Found problems: 233
2019 Taiwan TST Round 2, 1
Given any set $S$ of positive integers, show that at least one of the following two assertions holds:
(1) There exist distinct finite subsets $F$ and $G$ of $S$ such that $\sum_{x\in F}1/x=\sum_{x\in G}1/x$;
(2) There exists a positive rational number $r<1$ such that $\sum_{x\in F}1/x\neq r$ for all finite subsets $F$ of $S$.
2017 Bosnia and Herzegovina Junior BMO TST, 2
Let $A$ be a set $A=\{1,2,3,...,2017\}$. Subset $S$ of set $A$ is [i]good [/i] if for all $x\in A$ sum of remaining elements of set $S$ has same last digit as $x$. Prove that [i]good[/i] subset with $405$ elements is not possible.
2017 AMC 10, 12
Let $S$ be the set of points $(x,y)$ in the coordinate plane such that two of the three quantities $3$, $x+2$, and $y-4$ are equal and the third of the three quantities is no greater than this common value. Which of the following is a correct description of $S$?
$\textbf{(A) } \text{a single point} \qquad \textbf{(B) } \text{two intersecting lines} \\ \\ \textbf{(C) } \text{three lines whose pairwise intersections are three distinct points} \\ \\ \textbf{(D) } \text{a triangle} \qquad \textbf{(E) } \text{three rays with a common endpoint}$
2015 Bosnia And Herzegovina - Regional Olympiad, 4
Alice and Mary were searching attic and found scale and box with weights. When they sorted weights by mass, they found out there exist $5$ different groups of weights. Playing with the scale and weights, they discovered that if they put any two weights on the left side of scale, they can find other two weights and put on to the right side of scale so scale is in balance. Find the minimal number of weights in the box
2020 Israel Olympic Revenge, P2
Let $A, B\subset \mathbb{Z}$ be two sets of integers. We say that $A,B$ are [u]mutually repulsive[/u] if there exist positive integers $m,n$ and two sequences of integers $\alpha_1, \alpha_2, \dots, \alpha_n$ and $\beta_1, \beta_2, \dots, \beta_m$, for which there is a [b]unique[/b] integer $x$ such that the number of its appearances in the sequence of sets $A+\alpha_1, A+\alpha_2, \dots, A+\alpha_n$ is [u]different[/u] than the number of its appearances in the sequence of sets $B+\beta_1, \dots, B+\beta_m$.
For a given quadruple of positive integers $(n_1,d_1, n_2, d_2)$, determine whether the sets
\[A=\{d_1, 2d_1, \dots, n_1d_1\}\]
\[B=\{d_2, 2d_2, \dots, n_2d_2\}\]
are mutually repulsive.
For a set $X\subset \mathbb{Z}$ and $c\in \mathbb{Z}$, we define $X+c=\{x+c\mid x\in X\}$.
2019 China Western Mathematical Olympiad, 8
We call a set $S$ a [i]good[/i] set if $S=\{x,2x,3x\}(x\neq 0).$ For a given integer $n(n\geq 3),$ determine the largest possible number of the [i]good[/i] subsets of a set containing $n$ positive integers.
2018 Brazil Undergrad MO, 5
Consider the set $A = \left\{\frac{j}{4}+\frac{100}{j}|j=1,2,3,..\right\} $ What is the smallest number that belongs to the $ A $ set?
2013 Junior Balkan Team Selection Tests - Romania, 5
a) Prove that for every positive integer n, there exist $a, b \in R - Z$ such that
the set $A_n = \{a - b, a^2 - b^2, a^3 - b^3,...,a^n - b^n\}$ contains only positive integers.
b) Let $a$ and $b$ be two real numbers such that the set $A = \{a^k - b^k | k \in N*\}$ contains only positive integers.
Prove that $a$ and $b$ are integers.
2021-IMOC, N5
Find all sets $S$ of positive integers that satisfy all of the following.
$1.$ If $a,b$ are two not necessarily distinct elements in $S$, then $\gcd(a,b)$, $ab$ are also in $S$.
$2.$ If $m,n$ are two positive integers with $n\nmid m$, then there exists an element $s$ in $S$ such that $m^2\mid s$ and $n^2\nmid s$.
$3.$ For any odd prime $p$, the set formed by moduloing all elements in $S$ by $p$ has size exactly $\frac{p+1}2$.
2015 Greece JBMO TST, 4
Pupils of a school are divided into $112$ groups, of $11$ members each.
Any two groups have exactly one common pupil. Prove that:
a) there is a pupil that belongs to at least $12$ groups.
b) there is a pupil that belongs to all the groups.
2022 AMC 10, 14
Suppose that $S$ is a subset of $\{1, 2, 3,...,25\}$ such that the sum of any two (not necessarily distinct) elements of $S$ is never an element of $S$. What is the maximum number of elements $S$ may contain?
$\textbf{(A) }12 \qquad \textbf{(B) }13 \qquad \textbf{(C) }14 \qquad \textbf{(D) }15 \qquad \textbf{(E) }16$
2014 JBMO TST - Macedonia, 5
Prove that there exist infinitely many pairwisely disjoint sets $A(1), A(2),...,A(2014)$ which are not empty, whose union is the set of positive integers and which satisfy the following condition:
For arbitrary positive integers $a$ and $b$, at least two of the numbers $a$, $b$ and $GCD(a,b)$ belong to one of the sets $A(1), A(2),...,A(2014)$.
2015 Mathematical Talent Reward Programme, MCQ: P 11
$S=\{1,2, \ldots, 6\} .$ Then find out the number of unordered pairs of $(A, B)$ such that $A, B \subseteq S$ and $A \cap B=\phi$
[list=1]
[*] 360
[*] 364
[*] 365
[*] 366
[/list]
2015 Korea Junior Math Olympiad, 8
A positive integer $n$ is given. If there exist sets $F_1, F_2, \cdots F_m$ satisfying the following, prove that $m \le n$.
(For sets $A, B$, $|A|$ is the number of elements in $A$. $A-B$ is the set of elements that are in $A$ but not $B$)
(i): For all $1 \le i \le m$, $F_i \subseteq \{1,2,\cdots n\}$
(ii): $|F_1| \le |F_2| \le \cdots \le |F_m|$
(iii): For all $1 \le i < j \le m$, $|F_i-F_j|=1$.
1997 Estonia Team Selection Test, 1
$(a)$ Is it possible to partition the segment $[0,1]$ into two sets $A$ and $B$ and to define a continuous function $f$ such that for every $x\in A \ f(x)$ is in $B$, and for every $x\in B \ f(x)$ is in $A$?
$(b)$ The same question with $[0,1]$ replaced by $[0,1).$
2025 Philippine MO, P1
The set $S$ is a subset of $\{1, 2, \dots, 2025\}$ such that no two elements of $S$ differ by $2$ or by $7$. What is the largest number of elements that $S$ can have?
2020 Iran MO (3rd Round), 4
What is the maximum number of subsets of size $5$, taken from the set $A=\{1,2,3,...,20\}$ such that any $2$ of them share exactly $1$ element.
2006 Thailand Mathematical Olympiad, 16
Find the number of triples of sets $(A, B, C)$ such that $A \cup B \cup C = \{1, 2, 3, ... , 2549\}$
2024 AMC 10, 20
Let $S$ be a subset of $\{1, 2, 3, \dots, 2024\}$ such that the following two conditions hold:
- If $x$ and $y$ are distinct elements of $S$, then $|x-y| > 2$
- If $x$ and $y$ are distinct odd elements of $S$, then $|x-y| > 6$.
What is the maximum possible number of elements in $S$?
$
\textbf{(A) }436 \qquad
\textbf{(B) }506 \qquad
\textbf{(C) }608 \qquad
\textbf{(D) }654 \qquad
\textbf{(E) }675 \qquad
$
2001 VJIMC, Problem 1
Let $A$ be a set of positive integers such that for any $x,y\in A$,
$$x>y\implies x-y\ge\frac{xy}{25}.$$Find the maximal possible number of elements of the set $A$.
2024 239 Open Mathematical Olympiad, 1
We will say that two sets of distinct numbers are $\textit{linked}$ to each other if between any two numbers of each set lies at least one number of the other set. Is it possible to fill the cells of a $100 \times 200$ rectangle with distinct numbers so that any two rows of the rectangle are linked to one another, and any two columns of the rectangle are linked to one another?
2016 USAJMO, 4
Find, with proof, the least integer $N$ such that if any $2016$ elements are removed from the set ${1, 2,...,N}$, one can still find $2016$ distinct numbers among the remaining elements with sum $N$.
2014 IFYM, Sozopol, 6
Let $A$ and $B$ be two non-infinite sets of natural numbers, each of which contains at least 3 elements. Two numbers $a\in A$ and $b\in B$ are called [i]"harmonious"[/i], if they are not coprime. It is known that each element from $A$ is not [i]harmonious[/i] with at least one element from $B$ and each element from $B$ is harmonious with at least one from $A$. Prove that there exist $a_1,a_2\in A$ and $b_1,b_2\in B$ such that $(a_1,b_1)$ and $(a_2,b_2)$ are [i]harmonious[/i] but $(a_1,b_2)$ and $(a_2,b_1)$ are not.
2019 Peru IMO TST, 6
Let $p$ and $q$ two positive integers. Determine the greatest value of $n$ for which there exists sets $A_1,\ A_2,\ldots,\ A_n$ and $B_1,\ B_2,\ldots,\ B_n$ such that:
[LIST]
[*] The sets $A_1,\ A_2,\ldots,\ A_n$ have $p$ elements each one. [/*]
[*] The sets $B_1,\ B_2,\ldots,\ B_n$ have $q$ elements each one. [/*]
[*] For all $1\leq i,\ j \leq n$, sets $A_i$ and $B_j$ are disjoint if and only if $i=j$.
[/LIST]
1989 Romania Team Selection Test, 4
A family of finite sets $\left\{ A_{1},A_{2},.......,A_{m}\right\} $is called [i]equipartitionable [/i] if there is a function $\varphi:\cup_{i=1}^{m}$$\rightarrow\left\{ -1,1\right\} $ such that $\sum_{x\in A_{i}}\varphi\left(x\right)=0$ for every $i=1,.....,m.$ Let $f\left(n\right)$ denote the smallest possible number of $n$-element sets which form a non-equipartitionable family. Prove that
a) $f(4k +2) = 3$ for each nonnegative integer $k$,
b) $f\left(2n\right)\leq1+m d\left(n\right)$, where $m d\left(n\right)$ denotes the least positive non-divisor of $n.$