Found problems: 233
2018 Bosnia And Herzegovina - Regional Olympiad, 1
$a)$ Prove that for all positive integers $n \geq 3$ holds:
$$\binom{n}{1}+\binom{n}{2}+...+\binom{n}{n-1}=2^n-2$$ where $\binom{n}{k}$ , with integer $k$ such that $n \geq k \geq 0$, is binomial coefficent
$b)$ Let $n \geq 3$ be an odd positive integer. Prove that set $A=\left\{ \binom{n}{1},\binom{n}{2},...,\binom{n}{\frac{n-1}{2}} \right\}$ has odd number of odd numbers
2015 NIMO Summer Contest, 6
Let $S_0 = \varnothing$ denote the empty set, and define $S_n = \{ S_0, S_1, \dots, S_{n-1} \}$ for every positive integer $n$. Find the number of elements in the set
\[ (S_{10} \cap S_{20}) \cup (S_{30} \cap S_{40}). \]
[i] Proposed by Evan Chen [/i]
2011 VTRMC, Problem 4
Let $m,n$ be positive integers and let $[a]$ denote the residue class$\pmod{mn}$ of the integer $a$ (thus $\{[r]|r\text{ is an integer}\}$ has exactly $mn$ elements). Suppose the set $\{[ar]|r\text{ is an integer}\}$ has exactly $m$ elements. Prove that there is a positive integer $q$ such that $q$ is coprime to $mn$ and $[nq]=[a]$.
2015 Dutch Mathematical Olympiad, 1
We make groups of numbers. Each group consists of [i]fi ve[/i] distinct numbers. A number may occur in multiple groups. For any two groups, there are exactly four numbers that occur in both groups.
(a) Determine whether it is possible to make $2015$ groups.
(b) If all groups together must contain exactly [i]six [/i] distinct numbers, what is the greatest number of groups that you can make?
(c) If all groups together must contain exactly [i]seven [/i] distinct numbers, what is the greatest number of groups that you can make?
2015 Indonesia MO Shortlist, C2
Given $2n$ natural numbers, so that the average arithmetic of those $2n$ number is $2$. If all the number is not more than $2n$. Prove we can divide those $2n$ numbers into $2$ sets, so that the sum of each set to be the same.
2025 Romania Team Selection Tests, P4
Determine the sets $S{}$ of positive integers satisfying the following two conditions:
[list=a]
[*]For any positive integers $a, b, c{}$, if $ab + bc + ca{}$ is in $S$, then so are $a + b + c{}$ and $abc$; and
[*]The set $S{}$ contains an integer $N \geqslant 160$ such that $N-2$ is not divisible by $4$.
[/list]
[i]Bogdan Blaga, United Kingdom[/i]
2008 Korea Junior Math Olympiad, 8
There are $12$ members in a club. The members created some small groups, which satisfy the following:
- The small group consists of $3$ or $4$ people.
- Also, for two arbitrary members, there exists exactly one small group that has both members.
Prove that all members are in the same number of small groups.
2016 Dutch IMO TST, 4
Determine the number of sets $A = \{a_1,a_2,...,a_{1000}\}$ of positive integers satisfying $a_1 < a_2 <...< a_{1000} \le 2014$, for which we have that the set
$S = \{a_i + a_j | 1 \le i, j \le 1000$ with $i + j \in A\}$ is a subset of $A$.
2015 Cono Sur Olympiad, 6
Let $S = \{1, 2, 3, \ldots , 2046, 2047, 2048\}$. Two subsets $A$ and $B$ of $S$ are said to be [i]friends[/i] if the following conditions are true:
[list]
[*] They do not share any elements.
[*] They both have the same number of elements.
[*] The product of all elements from $A$ equals the product of all elements from $B$.
[/list]
Prove that there are two subsets of $S$ that are [i]friends[/i] such that each one of them contains at least $738$ elements.
2021 Junior Balkan Team Selection Tests - Romania, P4
Let $M$ be a set of $13$ positive integers with the property that $\forall \ m\in M, \ 100\leq m\leq 999$. Prove that there exists a subset $S\subset M$ and a combination of arithmetic operations (addition, subtraction, multiplication, division – without using parentheses) between the elements of $S$, such that the value of the resulting expression is a rational number in the interval $(3,4)$.
1997 Bosnia and Herzegovina Team Selection Test, 5
$a)$ Prove that for all positive integers $n$ exists a set $M_n$ of positive integers with exactly $n$ elements and:
$i)$ Arithmetic mean of arbitrary non-empty subset of $M_n$ is integer
$ii)$ Geometric mean of arbitrary non-empty subset of $M_n$ is integer
$iii)$ Both arithmetic mean and geometry mean of arbitrary non-empty subset of $M_n$ is integer
$b)$ Does there exist infinite set $M$ of positive integers such that arithmetic mean of arbitrary non-empty subset of $M$ is integer
2013 VJIMC, Problem 3
Let $S$ be a finite set of integers. Prove that there exists a number $c$ depending on $S$ such that for each non-constant polynomial $f$ with integer coefficients the number of integers $k$ satisfying $f(k)\in S$ does not exceed $\max(\deg f,c)$.
2010 Contests, 1
We write $\{a,b,c\}$ for the set of three different positive integers $a, b$, and $c$. By choosing some or all of the numbers a, b and c, we can form seven nonempty subsets of $\{a,b,c\}$. We can then calculate the sum of the elements of each subset. For example, for the set $\{4,7,42\}$ we will find sums of $4, 7, 42,11, 46, 49$, and $53$ for its seven subsets. Since $7, 11$, and $53$ are prime, the set $\{4,7,42\}$ has exactly three subsets whose sums are prime. (Recall that prime numbers are numbers with exactly two different factors, $1$ and themselves. In particular, the number $1$ is not prime.)
What is the largest possible number of subsets with prime sums that a set of three different positive integers can have? Give an example of a set $\{a,b,c\}$ that has that number of subsets with prime sums, and explain why no other three-element set could have more.
2016 IMC, 4
Let $n\ge k$ be positive integers, and let $\mathcal{F}$ be a family of finite sets with the following properties:
(i) $\mathcal{F}$ contains at least $\binom{n}{k}+1$ distinct sets containing exactly $k$ elements;
(ii) for any two sets $A, B\in \mathcal{F}$, their union $A\cup B$ also belongs to $\mathcal{F}$.
Prove that $\mathcal{F}$ contains at least three sets with at least $n$ elements.
(Proposed by Fedor Petrov, St. Petersburg State University)
2010 Peru IMO TST, 5
Let $\Bbb{N}$ be the set of positive integers. For each subset $\mathcal{X}$ of $\Bbb{N}$ we define the set $\Delta(\mathcal{X})$ as the set of all numbers $| m - n |,$ where $m$ and $n$ are elements of $\mathcal{X}$, ie: $$\Delta (\mathcal{X}) = \{ |m-n| \ | \ m, n \in \mathcal{X} \}$$ Let $\mathcal A$ and $\mathcal B$ be two infinite, disjoint sets whose union is $\Bbb{N.}$
a) Prove that the set $\Delta (\mathcal A) \cap \Delta (\mathcal B)$ has infinitely many elements.
b) Prove that there exists an infinite subset $\mathcal C$ of $\Bbb{N}$ such that $\Delta (\mathcal C)$ is a subset of $\Delta (\mathcal A) \cap \Delta (\mathcal B).$
2018 Danube Mathematical Competition, 4
Let $M$ be the set of positive odd integers.
For every positive integer $n$, denote $A(n)$ the number of the subsets of $M$ whose sum of elements equals $n$.
For instance, $A(9) = 2$, because there are exactly two subsets of $M$ with the sum of their elements equal to $9$: $\{9\}$ and $\{1, 3, 5\}$.
a) Prove that $A(n) \le A(n + 1)$ for every integer $n \ge 2$.
b) Find all the integers $n \ge 2$ such that $A(n) = A(n + 1)$
2017 Canada National Olympiad, 3
Define $S_n$ as the set ${1,2,\cdots,n}$. A non-empty subset $T_n$ of $S_n$ is called $balanced$ if the average of the elements of $T_n$ is equal to the median of $T_n$. Prove that, for all $n$, the number of balanced subsets $T_n$ is odd.
2020 China Northern MO, BP5
It is known that subsets $A_1,A_2, \cdots , A_n$ of set $I=\{1,2,\cdots ,101\}$ satisfy the following condition
$$\text{For any } i,j \text{ } (1 \leq i < j \leq n) \text{, there exists } a,b \in A_i \cap A_j \text{ so that } (a,b)=1$$
Determine the maximum positive integer $n$.
*$(a,b)$ means $\gcd (a,b)$
2012 IFYM, Sozopol, 3
Let $A$ be a set of natural numbers, for which for $\forall n\in \mathbb{N}$ exactly one of the numbers $n$, $2n$, and $3n$ is an element of $A$. If $2\in A$, show whether $13824\in A$.
2024 Romanian Master of Mathematics, 3
Given a positive integer $n$, a collection $\mathcal{S}$ of $n-2$ unordered triples of integers in $\{1,2,\ldots,n\}$ is [i]$n$-admissible[/i] if for each $1 \leq k \leq n - 2$ and each choice of $k$ distinct $A_1, A_2, \ldots, A_k \in \mathcal{S}$ we have $$ \left|A_1 \cup A_2 \cup \cdots A_k \right| \geq k+2.$$
Is it true that for all $n > 3$ and for each $n$-admissible collection $\mathcal{S}$, there exist pairwise distinct points $P_1, \ldots , P_n$ in the plane such that the angles of the triangle $P_iP_jP_k$ are all less than $61^{\circ}$ for any triple $\{i, j, k\}$ in $\mathcal{S}$?
[i]Ivan Frolov, Russia[/i]
2021 Junior Balkan Team Selection Tests - Romania, P3
Let $p,q$ be positive integers. For any $a,b\in\mathbb{R}$ define the sets $$P(a)=\bigg\{a_n=a \ + \ n \ \cdot \ \frac{1}{p} : n\in\mathbb{N}\bigg\}\text{ and }Q(b)=\bigg\{b_n=b \ + \ n \ \cdot \ \frac{1}{q} : n\in\mathbb{N}\bigg\}.$$
The [i]distance[/i] between $P(a)$ and $Q(b)$ is the minimum value of $|x-y|$ as $x\in P(a), y\in Q(b)$. Find the maximum value of the distance between $P(a)$ and $Q(b)$ as $a,b\in\mathbb{R}$.
1983 Austrian-Polish Competition, 4
The set $N$ has been partitioned into two sets A and $B$. Show that for every $n \in N$ there exist distinct integers $a, b > n$ such that $a, b, a + b$ either all belong to $A$ or all belong to $B$.
1996 Tuymaada Olympiad, 2
Given a finite set of real numbers $A$, not containing $0$ and $1$ and possessing the property: if the number a belongs to $A$, then numbers $\frac{1}{a}$ and $1-a$ also belong to $A$. How many numbers are in the set $A$?
2013 German National Olympiad, 5
Five people form several commissions to prepare a competition. Here any commission must be nonempty and any two commissions cannot contain the same members. Moreover, any two commissions have at least one common member.
There are already $14$ commissions. Prove that at least one additional commission can be formed.
2015 Ukraine Team Selection Test, 12
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.