression. Find all $n$ which satisfies part [b]b.)[/b]

IMOC is a small country without any lake. One day, the king decides to divide IMOC into many regions so that each region borders the sea. Prove that the map is $3$-colorable.

Each of $1000$ elves has a hat, red on the inside and blue on the outside or vise versa. An elf with a hat that is red outside can only lie, and an elf with a hat that is blue outside can only tell the truth. One day every elf tells every other elf, “Your hat is red on the outside.” During that day, some of the elves turn their hats inside out at any time during the day. (An elf can do that more than once per day.) Find the smallest possible number of times any hat is turned inside out.

In an international meeting of $n \geq 3$ participants, 14 languages are spoken. We know that: - Any 3 participants speak a common language. - No language is spoken more that by the half of the participants. What is the least value of $n$?

The positive integers are arranged in a row in some order, each occuring exactly once. Does there always exist an adjacent block of at least two numbers somewhere in this row such that the sum of the numbers in the block is a prime number?

Let $n_1,n_2, \cdots, n_{26}$ be pairwise distinct positive integers satisfying (1) for each $n_i$, its digits belong to the set $\{1,2\}$; (2) for each $i,j$, $n_i$ can't be obtained from $n_j$ by adding some digits on the right. Find the smallest possible value of $\sum_{i=1}^{26} S(n_i)$, where $S(m)$ denotes the sum of all digits of a positive integer $m$.

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.

[b]p1.[/b] A number of Exonians took a math test. If all of their scores were positive integers and the mean of their scores was $8.6$, find the minimum possible number of students. [b]p2.[/b] Find the least composite positive integer that is not divisible by any of $3, 4$, and $5$. [b]p3.[/b] Five checkers are on the squares of an $8\times 8$ checkerboard such that no two checkers are in the same row or the same column. How many squares on the checkerboard share neither a row nor a column with any of the five checkers? [b]p4.[/b] Let the operation $x@y$ be $y - x$. Compute $((... ((1@2)@3)@ ...@ 2013)@2014)@2015$. [b]p5.[/b] In a town, each family has either one or two children. According to a recent survey, $40\%$ of the children in the town have a sibling. What fraction of the families in the town have two children? [b]p6.[/b] Equilateral triangles $ABE$, $BCF$, $CDG$ and $DAH$ are constructed outside the unit square $ABCD$. Eliza wants to stand inside octagon $AEBFCGDH$ so that she can see every point in the octagon without being blocked by an edge. What is the area of the region in which she can stand? [b]p7.[/b] Let $S$ be the string $0101010101010$. Determine the number of substrings containing an odd number of $1$'s. (A substring is defined by a pair of (not necessarily distinct) characters of the string and represents the characters between, inclusively, the two elements of the string.) [b]p8.[/b] Let the positive divisors of $n$ be $d_1, d_2, ...$ in increasing order. If $d_6 = 35$, determine the minimum possible value of $n$. [b]p9.[/b] The unit squares on the coordinate plane that have four lattice point vertices are colored black or white, as on a chessboard, shown on the diagram below. [img]https://cdn.artofproblemsolving.com/attachments/6/4/f400d52ae9e8131cacb90b2de942a48662ea8c.png[/img] For an ordered pair $(m, n)$, let $OXZY$ be the rectangle with vertices $O = (0, 0)$, $X = (m, 0)$, $Z = (m, n)$ and $Y = (0, n)$. How many ordered pairs $(m, n)$ of nonzero integers exist such that rectangle $OXZY$ contains exactly $32$ black squares? [b]p10.[/b] In triangle $ABC$, $AB = 2BC$. Given that $M$ is the midpoint of $AB$ and $\angle MCA = 60^o$, compute $\frac{CM}{AC}$ . PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all integers $n \geq 2$ for which there exists a sequence of $2n$ pairwise distinct points $(P_1, \dots, P_n, Q_1, \dots, Q_n)$ in the plane satisfying the following four conditions: [list=i] [*]no three of the $2n$ points are collinear; [*] $P_iP_{i+1} \ge 1$ for all $i = 1, 2, \dots ,n$, where $P_{n+1}=P_1$; [*] $Q_iQ_{i+1} \ge 1$ for all $i = 1, 2, \dots, n$, where $Q_{n+1} = Q_1$; and [*] $P_iQ_j \le 1$ for all $i = 1, 2, \dots, n$ and $j = 1, 2, \dots, n$.[/list] [i]Ray Li[/i]

Gleb picked positive integers $N$ and $a$ ($a < N$). He wrote the number $a$ on a blackboard. Then each turn he did the following: he took the last number on the blackboard, divided the number $N$ by this last number with remainder and wrote the remainder onto the board. When he wrote the number $0$ onto the board, he stopped. Could he pick $N$ and $a$ such that the sum of the numbers on the blackboard would become greater than $100N$ ? Ivan Mitrofanov

a) The product of $n$ integers equals $n$, and their sum is zero. Prove that $n$ is divisible by $4$. b) Let $n$ is divisible by $4$. Prove that there exist $n$ integers such, that their product equals $n$, and their sum is zero.