Found problems: 14842
2000 Tuymaada Olympiad, 7
Every two of five regular pentagons on the plane have a common point.
Is it true that some of these pentagons have a common point?
2022 Princeton University Math Competition, A2 / B4
Ten evenly spaced vertical lines in the plane are labeled $\ell_1,\ell_2, \ldots,\ell_{10}$ from left to right. A set $\{a,b,c,d\}$ of four distinct integers $a,b,c,d \in \{1,2,\ldots,10\}$ is [i]squarish[/i] if some square has one vertex on each of the lines $\ell_a,\ell_b,\ell_c,$ and $\ell_d.$ Find the number of squarish sets.
2004 Bulgaria Team Selection Test, 2
The edges of a graph with $2n$ vertices ($n \ge 4$) are colored in blue and red such that there is no blue triangle and there is no red complete subgraph with $n$ vertices. Find the least possible number of blue edges.
2005 Junior Balkan Team Selection Tests - Romania, 3
In a country 6 cities are connected two by two with round-trip air routes operated by exactly one of the two air companies in that country.
Prove that there exist 4 cities $A$, $B$, $C$ and $D$ such that each of the routes $A\leftrightarrow B$, $B\leftrightarrow C$, $C\leftrightarrow D$ and $D\leftrightarrow A$ are operated by the same company.
[i]Dan Schwartz[/i]
2010 ELMO Shortlist, 2
For a positive integer $n$, let $s(n)$ be the number of ways that $n$ can be written as the sum of strictly increasing perfect $2010^{\text{th}}$ powers. For instance, $s(2) = 0$ and $s(1^{2010} + 2^{2010}) = 1$. Show that for every real number $x$, there exists an integer $N$ such that for all $n > N$,
\[\frac{\max_{1 \leq i \leq n} s(i)}{n} > x.\]
[i]Alex Zhu.[/i]
2004 Estonia National Olympiad, 3
From $25$ points in a plane, both of whose coordinates are integers of the set $\{-2,-1, 0, 1, 2\}$, some $17$ points are marked. Prove that there are three points on one line, one of them is the midpoint of two others.
2005 VTRMC, Problem 3
We wish to tile a strip of $n$ $1$-inch by $1$-inch squares. We can use dominos which are made up of two tiles that cover two adjacent squares, or $1$-inch square tiles which cover one square. We may cover each square with one or two tiles and a tile can be above or below a domino on a square, but no part of a domino can be placed on any part of a different domino. We do not distinguish whether a domino is above or below a tile on a given square. Let $t(n)$ denote the number of ways the strip can be tiled according to the above rules. Thus for example, $t(1)=2$ and $t(2)=8$. Find a recurrence relation for $t(n)$, and use it to compute $t(6)$.
2006 VJIMC, Problem 3
Two players play the following game: Let $n$ be a fixed integer greater than $1$. Starting from number $k=2$, each player has two possible moves: either replace the number $k$ by $k+1$ or by $2k$. The player who is forced to write a number greater than $n$ loses the game. Which player has a winning strategy for which $n$?
2016 India IMO Training Camp, 3
An equilateral triangle with side length $3$ is divided into $9$ congruent triangular cells as shown in the figure below. Initially all the cells contain $0$. A [i]move[/i] consists of selecting two adjacent cells (i.e., cells sharing a common boundary) and either increasing or decreasing the numbers in both the cells by $1$ simultaneously. Determine all positive integers $n$ such that after performing several such moves one can obtain $9$ consecutive numbers $n,(n+1),\cdots ,(n+8)$ in some order.
[asy] size(3cm);
pair A=(0,0),D=(1,0),B,C,E,F,G,H,I;
G=rotate(60,A)*D;
B=(1/3)*D; C=2*B;I=(1/3)*G;H=2*I;E=C+I-A;F=H+B-A;
draw(A--D--G--A^^B--F--H--C--E--I--B,black);[/asy]
2011 Danube Mathematical Competition, 4
Given a positive integer number $n$, determine the maximum number of edges a triangle-free Hamiltonian simple graph on $n$ vertices may have.
2017 Hanoi Open Mathematics Competitions, 10
Consider all words constituted by eight letters from $\{C ,H,M, O\}$. We arrange the words in an alphabet sequence.
Precisely, the first word is $CCCCCCCC$, the second one is $CCCCCCCH$, the third is $CCCCCCCM$, the fourth one is $CCCCCCCO, ...,$ and the last word is $OOOOOOOO$.
a) Determine the $2017$th word of the sequence?
b) What is the position of the word $HOMCHOMC$ in the sequence?
2018 Costa Rica - Final Round, F2
Consider $f (n, m)$ the number of finite sequences of $ 1$'s and $0$'s such that each sequence that starts at $0$, has exactly n $0$'s and $m$ $ 1$'s, and there are not three consecutive $0$'s or three $ 1$'s. Show that if $m, n> 1$, then
$$f (n, m) = f (n-1, m-1) + f (n-1, m-2) + f (n-2, m-1) + f (n-2, m-2)$$
2022 Bulgaria National Olympiad, 6
Let $n\geq 2$ be a positive integer. The sets $A_{1},A_{2},\ldots, A_{n}$ and $B_{1},B_{2},\ldots, B_{n}$ of positive integers are such that $A_{i}\cap B_{j}$ is non-empty $\forall i,j\in\{1,2,\ldots ,n\}$ and $A_{i}\cap A_{j}=\o$, $B_{i}\cap B_{j}=\o$ $\forall i\neq j\in \{1,2,\ldots, n\}$. We put the elements of each set in a descending order and calculate the differences between consecutive elements in this new order. Find the least possible value of the greatest of all such differences.
2013 Saudi Arabia IMO TST, 4
Determine whether it is possible to place the integers $1, 2,...,2012$ in a circle in such a way that the $2012$ products of adjacent pairs of numbers leave pairwise distinct remainders when divided by $2013$.
2021 Kyiv Mathematical Festival, 5
Bilbo composes a number triangle of zeroes and ones in such a way: he fills the topmost row with any $n$ digits, and in other rows he always writes $0$ under consecutive equal digits and writes $1$ under consecutive distinct digits. (An example of a triangle for $n=5$ is shown below.) In how many ways can Bilbo fill the topmost row for $n=100$ so that each of $n$ rows of the triangle contains even number of ones?\[\begin{smallmatrix}1\,0\,1\,1\,0\\1\,1\,0\,1\\0\,1\,1\\1\,0\\1\end{smallmatrix}\] (O. Rudenko and V. Brayman)
2019 ELMO Shortlist, C1
Elmo and Elmo's clone are playing a game. Initially, $n\geq 3$ points are given on a circle. On a player's turn, that player must draw a triangle using three unused points as vertices, without creating any crossing edges. The first player who cannot move loses. If Elmo's clone goes first and players alternate turns, who wins? (Your answer may be in terms of $n$.)
[i]Proposed by Milan Haiman[/i]
2018 Nepal National Olympiad, 4c
[b]Problem Section #4
c) A special deck of cards contains $49$ cards, each labeled with a number from $1$ to $7$ and colored with
one of seven colors. Each number-color combination appears on exactly one card. John will select
a set of eight cards from the deck at random. Given that he gets at least one card of each color and
at least one card with each number, the probability that John can discard one of his cards and still
have at least one card of each color and at least one card with each number is $\frac{p}{q}$, where $p$ and $q$ are
relatively prime positive integers. Find $p+q$.
2012 Chile National Olympiad, 1
What is the minimum number of movements that a horse must carry out on chess, on an $8\times 8$ board, to reach the upper right square starting at the lower left? Remember that the horse moves in the usual $L$-shaped manner.
2021 Iran RMM TST, 1
A polyomino is region with connected interior that is a union of a finite number of squares from a grid of unit squares. Do there exist a positive integer $n>4$ and a polyomino $P$ contained entirely within and $n$-by-$n$ grid such that $P$ contains exactly $3$ unit squares in every row and every column of the grid?
Proposed by [i]Nikolai Beluhov[/i]
2019 Iran RMM TST, 3
An infinite network is partitioned with dominos. Prove there exist three other tilings with dominos, have neither common domino with the existing tiling nor with each other.
Clarifications for network: It means an infinite board consisting of square cells.
2016 Tuymaada Olympiad, 1
Tanya and Serezha have a heap of $2016$ candies. They make moves in turn, Tanya moves first. At each move a player can eat either one candy or (if the number of candies is even at the moment) exactly half of all candies. The player that cannot move loses. Which of the players has a winning strategy?
1985 Yugoslav Team Selection Test, Problem 1
Suppose each element $i\in S=\{1,2,\ldots,n\}$ is assigned a nonempty set $S_i\subseteq S$ so that the following conditions are fulfilled:
(i) for any $i,j\in S$, if $j\in S_i$ then $i\in S_j$;
(ii) for any $i,j\in S$, if $|S_i|=|S_j|$ then $S_i\cap S_j=\emptyset$.
Prove that there exists $k\in S$ for which $|S_k|=1$.
2011 Chile National Olympiad, 4
It is intended to make a map locating $30$ different cities on it. For this, all the distances between these cities are available as data (each of these distances is considered as a “data”). Three of these cities are already laid out on the map, and they turn out to be non-collinear. How much data must be used as a minimum to complete the map?
2017 Sharygin Geometry Olympiad, 7
Let $a$ and $b$ be parallel lines with $50$ distinct points marked on $a$ and $50$ distinct points marked on $b$. Find the greatest possible number of acute-angled triangles all of whose vertices are marked.
1995 ITAMO, 1
Determine for which values of $n$ it is possible to tile a square of side $n$ with figures of the type shown in the picture
[asy]
unitsize(0.4 cm);
draw((0,0)--(5,0));
draw((0,1)--(5,1));
draw((1,2)--(4,2));
draw((2,3)--(3,3));
draw((0,0)--(0,1));
draw((1,0)--(1,2));
draw((2,0)--(2,3));
draw((3,0)--(3,3));
draw((4,0)--(4,2));
draw((5,0)--(5,1));
[/asy]