Found problems: 14842
Kvant 2023, M2745
Two 100-digit binary sequences are given. In one operation, one may insert (possibly at the beggining or end) or remove one or more identical digits from a sequence. What is the smallest $k{}$ for which we can transform the first sequence into the second one in no more than $k{}$ operations?
[i]Proposed by V. Novikov[/i]
2008 Spain Mathematical Olympiad, 3
Let $p\ge 3$ be a prime number. Each side of a triangle is divided into $p$ equal parts, and we draw a line from each division point to the opposite vertex. Find the maximum number of regions, every two of them disjoint, that are formed inside the triangle.
2023 BMT, 2
Three people, Pranav, Sumith, and Jacklyn, are attending a fair. Every time a person enters or exits, the groundskeeper writes their name down in chronological order. If each person enters and exits the fairgrounds exactly once, in how many ways can the groundskeeper write down their names?
2001 Tournament Of Towns, 3
On an east-west shipping lane are ten ships sailing individually. The first five from the west are sailing eastwards while the other five ships are sailing westwards. They sail at the same constant speed at all times. Whenever two ships meet, each turns around and sails in the opposite direction. When all ships have returned to port, how many meetings of two ships have taken place?
2001 IberoAmerican, 3
Let $S$ be a set of $n$ elements and $S_1,\ S_2,\dots,\ S_k$ are subsets of $S$ ($k\geq2$), such that every one of them has at least $r$ elements.
Show that there exists $i$ and $j$, with $1\leq{i}<j\leq{k}$, such that the number of common elements of $S_i$ and $S_j$ is greater or equal to: $r-\frac{nk}{4(k-1)}$
1989 IMO Longlists, 76
Poldavia is a strange kingdom. Its currency unit is the bourbaki and there exist only two types of coins: gold ones and silver ones. Each gold coin is worth $ n$ bourbakis and each silver coin is worth $ m$ bourbakis ($ n$ and $ m$ are positive integers). Using gold and silver coins, it is possible to obtain sums such as 10000 bourbakis, 1875 bourbakis, 3072 bourbakis, and so on. But Poldavia’s monetary system is not as strange as it seems:
[b](a)[/b] Prove that it is possible to buy anything that costs an integral number of bourbakis, as long as one can receive change.
[b](b)[/b] Prove that any payment above $ mn\minus{}2$ bourbakis can be made without the need to receive change.
2021 Ecuador NMO (OMEC), 4
In a board $8$x$8$, the unit squares have numbers $1-64$, as shown. The unit square with a multiple of $3$ on it are red. Find the minimum number of chess' bishops that we need to put on the board such that any red unit square either has a bishop on it or is attacked by at least one bishop.
Note: A bishops moves diagonally.
[img]https://i.imgur.com/03baBwp.jpeg[/img]
2018 Brazil National Olympiad, 6
Consider $4n$ points in the plane, with no three points collinear. Using these points as vertices, we form $\binom{4n}{3}$ triangles. Show that there exists a point $X$ of the plane that belongs to the interior of at least $2n^3$ of these triangles.
2013 Saudi Arabia Pre-TST, 3.3
The points of the plane have been colored by $2013$ different colors. We say that a triangle $\vartriangle ABC$ has the color $X$ if its three vertices $A,B,C$ has the color $X$. Prove that there are in nitely many triangles with the same color and the same area.
2007 Indonesia MO, 4
A 10-digit arrangement $ 0,1,2,3,4,5,6,7,8,9$ is called [i]beautiful[/i] if (i) when read left to right, $ 0,1,2,3,4$ form an increasing sequence, and $ 5,6,7,8,9$ form a decreasing sequence, and (ii) $ 0$ is not the leftmost digit. For example, $ 9807123654$ is a beautiful arrangement. Determine the number of beautiful arrangements.
2002 Swedish Mathematical Competition, 1
$268$ numbers are written around a circle. The $17$th number is $3$, the $83$rd is $4$ and the $144$th is $9$. The sum of every $20$ consecutive numbers is $72$. Find the $210$th number.
2010 Ukraine Team Selection Test, 9
Five identical empty buckets of $2$-liter capacity stand at the vertices of a regular pentagon. Cinderella and her wicked Stepmother go through a sequence of rounds: At the beginning of every round, the Stepmother takes one liter of water from the nearby river and distributes it arbitrarily over the five buckets. Then Cinderella chooses a pair of neighbouring buckets, empties them to the river and puts them back. Then the next round begins. The Stepmother goal's is to make one of these buckets overflow. Cinderella's goal is to prevent this. Can the wicked Stepmother enforce a bucket overflow?
[i]Proposed by Gerhard Woeginger, Netherlands[/i]
1995 Tournament Of Towns, (451) 7
A team of geologists on a field expedition have taken with them $80$ tin cans of provisions. The $80$ cans have different weights, which are known (there is a list). After a while the names of the contents of the cans have become illegible. The cook knows what is in each can and claims that he can prove it without opening any can and only using the list and a balance which indicates the difference of weight of the objects placed on its two pans. Show that in order to do so,
(a) four weight measurements will be enough,
(b) three will not
(AK Tolpygo)
2012 ELMO Shortlist, 7
Consider a graph $G$ with $n$ vertices and at least $n^2/10$ edges. Suppose that each edge is colored in one of $c$ colors such that no two incident edges have the same color. Assume further that no cycles of size $10$ have the same set of colors. Prove that there is a constant $k$ such that $c$ is at least $kn^\frac{8}{5}$ for any $n$.
[i]David Yang.[/i]
1995 Spain Mathematical Olympiad, 1
Consider all sets $A$ of one hundred different natural numbers with the property that any three elements $a,b,c \in A$ (not necessarily different) are the sides of a non-obtuse triangle. Denote by $S(A)$ the sum of the perimeters of all such triangles. Compute the smallest possible value of $S(A)$.
2024 IMO, 3
Let $a_1, a_2, a_3, \dots$ be an infinite sequence of positive integers, and let $N$ be a positive integer. Suppose that, for each $n > N$, $a_n$ is equal to the number of times $a_{n-1}$ appears in the list $a_1, a_2, \dots, a_{n-1}$.
Prove that at least one of the sequence $a_1, a_3, a_5, \dots$ and $a_2, a_4, a_6, \dots$ is eventually periodic.
(An infinite sequence $b_1, b_2, b_3, \dots$ is eventually periodic if there exist positive integers $p$ and $M$ such that $b_{m+p} = b_m$ for all $m \ge M$.)
2006 China Girls Math Olympiad, 5
The set $S = \{ (a,b) \mid 1 \leq a, b \leq 5, a,b \in \mathbb{Z}\}$ be a set of points in the plane with integeral coordinates. $T$ is another set of points with integeral coordinates in the plane. If for any point $P \in S$, there is always another point $Q \in T$, $P \neq Q$, such that there is no other integeral points on segment $PQ$. Find the least value of the number of elements of $T$.
2023 China National Olympiad, 6
There are $n(n\ge 8)$ airports, some of which have one-way direct routes between them. For any two airports $a$ and $b$, there is at most one one-way direct route from $a$ to $b$ (there may be both one-way direct routes from $a$ to $b$ and from $b$ to $a$). For any set $A$ composed of airports $(1\le | A| \le n-1)$, there are at least $4\cdot \min \{|A|,n-|A| \}$ one-way direct routes from the airport in $A$ to the airport not in $A$.
Prove that: For any airport $x$, we can start from $x$ and return to the airport by no more than $\sqrt{2n}$ one-way direct routes.
2020 HMNT (HMMO), 7
While waiting for their food at a restaurant in Harvard Square, Ana and Banana draw $3$ squares $\square_1, \square_2, \square_3$ on one of their napkins. Starting with Ana, they take turns filling in the squares with integers from the set $\{1,2,3,4,5\}$ such that no integer is used more than once. Ana's goal is to minimize the minimum value that the polynomial $a_1x^2 + a_2x + a_3$ attains over all real $x$, where $a_1, a_2, a_3$ are the integers written in $\square_1, \square_2, \square_3$ respectively. Banana aims to maximize $M$. Assuming both play optimally, compute the final value of $100a_1+10a_2+a_3$.
1994 North Macedonia National Olympiad, 4
$1994$ points from the plane are given so that any $100$ of them can be selected $98$ that can be rounded (some points may be at the border of the circle) with a diameter of $1$. Determine the smallest number of circles with radius $1$, sufficient to cover all $1994$
2012 Germany Team Selection Test, 1
Find the least integer $k$ such that for any $2011 \times 2011$ table filled with integers Kain chooses, Abel be able to change at most $k$ cells to achieve a new table in which $4022$ sums of rows and columns are pairwise different.
1943 Eotvos Mathematical Competition, 1
Prove that in any group of people, the number of those who know an odd number of the others in the group is even. Assume that knowing is a symmetric relation.
2010 International Zhautykov Olympiad, 3
A rectangle formed by the lines of checkered paper is divided into figures of three kinds: isosceles right triangles (1) with base of two units, squares (2) with unit side, and parallelograms (3) formed by two sides and two diagonals of unit squares (figures may be oriented in any way). Prove that the number of figures of the third kind is even.
[img]http://up.iranblog.com/Files7/dda310bab8b6455f90ce.jpg[/img]
2017 Princeton University Math Competition, A1/B3
Let $X =\{1, 2, ... , 2017\}$. Let $k$ be a positive integer. Given any $r$ such that $1\le r \le k$, there exist $k$ subsets of $X$ such that the union of any $ r$ of them is equal to $X$ , but the union of any fewer than $r$ of them is not equal to $X$ . Find, with proof, the greatest possible value for $k$.
2020 BMT Fall, 7
A fair six-sided die is rolled five times. The probability that the five die rolls form an increasing sequence where each value is strictly larger than the one that preceded can be written in the form $m/n$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.