Found problems: 14842
2012 Miklós Schweitzer, 3
There is a simple graph which chromatic number is equal to $k$. We painted all of the edges of graph using two colors. Prove that there exist a monochromatic tree with $k$ vertices
1997 All-Russian Olympiad Regional Round, 10.1
The microcalculator ''MK-97'' can work out the numbers entered in memory, perform only three operations:
a) check whether the selected two numbers are equal;
b) add the selected numbers;
c) using the selected numbers $a$ and $b$, find the equation $x^2 +ax+b = 0$, and if there are no roots, display a message about this.
The results of all actions are stored in memory. Initially, one number $x$ is stored in memory. How to use ''MK-97'' to find out whether is this number one?
2011 Middle European Mathematical Olympiad, 3
For an integer $n \geq 3$, let $\mathcal M$ be the set $\{(x, y) | x, y \in \mathbb Z, 1 \leq x \leq n, 1 \leq y \leq n\}$ of points in the plane.
What is the maximum possible number of points in a subset $S \subseteq \mathcal M$ which does not contain three distinct points being the vertices of a right triangle?
2011 IMAC Arhimede, 3
Place $n$ points on a circle and draw all possible chord joining these points. If no three chord are concurent, find the number of disjoint regions created.
[color=#008000]Moderator says: http://www.artofproblemsolving.com/Forum/viewtopic.php?f=151&t=260926&hilit=circle+points+segments+regions[/color]
2021 BMT, 23
Shivani has a single square with vertices labeled $ABCD$. She is able to perform the following transformations:
$\bullet$ She does nothing to the square.
$\bullet$ She rotates the square by $90$, $180$, or $270$ degrees.
$\bullet$ She reflects the square over one of its four lines of symmetry.
For the first three timesteps, Shivani only performs reflections or does nothing. Then for the next three timesteps, she only performs rotations or does nothing. She ends up back in the square’s original configuration. Compute the number of distinct ways she could have achieved this.
2022/2023 Tournament of Towns, P1
Is it possible to arrange $36$ distinct numbers in the cells of a $6 \times 6$ table, so that in each $1\times 5$ rectangle (both vertical and horizontal) the sum of the numbers equals $2022$ or $2023$?
2024 Romania EGMO TST, P2
Given a graph with $99$ vertices and degrees in $\{81,82,\dots,90\}$, prove that there exist $10$ vertices of this graph with equal degrees and a common neighbour.
[i]Proposed by Alireza Alipour[/i]
1966 IMO Shortlist, 52
A figure with area $1$ is cut out of paper. We divide this figure into $10$ parts and color them in $10$ different colors. Now, we turn around the piece of paper, divide the same figure on the other side of the paper in $10$ parts again (in some different way). Show that we can color these new parts in the same $10$ colors again (hereby, different parts should have different colors) such that the sum of the areas of all parts of the figure colored with the same color on both sides is $\geq \frac{1}{10}.$
2020 CMIMC Combinatorics & Computer Science, 2
David is taking a true/false exam with $9$ questions. Unfortunately, he doesn’t know the answer to any of the questions, but he does know that exactly $5$ of the answers are True. In accordance with this, David guesses the answers to all $9$ questions, making sure that exactly $5$ of his answers are True. What is the probability he answers at least $5$ questions correctly?
2017 China Team Selection Test, 3
Find the numbers of ordered array $(x_1,...,x_{100})$ that satisfies the following conditions:
($i$)$x_1,...,x_{100}\in\{1,2,..,2017\}$;
($ii$)$2017|x_1+...+x_{100}$;
($iii$)$2017|x_1^2+...+x_{100}^2$.
2022 Bolivia IMO TST, P4
Let $S$ be an infinite set of positive integers, such that there exist four pairwise distinct $a,b,c,d \in S$ with $\gcd(a,b) \neq \gcd(c,d)$. Prove that there exist three pairwise distinct $x,y,z \in S$ such that $\gcd(x,y)=\gcd(y,z) \neq \gcd(z,x)$.
1998 Iran MO (3rd Round), 4
Let be given $r_1,r_2,\ldots,r_n \in \mathbb R$. Show that there exists a subset $I$ of $\{1,2,\ldots,n \}$ which which has one or two elements in common with the sets $\{i,i + 1,i + 2\} , (1 \leq i \leq n- 2)$ such that
\[\left| {\mathop \sum \limits_{i \in I} {r_i}} \right| \geqslant \frac{1}{6}\mathop \sum \limits_{i = 1}^n \left| {{r_i}} \right|.\]
2009 Peru MO (ONEM), 3
a) On a circumference $8$ points are marked. We say that Juliana does an “T-operation ” if she chooses three of these points and paint the sides of the triangle that they determine, so that each painted triangle has at most one vertex in common with a painted triangle previously. What is the greatest number of “T-operations” that Juliana can do?
b) If in part (a), instead of considering $8$ points, $7$ points are considered, what is the greatest number of “T operations” that Juliana can do?
2008 Brazil Team Selection Test, 2
Let $n$ be a positive integer. A sequence $(a, b, c)$ of $a, b, c \in \{1, 2, . . . , 2n\}$ is called [i]joke [/i] if its shortest term is odd and if only that smallest term, or no term, is repeated. For example, the sequences $(4, 5, 3)$ and $(3, 8, 3)$ are jokes, but $(3, 2, 7)$ and $(3, 8, 8)$ are not. Determine the number of joke sequences in terms of $n$.
1975 All Soviet Union Mathematical Olympiad, 218
The world and the european champion are determined in the same tournament carried in one round. There are $20$ teams and $k$ of them are european. The european champion is determined according to the results of the games only between those $k$ teams. What is the greatest $k$ such that the situation, when the single european champion is the single world outsider, is possible if:
a) it is hockey (draws allowed)?
b) it is volleyball (no draws)?
2010 Paenza, 5
In $4$-dimensional space, a set of $1 \times 2 \times 3 \times 4$ bricks is given. Decide whether it is possible to build boxes of the following sizes using these bricks:
[list]i) $2 \times 5 \times 7 \times 12$
ii) $5 \times 5 \times 10 \times 12$
iii) $6 \times 6 \times 6 \times 6$.[/list]
2007 Pre-Preparation Course Examination, 2
There is a WORD game with the following rules. There are finite number of relations $U_{i}\longrightarrow V_{i}$($U_{i},V_{i}$ are words). There is are two words $A,B$. We start from $A$, and we want to reach to $B$. At each step we can change one subword $U_{i}$ to $V_{i}$. Prove that there does not exist an algorithm that picks up $A,B$ and $U_{i}$'s,$V_{i}$'s and decides whether we can reach from $A$ to $B$ or not.
2013 Ukraine Team Selection Test, 7
$2013$ users have registered on the social network "Graph". Some users are friends, and friendship in "Graph" is mutual. It is known that among network users there are no three, each of whom would be friends. Find the biggest one possible number of pairs of friends in "Graph".
LMT Speed Rounds, 12
Sam and Jonathan play a game where they take turns flipping a weighted coin, and the game ends when one of them wins. The coin has a $\frac89$ chance of landing heads and a $\frac19$ chance of landing tails. Sam wins when he flips heads, and Jonathan wins when he flips tails. Find the probability that Samwins, given that he takes the first turn.
[i]Proposed by Samuel Tsui[/i]
2016 Middle European Mathematical Olympiad, 4
An exam was taken by some students. Each problem was worth $1$ point for the correct answer, and $0$ points for an incorrect one.
For each question, at least one student answered it correctly. Also, there are two students with different scores on the exam.
Prove that there exists a question for which the following holds:
The average score of the students who answered the question correctly is greater than the average score of the students who didn't.
2017 Iberoamerican, 3
Consider the configurations of integers
$a_{1,1}$
$a_{2,1} \quad a_{2,2}$
$a_{3,1} \quad a_{3,2} \quad a_{3,3}$
$\dots \quad \dots \quad \dots$
$a_{2017,1} \quad a_{2017,2} \quad a_{2017,3} \quad \dots \quad a_{2017,2017}$
Where $a_{i,j} = a_{i+1,j} + a_{i+1,j+1}$ for all $i,j$ such that $1 \leq j \leq i \leq 2016$.
Determine the maximum amount of odd integers that such configuration can contain.
2010 IFYM, Sozopol, 5
Each vertex of a right $n$-gon $(n\geq 3)$ is colored in yellow, blue or red. On each turn are chosen two adjacent vertices in different color and then are recolored in the third. For which $n$ can we get from an arbitrary coloring of the $n$-gon a monochromatic one (in one color)?
2011 Princeton University Math Competition, A2 / B3
A set of $n$ dominoes, each colored with one white square and one black square, is used to cover a $2 \times n$ board of squares. For $n = 6$, how many different patterns of colors can the board have? (For $n = 2$, this number is $6$.)
1990 IMO Longlists, 99
Given a $10 \times 10$ chessboard colored as black-and-white alternately. Prove that for any $46$ unit squares without common edges, there are at least $30$ unit squares with the same color.
2007 Swedish Mathematical Competition, 5
Anna and Brian play a game where they put the domino tiles (of size $2 \times 1$) in a boards composed of $n \times 1$ boxes. Tiles must be placed so that they cover exactly two boxes. Players take turnslaying each tile and the one laying last tile wins. They play once for each $n$, where $n = 2, 3,\dots,2007$. Show that Anna wins at least $1505$ of the games if she always starts first and they both always play optimally, ie if they do their best to win in every move.