Found problems: 14842
2024 Bosnia and Herzegovina Junior BMO TST, 4.
Let $m$ and $n$ be natural numbers. Every one of the $m*n$ squares of the $m*n$ board is colored either black or white, so that no 2 neighbouring squares are the same color(the board is colored like in chess").In one step we can pick 2 neighbouring squares and change their colors like this:
[b]- [/b]a white square becomes black;
[b]-[/b]a black square becomes blue;
[b]-[/b]a blue square becomes white.
For which $m$ and $n$ can we ,in a finite sequence of these steps, switch the starting colors from white to black and vice versa.
2020-21 KVS IOQM India, 21
Let $A = \{1,2,3,4,5,6,7,8\}$, $B = \{9,10,11,12,13,14,15,16\}$ and $C =\{17,18,19,20,21,22,23,24\}$. Find the number of triples $(x, y, z)$ such that $x \in A, y \in B, z \in C $ and $x + y + z = 36$.
2019 Baltic Way, 10
There are $2019$ points given in the plane. A child wants to draw $k$ (closed) discs in such a manner, that for any two distinct points there exists a disc that contains exactly one of these two points. What is the minimal $k$, such that for any initial configuration of points it is possible to draw $k$ discs with the above property?
OMMC POTM, 2023 1
Define a $100 \times 100$ square grid $G$. Initially color all cells of $G$ white. A move consists of selecting a $1 \times 7$ or $7 \times 1$ subgrid of $G$ and flipping the colors of all cells in this subgrid from white to black or vice versa. Is it possible that after a series of moves, all cells are colored black?
[i]Proposed by Evan Chang (squareman), USA[/i]
2016 Estonia Team Selection Test, 1
There are $k$ heaps on the table, each containing a different positive number of stones. Juri and Mari make moves alternatingly, Juri starts. On each move, the player making the move has to pick a heap and remove one or more stones in it from the table; in addition, the player is allowed to distribute any number of remaining stones from that heap in any way between other non-empty heaps. The player to remove the last stone from the table wins. For which positive integers $k$ does Juri have a winning strategy for any initial state that satisfies the conditions?
Kvant 2019, M2582
An integer $1$ is written on the blackboard. We are allowed to perform the following operations:to change the number $x$ to $3x+1$ of to $[\frac{x}{2}]$. Prove that we can get all positive integers using this operations.
2022 Dutch IMO TST, 4
In a sequence $a_1, a_2, . . . , a_{1000}$ consisting of $1000$ distinct numbers a pair $(a_i, a_j )$ with $i < j$ is called [i]ascending [/i] if $a_i < a_j$ and [i]descending[/i] if $a_i > a_j$ . Determine the largest positive integer $k$ with the property that every sequence of $1000$ distinct numbers has at least $k$ non-overlapping ascending pairs or at least $k$ non-overlapping descending pairs.
2007 Tournament Of Towns, 4
Each cell of a $29 \times 29$ table contains one of the integers $1, 2, 3, \ldots , 29$, and each of these integers appears $29$ times. The sum of all the numbers above the main diagonal is equal to three times the sum of all the numbers below this diagonal. Determine the number in the central cell of the table.
Russian TST 2019, P2
For each permutation $\sigma$ of the set $\{1, 2, \ldots , N\}$ we define its [i]correctness[/i] as the number of triples $1 \leqslant i < j < k \leqslant N$ such that the number $\sigma(j)$ lies between the numbers $\sigma(i)$ and $\sigma(k)$. Find the difference between the number of permutations with even correctness and the number of permutations with odd correctness if a) $N = 2018$ and b) $N = 2019$.
1974 Bulgaria National Olympiad, Problem 4
Find the maximal count of shapes that can be placed over a chessboard with size $8\times8$ in such a way that no three shapes are not on two squares, lying next to each other by diagonal parallel $A1-H8$ ($A1$ is the lowest-bottom left corner of the chessboard, $H8$ is the highest-upper right corner of the chessboard).
[i]V. Chukanov[/i]
2018 IFYM, Sozopol, 3
We will call one of the cells of a rectangle 11 x 13 “[i]peculiar[/i]” , if after removing it the remaining figure can be cut into squares 2 x 2 and 3 x 3. How many of the 143 cells are “[i]peculiar[/i]”?
Kvant 2021, M2675
There was a rook at some square of a $10 \times 10{}$ chessboard. At each turn it moved to a square adjacent by side. It visited each square exactly once. Prove that for each main diagonal (the diagonal between the corners of the board) the following statement is true: in the rook’s path there were two consecutive steps at which the rook first stepped away from the diagonal and then returned back to the diagonal.
[i]Alexandr Gribalko[/i]
2019 Saint Petersburg Mathematical Olympiad, 2
In the city built are $2019$ metro stations. Some pairs of stations are connected. tunnels, and from any station through the tunnels you can reach any other. The mayor ordered to organize several metro lines: each line should include several different stations connected in series by tunnels (several lines can pass through the same tunnel), and in each station must lie at least on one line. To save money no more than $k$ lines should be made. It turned out that the order of the mayor is not feasible. What is the largest $k$ it could to happen?
2012 China Team Selection Test, 1
In a simple graph $G$, we call $t$ pairwise adjacent vertices a $t$[i]-clique[/i]. If a vertex is connected with all other vertices in the graph, we call it a [i]central[/i] vertex. Given are two integers $n,k$ such that $\dfrac {3}{2} \leq \dfrac{1}{2} n < k < n$. Let $G$ be a graph on $n$ vertices such that
[b](1)[/b] $G$ does not contain a $(k+1)$-[i]clique[/i];
[b](2)[/b] if we add an arbitrary edge to $G$, that creates a $(k+1)$-[i]clique[/i].
Find the least possible number of [i]central[/i] vertices in $G$.
1995 Taiwan National Olympiad, 3
Suppose that $n$ persons meet in a meeting, and that each of the persons is acquainted to exactly $8$ others. Any two acquainted persons have exactly $4$ common acquaintances, and any two non-acquainted persons have exactly $2$ common acquaintances. Find all possible values of $n$.
2017 Romanian Masters In Mathematics, 5
Fix an integer $n \geq 2$. An $n\times n$ sieve is an $n\times n$ array with $n$ cells removed so that exactly one cell is removed from every row and every column. A stick is a $1\times k$ or $k\times 1$ array for any positive integer $k$. For any sieve $A$, let $m(A)$ be the minimal number of sticks required to partition $A$. Find all possible values of $m(A)$, as $A$ varies over all possible $n\times n$ sieves.
[i]Palmer Mebane[/i]
2010 HMNT, 8
Allison has a coin which comes up heads $\frac23$ of the time. She flips it $5$ times. What is the probability that she sees more heads than tails?
2018 Costa Rica - Final Round, 1
There are $10$ points on a circle and all possible segments are drawn on the which two of these points are the endpoints. Determine the probability that selecting two segments randomly, they intersect at some point (it could be on the circumference).
2009 All-Russian Olympiad, 7
We call any eight squares in a diagonal of a chessboard as a fence. The rook is moved on the chessboard in such way that he stands neither on each square over one time nor on the squares of the fences (the squares which the rook passes is not considered ones it has stood on). Then what is the maximum number of times which the rook jumped over the fence?
2000 Moldova National Olympiad, Problem 3
For every nonempty subset $X$ of $M=\{1,2,\ldots,2000\}$, $a_X$ denotes the sum of the minimum and maximum element of $X$. Compute the arithmetic mean of the numbers $a_X$ when $X$ goes over all nonempty subsets $X$ of $M$.
2003 Gheorghe Vranceanu, 4
Prove that among any $ 16 $ numbers smaller than $ 101 $ there are four of them that have the property that the sum of two of them is equal to the sum of the other two.
1999 Taiwan National Olympiad, 3
There are $1999$ people participating in an exhibition. Among any $50$ people there are two who don't know each other. Prove that there are $41$ people, each of whom knows at most $1958$ people.
2003 Irish Math Olympiad, 5
(a) In how many ways can $1003$ distinct integers be chosen from the set $\{1, 2, ... , 2003\}$ so that no two of the chosen integers di ffer by $10?$
(b) Show that there are $(3(5151) + 7(1700)) 101^7$ ways to choose $1002$ distinct integers from the set $\{1, 2, ... , 2003\}$ so that no two of the chosen integers diff er by $10.$
1998 Belarus Team Selection Test, 1
Any of $6$ gossips has her own news. From time to time one of them makes a telephone call to some other gossip and they discuss fill the news they know. What the minimum number of the calls is necessary so as (for) all of them to know all the news?
2004 Tournament Of Towns, 2
A box contains red, green, blue, and white balls, 111 balls in all. If you take out 100 balls without looking, then there will always be 4 balls of different colors among them. What is the smallest number of balls you must take out without looking to guarantee that among them there will always be balls of at least 3 different colors?