Olympiad problems

2023 Bulgaria JBMO TST, 4

Given is a set of $n\ge5$ people and $m$ commissions with $3$ persons in each. Let all the commissions be [i]nice[/i] if there are no two commissions $A$ and $B$, such that $\mid A\cap B\mid=1$. Find the biggest possible $m$ (as a function of $n$).

1992 IMO Longlists, 75

Tags: number theory , Sequence , Subset , IMO Shortlist , IMO Longlist , floor function

A sequence $\{an\}$ of positive integers is defined by \[a_n=\left[ n +\sqrt n + \frac 12 \right] , \qquad \forall n \in \mathbb N\] Determine the positive integers that occur in the sequence.

2015 Indonesia MO Shortlist, N6

Tags: number theory , Sets , Subset , divisible

Defined as $N_0$ as the set of all non-negative integers. Set $S \subset N_0$ with not so many elements is called beautiful if for every $a, b \in S$ with $a \ge b$ ($a$ and $b$ do not have to be different), exactly one of $a + b$ or $a - b$ is in $S$. Set $T \subset N_0$ with not so many elements is called charming if the largest number $k$ such that up to 3$^k | a$ is the same for each element $a \in T$. Prove that each beautiful set must be charming.

2016 India PRMO, 14

Tags: minimum , Sets , Subset

Find the minimum value of $m$ such that any $m$-element subset of the set of integers $\{1,2,...,2016\}$ contains at least two distinct numbers $a$ and $b$ which satisfy $|a - b|\le 3$.

1994 IMO Shortlist, 1

Tags: number theory , Extremal combinatorics , Perfect Squares , Subset , IMO Shortlist

$ M$ is a subset of $ \{1, 2, 3, \ldots, 15\}$ such that the product of any three distinct elements of $ M$ is not a square. Determine the maximum number of elements in $ M.$

2023 Korea - Final Round, 3

Tags: number theory , FKMO , Subset

Let $p$ be an odd prime. Let $A(n)$ be the number of subsets of $\{1,2,...,n\}$ such that the sum of elements of the subset is a multiple of $p$. Prove that if $2^{p-1}-1$ is not a multiple of $p^2$, there exists infinitely many positive integer $m$ for any integer $k$ that satisfies the following. (The sum of elements of the empty set is 0.) $$\frac{A(m)-k}{p}\in\mathbb{Z}$$

1994 Korea National Olympiad, Problem 2

Tags: Integer sequence , Subset , set , number theory

Given a set $S \subset N$ and a positive integer n, let $S\oplus \{n\} = \{s+n / s \in S\}$. The sequence $S_k$ of sets is defined inductively as follows: $S_1 = {1}$, $S_k=(S_{k-1} \oplus \{k\}) \cup \{2k-1\}$ for $k = 2,3,4, ...$ (a) Determine $N - \cup _{k=1}^{\infty} S_k$. (b) Find all $n$ for which $1994 \in S_n$.

1985 Spain Mathematical Olympiad, 2

Tags: Subset , Integers , algebra

Determine if there exists a subset $E$ of $Z \times Z$ with the properties: (i) $E$ is closed under addition, (ii) $E$ contains $(0,0),$ (iii) For every $(a,b) \ne (0,0), E$ contains exactly one of $(a,b)$ and $-(a,b)$. Remark: We define $(a,b)+(a',b') = (a+a',b+b')$ and $-(a,b) = (-a,-b)$.

2024 Indonesia TST, C

Tags: Tstst , combinatorics , Subset

Let $A$ be a set with $1000$ members and $\mathcal F =${$A_1,A_2,\ldots,A_n$} a family of subsets of A such that (a) Each element in $\mathcal F$ consists of 3 members (b) For every five elements in $\mathcal F$, the union of them all will have at least $12$ members Find the largest value of $n$

2025 Romania EGMO TST, P2

Tags: divisible , number theory , inequalities , Subset

Let $m$ and $n$ be positive integers with $m > n \ge 2$. Set $S =\{1,2,...,m\}$, and set $T = \{a_1,a_2,...,a_n\}$ is a subset of $S$ such that every element of $S$ is not divisible by any pair of distinct elements of $T$. Prove that $$\frac{1}{a_1}+\frac{1}{a_2}+ ...+ \frac{1}{a_n} < \frac{m+n}{m}$$

1991 All Soviet Union Mathematical Olympiad, 556

Tags: Subset , combinatorics , minimum

$X$ is a set with $100$ members. What is the smallest number of subsets of $X$ such that every pair of elements belongs to at least one subset and no subset has more than $50$ members? What is the smallest number if we also require that the union of any two subsets has at most $80$ members?

2022 Korea Winter Program Practice Test, 4

Tags: combinatorics , Subset , algebra

For a finite set $A$ of positive integers and its subset $B$, call $B$ a [i]half subset[/i] of $A$ when it satisfies the equation $\sum_{a\in A}a=2\sum_{b\in B}b$. For example, if $A=\{1,2,3\}$, then $\{1,2\}$ and $\{3\}$ are half subset of $A$. Determine all positive integers $n$ such that there exists a finite set $A$ which has exactly $n$ half subsets.

1954 Moscow Mathematical Olympiad, 286

Tags: number theory , combinatorics , Subset , Sum , 10-digit

Consider the set of all $10$-digit numbers expressible with the help of figures $1$ and $2$ only. Divide it into two subsets so that the sum of any two numbers of the same subset is a number which is written with not less than two $3$’s.

1989 All Soviet Union Mathematical Olympiad, 494

Tags: Sum of Squares , Subset , number theory , 5-digit , algebra

Show that the $120$ five digit numbers which are permutations of $12345$ can be divided into two sets with each set having the same sum of squares.

2001 Bosnia and Herzegovina Team Selection Test, 3

Tags: maximum , Subset , combinatorics

Find maximal value of positive integer $n$ such that there exists subset of $S=\{1,2,...,2001\}$ with $n$ elements, such that equation $y=2x$ does not have solutions in set $S \times S$

1987 Bundeswettbewerb Mathematik, 2

Tags: combinatorics , Subset , cardinality , number

Let $n$ be a positive integer and $M=\{1,2,\ldots, n\}.$ A subset $T\subset M$ is called [i]heavy[/i] if each of its elements is greater or equal than $|T|.$ Let $f(n)$ denote the number of heavy subsets of $M.$ Describe a method for finding $f(n)$ and use it to calculate $f(32).$

2004 Regional Olympiad - Republic of Srpska, 4

Tags: Sets , combinatorics , Division , disjoint , Subset

Set $S=\{1,2,...,n\}$ is firstly divided on $m$ disjoint nonempty subsets, and then on $m^2$ disjoint nonempty subsets. Prove that some $m$ elements of set $S$ were after first division in same set, and after the second division were in $m$ different sets

2015 Indonesia MO Shortlist, C7

Tags: combinatorics , Sum , Subset

Show that there is a subset of $A$ from $\{1,2, 3,... , 2014\}$ such that : (i) $|A| = 12$ (ii) for each coloring number in $A$ with red or white , we can always find some numbers colored in $A$ whose sum is $2015$.

2006 MOP Homework, 6

Tags: divisible , number theory , inequalities , Subset

Let $m$ and $n$ be positive integers with $m > n \ge 2$. Set $S =\{1,2,...,m\}$, and set $T = \{a_1,a_2,...,a_n\}$ is a subset of $S$ such that every element of $S$ is not divisible by any pair of distinct elements of $T$. Prove that $$\frac{1}{a_1}+\frac{1}{a_2}+ ...+ \frac{1}{a_n} < \frac{m+n}{m}$$

1999 Tournament Of Towns, 4

Tags: combinatorics , Subset , partition

Is it possible to divide the integers from $1$ to $100$ inclusive into $50$ pairs such that for $1\le k\le 50$, the difference between the two numbers in the $k$-th pair is $k$? (V Proizvolov)

2015 Ukraine Team Selection Test, 12

Tags: Sets , arithmetic sequence , Subset , algebra

For a given natural $n$, we consider the set $A\subset \{1,2, ..., n\}$, which consists of at least $\left[\frac{n+1}{2}\right]$ items. Prove that for $n \ge 2015$ the set $A$ contains a three-element arithmetic sequence.

2024 Indonesia TST, C

Tags: Tstst , combinatorics , Subset

Let $A$ be a set with $1000$ members and $\mathcal F =${$A_1,A_2,\ldots,A_n$} a family of subsets of A such that (a) Each element in $\mathcal F$ consists of 3 members (b) For every five elements in $\mathcal F$, the union of them all will have at least $12$ members Find the largest value of $n$

2002 Brazil National Olympiad, 1

Tags: number theory , Perfect Squares , perfect cube , Perfect Powers , Subset , Sum

Show that there is a set of $2002$ distinct positive integers such that the sum of one or more elements of the set is never a square, cube, or higher power.

1977 Chisinau City MO, 136

Tags: algebra , Subset , Subsets

We represent the number line $R$ as the union of two non-empty sets $A, B$ different from $R$. Prove that one of the sets $A, B$ does not have the following property: the difference of any elements of the set belongs to the same set.

2007 Dutch Mathematical Olympiad, 2

Tags: partition , combinatorics , Sets , Subset , Sum

Is it possible to partition the set $A = \{1, 2, 3, ... , 32, 33\}$ into eleven subsets that contain three integers each, such that for every one of these eleven subsets, one of the integers is equal to the sum of the other two? If so, give such a partition, if not, prove that such a partition cannot exist.