Found problems: 14842
2023 Purple Comet Problems, 20
Nine light bulbs are equally spaced around a circle. When the power is turned on, each of the nine light bulbs turns blue or red, where the color of each bulb is determined randomly and independently of the colors of the other bulbs. Each time the power is turned on, the probability that the color of each bulb will be the same as at least one of the two adjacent bulbs on the circle is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
2004 China National Olympiad, 3
Let $M$ be a set consisting of $n$ points in the plane, satisfying:
i) there exist $7$ points in $M$ which constitute the vertices of a convex heptagon;
ii) if for any $5$ points in $M$ which constitute the vertices of a convex pentagon, then there is a point in $M$ which lies in the interior of the pentagon.
Find the minimum value of $n$.
[i]Leng Gangsong[/i]
1984 IMO Longlists, 23
A $2\times 2\times 12$ box fixed in space is to be filled with twenty-four $1 \times 1 \times 2$ bricks. In how many ways can this be done?
2022 Auckland Mathematical Olympiad, 6
Eight pieces are placed on a chessboard so that each row and each column contains exactly one piece. Prove that there are an even number of pieces on the black squares of the board.
1986 All Soviet Union Mathematical Olympiad, 435
All the fields of a square $n\times n$ (n>2) table are filled with $+1$ or $-1$ according to the rules:
[i]At the beginning $-1$ are put in all the boundary fields. The number put in the field in turn (the field is chosen arbitrarily) equals to the product of the closest, from the different sides, numbers in its row or in its column.
[/i]
a) What is the minimal
b) What is the maximal
possible number of $+1$ in the obtained table?
2019 Durer Math Competition Finals, 9
A cube has been divided into $27$ equal-sized sub-cubes. We take a line that passes through the interiors of as many sub-cubes as possible. How many does it pass through?
1997 All-Russian Olympiad, 4
An $n\times n\times n$ cube is divided into unit cubes. We are given a closed non-self-intersecting polygon (in space), each of whose sides joins the centers of two unit cubes sharing a common face. The faces of unit cubes which intersect the polygon are said to be distinguished. Prove that the edges of the unit cubes may be colored in two colors so that each distinguished face has an odd number of edges of each color, while each nondistinguished face has an even number of edges of each color.
[i]M. Smurov[/i]
2014 International Zhautykov Olympiad, 2
Let $U=\{1, 2,\ldots, 2014\}$. For positive integers $a$, $b$, $c$ we denote by $f(a, b, c)$ the number of ordered 6-tuples of sets $(X_1,X_2,X_3,Y_1,Y_2,Y_3)$ satisfying the following conditions:
(i) $Y_1 \subseteq X_1 \subseteq U$ and $|X_1|=a$;
(ii) $Y_2 \subseteq X_2 \subseteq U\setminus Y_1$ and $|X_2|=b$;
(iii) $Y_3 \subseteq X_3 \subseteq U\setminus (Y_1\cup Y_2)$ and $|X_3|=c$.
Prove that $f(a,b,c)$ does not change when $a$, $b$, $c$ are rearranged.
[i]Proposed by Damir A. Yeliussizov, Kazakhstan[/i]
1982 Dutch Mathematical Olympiad, 3
Five marbles are distributed at a random among seven urns. What is the expected number of urns with exactly one marble?
2021 New Zealand MO, 1
A school offers three subjects: Mathematics, Art and Science. At least $80\%$ of students study both Mathematics and Art. At least $80\%$ of students study both Mathematics and Science. Prove that at least $80\%$ of students who study both Art and Science, also study Mathematics.
2015 CentroAmerican, Problem 6
$39$ students participated in a math competition. The exam consisted of $6$ problems and each problem was worth $1$ point for a correct solution and $0$ points for an incorrect solution. For any $3$ students, there is at most $1$ problem that was not solved by any of the three. Let $B$ be the sum of all of the scores of the $39$ students. Find the smallest possible value of $B$.
2020 Thailand TST, 1
The infinite sequence $a_0,a _1, a_2, \dots$ of (not necessarily distinct) integers has the following properties: $0\le a_i \le i$ for all integers $i\ge 0$, and \[\binom{k}{a_0} + \binom{k}{a_1} + \dots + \binom{k}{a_k} = 2^k\] for all integers $k\ge 0$. Prove that all integers $N\ge 0$ occur in the sequence (that is, for all $N\ge 0$, there exists $i\ge 0$ with $a_i=N$).
1986 Swedish Mathematical Competition, 6
The interval $[0,1]$ is covered by a finite number of intervals. Show that one can choose a number of these intervals which are pairwise disjoint and have the total length at least $1/2$.
2004 All-Russian Olympiad Regional Round, 8.2
There is a set of weights with the following properties:
1) It contains 5 weights, pairs of different weights.
2) For any two weights, there are two other weights of the same total weight.
What is the smallest number of weights that can be in this set?
2014 Peru Iberoamerican Team Selection Test, P2
Let $n\ge 4$ be an integer. You have two $n\times n$ boards. Each board contains the numbers $1$ to $n^2$ inclusive, one number per square, arbitrarily arranged on each board. A move consists of exchanging two rows or two columns on the first board (no moves can be made on the second board). Show that it is possible to make a sequence of moves such that for all $1 \le i \le n$ and $1 \le j \le n$, the number that is in the $i-th$ row and $j-th$ column of the first board is different from the number that is in the $i-th$ row and $j-th$ column of the second board.
2021 USEMO, 1
Let $n$ be a fixed positive integer and consider an $n\times n$ grid of real numbers. Determine the greatest possible number of cells $c$ in the grid such that the entry in $c$ is both strictly greater than the average of $c$'s column and strictly less than the average of $c$'s row.
[i]Proposed by Holden Mui[/i]
2012 Kosovo Team Selection Test, 4
Each term in a sequence $1,0,1,0,1,0...$starting with the seventh is the sum of the last 6 terms mod 10 .Prove that the sequence $...,0,1,0,1,0,1...$ never occurs
2010 QEDMO 7th, 10
Let $a_1, a_2, ..., a_n$ be positive real numbers. Furthermore, let $S_n$ denote the set of all permutations of set $\{1, 2, ..., n\}$. Prove that
$$\sum_{\pi \in S_n} \frac{1}{a_{\pi(1)}(a_{\pi(1)}+a_{\pi(2)})...(a_{\pi(1)}+a_{\pi(2)}+...+a_{\pi(n)})}=\frac{1}{a_1 a_2 ... a_n}$$
2012 Saint Petersburg Mathematical Olympiad, 3
$25$ students are on exams. Exam consists of some questions with $5$ variants of answer. Every two students gave same answer for not more than $1$ question. Prove, that there are not more than $6$ questions in exam.
2000 All-Russian Olympiad, 4
Let $a_1, a_2, \cdots, a_n$ be a sequence of nonnegative integers. For $k=1,2,\cdots,n$ denote \[ m_k = \max_{1 \le l \le k} \frac{a_{k-l+1} + a_{k-l+2} + \cdots + a_k}{l}. \] Prove that for every $\alpha > 0$ the number of values of $k$ for which $m_k > \alpha$ is less than $\frac{a_1+a_2+ \cdots +a_n}{\alpha}.$
2014 Miklós Schweitzer, 3
We have $4n + 5$ points on the plane, no three of them are collinear. The points are colored with two colors. Prove that from the points we can form $n$ empty triangles (they have no colored points in their interiors) with pairwise disjoint interiors, such that all points occurring as vertices of the $n$ triangles have the same color.
2003 ITAMO, 2
A museum has the shape of a $n \times n$ square divided into $n^2$ rooms of the shape of a unit square $(n>1)$. Between every two adjacent rooms (i.e. sharing a wall) there is a door. A night guardian wants to organize an inspection journey through the museum according to the following rules. He starts from some room and, whenever he enters a room, he stays there for exactly one minute and then proceeds to another room. He is allowed to enter a room more than once, but at the end of his journey he must have spent exactly $k$ minutes in every room. Find all $n$ and $k$ for which it is possible to organize such a journey.
2023 Iran Team Selection Test, 3
Arman, starting from a number, calculates the sum of the cubes of the digits of that number, and again calculates the sum of the cubes of the digits of the resulting number and continues the same process. Arman calls a number $Good$ if it reaches $1$ after performing a number of steps. Prove that there is an arithmetic progression of length $1402$ of good numbers.
[i]Proposed by Navid Safaei [/i]
2018 IFYM, Sozopol, 1
Let $n > 4$ be an integer. A square is divided into $n^2$ smaller identical squares, in some of which were [b]1’s[/b] and in the other – [b]0's[/b]. It is not allowed in one row or column to have the following arrangements of adjacent digits in this order: $101$, $111$ or $1001$. What is the biggest number of [b]1’s[/b] in the table? (The answer depends on $n$.)
2006 Iran MO (3rd Round), 2
Let $B$ be a subset of $\mathbb{Z}_{3}^{n}$ with the property that for every two distinct members $(a_{1},\ldots,a_{n})$ and $(b_{1},\ldots,b_{n})$ of $B$ there exist $1\leq i\leq n$ such that $a_{i}\equiv{b_{i}+1}\pmod{3}$. Prove that $|B| \leq 2^{n}$.