Found problems: 14842
1999 All-Russian Olympiad Regional Round, 8.8
An open chain was made from $54$ identical single cardboard squares, connecting them hingedly at the vertices. Any square (except for the extreme ones) is connected to its neighbors by two opposite vertices. Is it possible to completely cover a $3\times 3 \times3$ surface with this chain of squares?
2022 EGMO, 5
For all positive integers $n$, $k$, let $f(n, 2k)$ be the number of ways an $n \times 2k$ board can be fully covered by $nk$ dominoes of size $2 \times 1$. (For example, $f(2, 2)=2$ and $f(3, 2)=3$.) Find all positive integers $n$ such that for every positive integer $k$, the number $f(n, 2k)$ is odd.
1978 Dutch Mathematical Olympiad, 2
One tiles a floor of $a \times b$ dm$^2$ with square tiles, $a,b \in N$. Tiles do not overlap, and sides of floor and tiles are parallel. Using tiles of $2\times 2$ dm$^2$ leaves the same amount of floor uncovered as using tiles of $4\times 4$ dm$^2$. Using $3\times 3$ dm$^2$ tiles leaves $29$ dm$^2$ floor uncovered. Determine $a$ and $b$.
2011 Indonesia TST, 3
Given a board consists of $n \times n$ unit squares ($n \ge 3$). Each unit square is colored black and white, resembling a chessboard. In each step, TOMI can choose any $2 \times 2$ square and change the color of every unit square chosen with the other color (white becomes black and black becomes white). Find every $n$ such that after a finite number of moves, every unit square on the board has a same color.
2016 Israel Team Selection Test, 4
A regular 60-gon is given. What is the maximum size of a subset of its vertices containing no isosceles triangles?
1998 AIME Problems, 13
If $\{a_1,a_2,a_3,\ldots,a_n\}$ is a set of real numbers, indexed so that $a_1<a_2<a_3<\cdots<a_n,$ its complex power sum is defined to be $a_1i+a_2i^2+a_3i^3+\cdots+a_ni^n,$ where $i^2=-1.$ Let $S_n$ be the sum of the complex power sums of all nonempty subsets of $\{1,2,\ldots,n\}.$ Given that $S_8=-176-64i$ and $S_9=p+qi,$ were $p$ and $q$ are integers, find $|p|+|q|.$
1977 IMO Longlists, 51
Several segments, which we shall call white, are given, and the sum of their lengths is $1$. Several other segments, which we shall call black, are given, and the sum of their lengths is $1$. Prove that every such system of segments can be distributed on the segment that is $1.51$ long in the following way: Segments of the same colour are disjoint, and segments of different colours are either disjoint or one is inside the other. Prove that there exists a system that cannot be distributed in that way on the segment that is $1.49$ long.
2016 CMIMC, 5
Let $\mathcal{S}$ be a regular 18-gon, and for two vertices in $\mathcal{S}$ define the $\textit{distance}$ between them to be the length of the shortest path along the edges of $\mathcal{S}$ between them (e.g. adjacent vertices have distance 1). Find the number of ways to choose three distinct vertices from $\mathcal{S}$ such that no two of them have distance 1, 8, or 9.
2011 IberoAmerican, 3
Let $k$ and $n$ be positive integers, with $k \geq 2$. In a straight line there are $kn$ stones of $k$ colours, such that there are $n$ stones of each colour. A [i]step[/i] consists of exchanging the position of two adjacent stones. Find the smallest positive integer $m$ such that it is always possible to achieve, with at most $m$ steps, that the $n$ stones are together, if:
a) $n$ is even.
b) $n$ is odd and $k=3$
2015 Tournament of Towns, 4
A convex$N-$gon with equal sides is located inside a circle. Each side is extended in both directions up to the intersection with the circle so that it contains two new segments outside the polygon. Prove that one can paint some of these new $2N$ segments in red and the rest in blue so that the sum of lengths of all the red segments would be the same as for the blue ones.
[i]($8$ points)[/i]
2020 IOM, 5
There is an empty table with $2^{100}$ rows and $100$ columns. Alice and Eva take turns filling the empty cells of the first row of the table, Alice plays first. In each move, Alice chooses an empty cell and puts a cross in it; Eva in each move chooses an empty cell and puts a zero. When no empty cells remain in the first row, the players move on to the second row, and so on (in each new row Alice plays first).
The game ends when all the rows are filled. Alice wants to make as many different rows in the table as possible, while Eva wants to make as few as possible. How many different rows will be there in the table if both follow their best strategies?
Proposed by Denis Afrizonov
2023 Belarus - Iran Friendly Competition, 5
Define $M_n = \{ 1, 2, \ldots , n \} $ for all positive integers $n$. A collection of $3$-element subsets
of $M_n$ is said to be fine if for any colouring of elements of $M_n$ in two colours there is a subset of the
collection all three elements of which are of the same colour. For each $n \geq 5$ find the minimal
possible number of the $3$-element subsets of a fine collection
1994 China Team Selection Test, 3
Find the smallest $n \in \mathbb{N}$ such that if any 5 vertices of a regular $n$-gon are colored red, there exists a line of symmetry $l$ of the $n$-gon such that every red point is reflected across $l$ to a non-red point.
2013 Stars Of Mathematics, 4
A set $S$ of unit cells of an $n\times n$ array, $n\geq 2$, is said [i]full[/i] if each row and each column of the array contain at least one element of $S$, but which has this property no more when any of its elements is removed. A full set having maximum cardinality is said [i]fat[/i], while a full set of minimum cardinality is said [i]meagre[/i].
i) Determine the cardinality $m(n)$ of the meagre sets, describe all meagre sets and give their count.
ii) Determine the cardinality $M(n)$ of the fat sets, describe all fat sets and give their count.
[i](Dan Schwarz)[/i]
Kvant 2020, M2626
An infinite number of participants gathered for the Olympiad, who were registered under the numbers $1, 2,\ldots$. It turns out that for every $n = 1, 2,\ldots$ a participant with number $n{}$ has at least $n{}$ friends among the remaining participants (note: friendship is mutual). There is a hotel with an infinite number of double rooms. Prove that the participants can be accommodated in double rooms so that there is a couple of friends in each room.
[i]Proposed by V. Bragin, P. Kozhevnikov[/i]
2025 Al-Khwarizmi IJMO, 8
There are $100$ cards on a table, flipped face down. Madina knows that on each card a single number is written and that the numbers are different integers from $1$ to $100$. In a move, Madina is allowed to choose any $3$ cards, and she is told a number that is written on one of the chosen cards, but not which specific card it is on. After several moves, Madina must determine the written numbers on as many cards as possible. What is the maximum number of cards Madina can ensure to determine?
[i]Shubin Yakov, Russia[/i]
VMEO II 2005, 2
Positive integers are colored in black and white. We know that the sum of two numbers of different colors is always black, and that there are infinitely many numbers that are white. Prove that the sum and product of two white numbers are also white numbers.
2021 Bundeswettbewerb Mathematik, 4
Consider a pyramid with a regular $n$-gon as its base. We colour all the segments connecting two of the vertices of the pyramid except for the sides of the base either red or blue.
Show that if $n=9$ then for each such colouring there are three vertices of the pyramid connecting by three segments of the same colour, and that this is not necessarily the case if $n=8$.
TNO 2023 Senior, 6
The points inside a circle \( \Gamma \) are painted with \( n \geq 1 \) colors. A color is said to be dense in a circle \( \Omega \) if every circle contained within \( \Omega \) has points of that color in its interior. Prove that there exists at least one color that is dense in some circle contained within \( \Gamma \).
2009 Princeton University Math Competition, 3
Using one straight cut we partition a rectangular piece of paper into two pieces. We call this one "operation". Next, we cut one of the two pieces so obtained once again, to partition [i]this piece[/i] into two smaller pieces (i.e. we perform the operation on any [i]one[/i] of the pieces obtained). We continue this process, and so, after each operation we increase the number of pieces of paper by $1$. What is the minimum number of operations needed to get $47$ pieces of $46$-sided polygons? [obviously there will be other pieces too, but we will have at least $47$ (not necessarily [i]regular[/i]) $46$-gons.]
2015 JBMO TST - Turkey, 3
In a country consisting of $2015$ cities, between any two cities there is exactly one direct round flight operated by some air company. Find the minimal possible number of air companies if direct flights between any three cities are operated by three different air companies.
II Soros Olympiad 1995 - 96 (Russia), 9.3
It is known that from these five segments it is possible to form four different right triangles. Find the ratio of the largest segment to the smallest.
2009 Irish Math Olympiad, 4
Given an $n$-tuple of numbers $(x_1,x_2,\dots ,x_n)$ where each $x_i=+1$ or $-1$, form a new $n$-tuple $$(x_1x_2,x_2x_3,x_3x_4,\dots ,x_nx_1),$$
and continue to repeat this operation. Show that if $n=2^k$ for some integer $k\ge 1$, then after a certain number of repetitions of the operation, we obtain the $n$-tuple $$(1,1,1,\dots ,1).$$
2015 NIMO Summer Contest, 6
Let $S_0 = \varnothing$ denote the empty set, and define $S_n = \{ S_0, S_1, \dots, S_{n-1} \}$ for every positive integer $n$. Find the number of elements in the set
\[ (S_{10} \cap S_{20}) \cup (S_{30} \cap S_{40}). \]
[i] Proposed by Evan Chen [/i]
2016 Regional Olympiad of Mexico Center Zone, 1
The grid shown below is completed by choosing nine of the following numbers without repeating: $4, 5, 6, 7, 8, 12, 13, 16, 18, 19$. If the sum of the five rows are equal to each other and the sum of the three columns are equal to each other, in how many different ways is it possible to fill the grid?
$ \[\begin {array} {| c | c | c |} \hline 10 & & \\ \hline & & 9 \\ \hline & 3 & \\ \hline 11 & & 17 \\ \hline & 20 & \\ \hline \end {array} \] $
Note: The sum of the rows and the sum of the columns are not necessarily equal.