Found problems: 149
1991 Czech And Slovak Olympiad IIIA, 6
The set $N$ is partitioned into three subsets $A_1,A_2,A_3$.
Prove that at least one of them has the following property: There exists a positive number $m$ such that for any $k$ one can find numbers $a_1 < a_2 < ... < a_k$ in that subset satisfying $a_{j+1} -a_j \le m$ for $j = 1,...,k -1$.
2001 Croatia Team Selection Test, 1
Consider $A = \{1, 2, ..., 16\}$. A partition of $A$ into nonempty sets $A_1, A_2,..., A_n$ is said to be good if none of the Ai contains elements $a, b, c$ (not necessarily distinct) such that $a = b + c$.
(a) Find a good partition $\{A_1, A_2, A_3, A_4\}$ of $A$.
(b) Prove that no partition $\{A_1, A_2, A_3\}$ of $A$ is good
2015 Moldova Team Selection Test, 4
Consider a positive integer $n$ and $A = \{ 1,2,...,n \}$. Call a subset $X \subseteq A$ [i][b]perfect[/b][/i] if $|X| \in X$. Call a perfect subset $X$ [i][b]minimal[/b][/i] if it doesn't contain another perfect subset. Find the number of minimal subsets of $A$.
2001 Dutch Mathematical Olympiad, 5
If you take a subset of $4002$ numbers from the whole numbers $1$ to $6003$, then there is always a subset of $2001$ numbers within that subset with the following property:
If you order the $2001$ numbers from small to large, the numbers are alternately even and odd (or odd and even).
Prove this.
2018 Thailand TST, 2
For finite sets $A,M$ such that $A \subseteq M \subset \mathbb{Z}^+$, we define $$f_M(A)=\{x\in M \mid x\text{ is divisible by an odd number of elements of }A\}.$$ Given a positive integer $k$, we call $M$ [i]k-colorable[/i] if it is possible to color the subsets of $M$ with $k$ colors so that for any $A \subseteq M$, if $f_M(A)\neq A$ then $f_M(A)$ and $A$ have different colors.
Determine the least positive integer $k$ such that every finite set $M \subset\mathbb{Z}^+$ is k-colorable.
2021 Junior Balkan Team Selection Tests - Romania, P2
For any non-empty subset $X$ of $M=\{1,2,3,...,2021\}$, let $a_X$ be the sum of the greatest and smallest elements of $X$. Determine the arithmetic mean of all the values of $a_X$, as $X$ covers all the non-empty subsets of $M$.
1986 Czech And Slovak Olympiad IIIA, 4
Let $C_1,C_2$, and $C_3$ be points inside a bounded convex planar set $M$. Rays $l_1,l_2,l_3$ emanating from $C_1,C_2,C_3$ respectively partition the complement of the set $M \cup l_1 \cup l_2 \cup l_3$ into three regions $D_1,D_2,D_3$. Prove that if the convex sets $A$ and $B$ satisfy $A\cap l_j =\emptyset = B\cap l_j$ and $A\cap D_j \ne \emptyset \ne B\cap D_j$ for $j = 1,2,3$, then $A\cap B \ne \emptyset$
1997 Pre-Preparation Course Examination, 1
Let $ k,m,n$ be integers such that $ 1 < n \leq m \minus{} 1 \leq k.$ Determine the maximum size of a subset $ S$ of the set $ \{1,2,3, \ldots, k\minus{}1,k\}$ such that no $ n$ distinct elements of $ S$ add up to $ m.$
1990 Romania Team Selection Test, 6
Prove that there are infinitely many n’s for which there exists a partition of $\{1,2,...,3n\}$ into subsets $\{a_1,...,a_n\}, \{b_1,...,b_n\}, \{c_1,...,c_n\}$ such that $a_i +b_i = c_i$ for all $i$, and prove that there are infinitely many $n$’s for which there is no such partition.
1974 Poland - Second Round, 1
Let $ Z $ be a set of $ n $ elements. Find the number of such pairs of sets $ (A, B) $ such that $ A $ is contained in $ B $ and $ B $ is contained in $ Z $. We assume that every set also contains itself and the empty set.
2015 NIMO Problems, 5
Compute the number of subsets $S$ of $\{0,1,\dots,14\}$ with the property that for each $n=0,1,\dots,
6$, either $n$ is in $S$ or both of $2n+1$ and $2n+2$ are in $S$.
[i]Proposed by Evan Chen[/i]
2014 Greece JBMO TST, 4
Givan the set $S = \{1,2,3,....,n\}$. We want to partition the set $S$ into three subsets $A,B,C$ disjoint (to each other) with $A\cup B\cup C=S$ , such that the sums of their elements $S_{A} S_{B} S_{C}$ to be equal .Examine if this is possible when:
a) $n=2014$
b) $n=2015 $
c) $n=2018$
2018 India PRMO, 22
A positive integer $k$ is said to be [i]good [/i] if there exists a partition of $ \{1, 2, 3,..., 20\}$ into disjoint proper subsets such that the sum of the numbers in each subset of the partition is $k$. How many [i]good [/i] numbers are there?
2016 Saudi Arabia GMO TST, 2
Let $n \ge 1$ be a fixed positive integer. We consider all the sets $S$ which consist of sub-sequences of the sequence $0, 1,2, ..., n$ satisfying the following conditions:
i) If $(a_i)_{i=0}^k$ belongs to $S$, then $a_0 = 0$, $a_k = n$ and $a_{i+1} - a_i \le 2$ for all $0 \le i \le k - 1$.
ii) If $(a_i)_{i=0}^k$ and $(b_j)^h_{j=0}$ both belong to $S$, then there exist $0 \le i_0 \le k - 1$ and $0 \le j_0 \le h - 1$ such that $a_{i_0} = b_{j_0}$ and $a_{i_0+1} = b_{j_0+1}$.
Find the maximum value of $|S|$ (among all the above-mentioned sets $S$).
2012 Danube Mathematical Competition, 4
Given a positive integer $n$, show that the set $\{1,2,...,n\}$ can be partitioned into $m$ sets, each with the same sum, if and only if m is a divisor of $\frac{n(n + 1)}{2}$ which does not exceed $\frac{n + 1}{2}$.
2008 Indonesia TST, 2
Let $S = \{1, 2, 3, ..., 100\}$ and $P$ is the collection of all subset $T$ of $S$ that have $49$ elements, or in other words: $$P = \{T \subset S : |T| = 49\}.$$ Every element of $P$ is labelled by the element of $S$ randomly (the labels may be the same). Show that there exist subset $M$ of $S$ that has $50$ members such that for every $x \in M$, the label of $M -\{x\}$ is not equal to $x$
1998 Bundeswettbewerb Mathematik, 2
Prove that there exist $16$ subsets of set $M = \{1,2,...,10000\}$ with the following property:
For every $z \in M$ there are eight of these subsets whose intersection is $\{z\}$.
1990 Czech and Slovak Olympiad III A, 6
Let $k\ge 1$ be an integer and $\mathsf S$ be a family of 2-element subsets of the index set $\{1,\ldots,2k\}$ with the following property: if $\mathsf M_1,\ldots,\mathsf M_{2k}$ are arbitrary sets such that \[\mathsf M_i\cap\mathsf M_j\neq\emptyset\quad\Leftrightarrow\quad\{i,j\}\in\mathsf S,\] then the union $\mathsf M_1\cup\ldots\cup\mathsf M_{2k}$ contains at least $k^2$ elements. Show that there is a suitable family $\mathsf S$ for any integer $k\ge1.$
1999 Greece Junior Math Olympiad, 4
Defi ne alternate sum of a set of real numbers $A =\{a_1,a_2,...,a_k\}$ with $a_1 < a_2 <...< a_k$, the number
$S(A) = a_k - a_{k-1} + a_{k-2} - ... + (-1)^{k-1}a_1$ (for example if $A = \{1,2,5, 7\}$ then $S(A) = 7 - 5 + 2 - 1$)
Consider the alternate sums, of every subsets of $A = \{1, 2, 3, 4, 5, 6, 7, 8,9, 10\}$ and sum them.
What is the last digit of the sum obtained?
2015 Irish Math Olympiad, 7
Let $n > 1$ be an integer and $\Omega=\{1,2,...,2n-1,2n\}$ the set of all positive integers that are not larger than $2n$.
A nonempty subset $S$ of $\Omega$ is called [i]sum-free[/i] if, for all elements $x, y$ belonging to $S, x + y$ does not belong to $S$. We allow $x = y$ in this condition.
Prove that $\Omega$ has more than $2^n$ distinct [i]sum-free[/i] subsets.
2005 Czech And Slovak Olympiad III A, 2
Determine for which $m$ there exist exactly $2^{15}$ subsets $X$ of $\{1,2,...,47\}$ with the following property: $m$ is the smallest element of $X$, and for every $x \in X$, either $x+m \in X$ or $x+m > 47$.
1986 Bundeswettbewerb Mathematik, 4
Given the finite set $M$ with $m$ elements and $1986$ further sets $M_1,M_2,M_3,...,M_{1986}$, each of which contains more than $\frac{m}{2}$ elements from $M$ . Show that no more than ten elements need to be marked in order for any set $M_i$ ($i =1, 2, 3,..., 1986$) contains at least one marked element.
1982 Spain Mathematical Olympiad, 7
Let $S$ be the subset of rational numbers that can be written in the form $a/b$, where $a$ is any integer and $b$ is an odd integer. Does the sum of two of its elements belong to the $S$ ? And the product? Are there elements in $S$ whose inverse belongs to $S$ ?
2004 Junior Tuymaada Olympiad, 4
Given the disjoint finite sets of natural numbers $ A $ and $ B $, consisting of $ n $ and $ m $ elements, respectively. It is known that every natural number belonging to $ A $ or $ B $ satisfies at least one of the conditions $ k + 17 \in A $, $ k-31 \in B $. Prove that $ 17n = 31m $
2024 European Mathematical Cup, 4
Let $\mathcal{F}$ be a family of (distinct) subsets of the set $\{1,2,\dots,n\}$ such that for all $A$, $B\in \mathcal{F}$,we have that $A^C\cup B\in \mathcal{F}$, where $A^C$ is the set of all members of ${1,2,\dots,n}$ that are not in $A$.
Prove that every $k\in {1,2,\dots,n}$ appears in at least half of the sets in $\mathcal{F}$.
[i]Stijn Cambie, Mohammad Javad Moghaddas Mehr[/i]