Found problems: 133
2002 Switzerland Team Selection Test, 10
Given an integer $m\ge 2$, find the smallest integer $k > m$ such that for any partition of the set $\{m,m + 1,..,k\}$ into two classes $A$ and $B$ at least one of the classes contains three numbers $a,b,c$ (not necessarily distinct) such that $a^b = c$.
1992 IMO Longlists, 39
Let $n \geq 2$ be an integer. Find the minimum $k$ for which there exists a partition of $\{1, 2, . . . , k\}$ into $n$ subsets $X_1,X_2, \cdots , X_n$ such that the following condition holds:
for any $i, j, 1 \leq i < j \leq n$, there exist $x_i \in X_1, x_j \in X_2$ such that $|x_i - x_j | = 1.$
2014 Contests, 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) $n=2018$
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.
1988 Tournament Of Towns, (177) 3
The set of all $10$-digit numbers may be represented as a union of two subsets: the subset $M$ consisting of all $10$-digit numbers, each of which may be represented as a product of two $5$-digit numbers, and the subset $N$ , containing the remaining $10$-digit numbers . Which of the sets $M$ and $N$ contains more elements?
(S. Fomin , Leningrad)
2019 Canadian Mathematical Olympiad Qualification, 4
Let $n$ be a positive integer. For a positive integer $m$, we partition the set $\{1, 2, 3,...,m\}$ into $n$ subsets, so that the product of two different elements in the same subset is never a perfect square. In terms of $n$, fi nd the largest positive integer $m$ for which such a partition exists.
1987 Austrian-Polish Competition, 5
The Euclidian three-dimensional space has been partitioned into three nonempty sets $A_1,A_2,A_3$. Show that one of these sets contains, for each $d > 0$, a pair of points at mutual distance $d$.
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 $j = 1,...,k -1$.
Oliforum Contest V 2017, 5
Find the smallest integer $n > 3$ such that, for each partition of $\{3, 4,..., n\}$ in two sets, at least one of these sets contains three (not necessarily distinct) numbers $ a, b, c$ for which $ab = c$.
(Alberto Alfarano)
2015 Korea National Olympiad, 4
For positive integers $n, k, l$, we define the number of $l$-tuples of positive integers $(a_1,a_2,\cdots a_l)$ satisfying the following as $Q(n,k,l)$.
(i): $n=a_1+a_2+\cdots +a_l$
(ii): $a_1>a_2>\cdots > a_l > 0$.
(iii): $a_l$ is an odd number.
(iv): There are $k$ odd numbers out of $a_i$.
For example, from $9=8+1=6+3=6+2+1$, we have $Q(9,1,1)=1$, $Q(9,1,2)=2$, $Q(9,1,3)=1$.
Prove that if $n>k^2$, $\sum_{l=1}^n Q(n,k,l)$ is $0$ or an even number.
1970 IMO, 1
Find all positive integers $n$ such that the set $\{n,n+1,n+2,n+3,n+4,n+5\}$ can be partitioned into two subsets so that the product of the numbers in each subset is equal.
1990 IMO Shortlist, 15
Determine for which positive integers $ k$ the set \[ X \equal{} \{1990, 1990 \plus{} 1, 1990 \plus{} 2, \ldots, 1990 \plus{} k\}\] can be partitioned into two disjoint subsets $ A$ and $ B$ such that the sum of the elements of $ A$ is equal to the sum of the elements of $ B.$
2010 IFYM, Sozopol, 6
We are given the natural numbers $1=a_1,\, \, a_2,...,a_n$, for which
$a_i\leq a_{i+1}\leq 2a_i$
for $i=1,2,...,n-1$ and the sum $\sum_{i=1}^n a_i$ is even. Prove that these numbers can be partitioned into two groups with equal sum.
1983 IMO Longlists, 71
Prove that every partition of $3$-dimensional space into three disjoint subsets has the following property: One of these subsets contains all possible distances; i.e., for every $a \in \mathbb R^+$, there are points $M$ and $N$ inside that subset such that distance between $M$ and $N$ is exactly $a.$
1988 IMO Shortlist, 20
Find the least natural number $ n$ such that, if the set $ \{1,2, \ldots, n\}$ is arbitrarily divided into two non-intersecting subsets, then one of the subsets contains 3 distinct numbers such that the product of two of them equals the third.
1998 North Macedonia National Olympiad, 2
Prove that the numbers $1,2,...,1998$ cannot be separated into three classes whose sums of elements are divisible by $2000,3999$, and $5998$, respectively.
2019 Belarusian National Olympiad, 9.4
The sum of several (not necessarily different) positive integers not exceeding $10$ is equal to $S$.
Find all possible values of $S$ such that these numbers can always be partitioned into two groups with the sum of the numbers in each group not exceeding $70$.
[i](I. Voronovich)[/i]
2014 Rioplatense Mathematical Olympiad, Level 3, 6
Let $n \in N$ such that $1 + 2 + ... + n$ is divisible by $3$. Integers $a_1\ge a_2\ge a_3\ge 2$ have sum $n$ and they satisfy $1 + 2 + ... + a_1\le \frac{1}{3}( 1 + 2 + ... + n ) $ and $1 + 2 + ... + (a_1+ a_2) \le \frac{2}{3}( 1 + 2 + ... + n )$.
Prove that there is a partition of $\{ 1 , 2 , ... , n\}$ in three subsets $A_1, A_2, A_3$ with cardinals $| A_i| = a_i, i = 1 , 2 , 3$, and with equal sums of their elements .
1989 IMO Shortlist, 22
Prove that in the set $ \{1,2, \ldots, 1989\}$ can be expressed as the disjoint union of subsets $ A_i, \{i \equal{} 1,2, \ldots, 117\}$ such that
[b]i.)[/b] each $ A_i$ contains 17 elements
[b]ii.)[/b] the sum of all the elements in each $ A_i$ is the same.
1999 IMO Shortlist, 4
Prove that the set of positive integers cannot be partitioned into three nonempty subsets such that, for any two integers $x,y$ taken from two different subsets, the number $x^2-xy+y^2$ belongs to the third subset.
Russian TST 2014, P2
Let $n$ be a positive integer, and let $A$ be a subset of $\{ 1,\cdots ,n\}$. An $A$-partition of $n$ into $k$ parts is a representation of n as a sum $n = a_1 + \cdots + a_k$, where the parts $a_1 , \cdots , a_k $ belong to $A$ and are not necessarily distinct. The number of different parts in such a partition is the number of (distinct) elements in the set $\{ a_1 , a_2 , \cdots , a_k \} $.
We say that an $A$-partition of $n$ into $k$ parts is optimal if there is no $A$-partition of $n$ into $r$ parts with $r<k$. Prove that any optimal $A$-partition of $n$ contains at most $\sqrt[3]{6n}$ different parts.
1972 IMO Shortlist, 4
Let $n_1, n_2$ be positive integers. Consider in a plane $E$ two disjoint sets of points $M_1$ and $M_2$ consisting of $2n_1$ and $2n_2$ points, respectively, and such that no three points of the union $M_1 \cup M_2$ are collinear. Prove that there exists a straightline $g$ with the following property: Each of the two half-planes determined by $g$ on $E$ ($g$ not being included in either) contains exactly half of the points of $M_1$ and exactly half of the points of $M_2.$
2018 Saudi Arabia BMO TST, 3
The partition of $2n$ positive integers into $n$ pairs is called [i]square-free[/i] if the product of numbers in each pair is not a perfect square.Prove that if for $2n$ distinct positive integers, there exists one square-free partition, then there exists at least $n!$ square-free partitions.
1990 IMO Longlists, 51
Determine for which positive integers $ k$ the set \[ X \equal{} \{1990, 1990 \plus{} 1, 1990 \plus{} 2, \ldots, 1990 \plus{} k\}\] can be partitioned into two disjoint subsets $ A$ and $ B$ such that the sum of the elements of $ A$ is equal to the sum of the elements of $ B.$
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