Found problems: 35
2018 Iran Team Selection Test, 1
Let $A_1, A_2, ... , A_k$ be the subsets of $\left\{1,2,3,...,n\right\}$ such that for all $1\leq i,j\leq k$:$A_i\cap A_j \neq \varnothing$. Prove that there are $n$ distinct positive integers $x_1,x_2,...,x_n$ such that for each $1\leq j\leq k$:
$$lcm_{i \in A_j}\left\{x_i\right\}>lcm_{i \notin A_j}\left\{x_i\right\}$$
[i]Proposed by Morteza Saghafian, Mahyar Sefidgaran[/i]
2016 IMAR Test, 1
Fix an integer $n \ge 3$ and let $a_0 = n$. Does there exist a permutation $a_1, a_2,..., a_{n-1}$ of the fi rst $n-1$ positive integers such that $\Sigma_{j=0}^{k-1} a_j$ is divisible by $a_k$ for all indices $k < n$?
2016 China Team Selection Test, 6
Let $m,n$ be naturals satisfying $n \geq m \geq 2$ and let $S$ be a set consisting of $n$ naturals. Prove that $S$ has at least $2^{n-m+1}$ distinct subsets, each whose sum is divisible by $m$. (The zero set counts as a subset).
2022 Bulgarian Spring Math Competition, Problem 11.4
Let $n \geq 2$ be a positive integer. The set $M$ consists of $2n^2-3n+2$ positive rational numbers. Prove that there exists a subset $A$ of $M$ with $n$ elements with the following property: $\forall$ $2 \leq k \leq n$ the sum of any $k$ (not necessarily distinct) numbers from $A$ is not in $A$.
2001 IMO Shortlist, 4
A set of three nonnegative integers $\{x,y,z\}$ with $x < y < z$ is called [i]historic[/i] if $\{z-y,y-x\} = \{1776,2001\}$. Show that the set of all nonnegative integers can be written as the union of pairwise disjoint historic sets.
2018 Iran Team Selection Test, 1
Let $A_1, A_2, ... , A_k$ be the subsets of $\left\{1,2,3,...,n\right\}$ such that for all $1\leq i,j\leq k$:$A_i\cap A_j \neq \varnothing$. Prove that there are $n$ distinct positive integers $x_1,x_2,...,x_n$ such that for each $1\leq j\leq k$:
$$lcm_{i \in A_j}\left\{x_i\right\}>lcm_{i \notin A_j}\left\{x_i\right\}$$
[i]Proposed by Morteza Saghafian, Mahyar Sefidgaran[/i]
2012 EGMO, 2
Let $n$ be a positive integer. Find the greatest possible integer $m$, in terms of $n$, with the following property: a table with $m$ rows and $n$ columns can be filled with real numbers in such a manner that for any two different rows $\left[ {{a_1},{a_2},\ldots,{a_n}}\right]$ and $\left[ {{b_1},{b_2},\ldots,{b_n}} \right]$ the following holds: \[\max\left( {\left| {{a_1} - {b_1}} \right|,\left| {{a_2} - {b_2}} \right|,...,\left| {{a_n} - {b_n}} \right|} \right) = 1\]
[i]Poland (Tomasz Kobos)[/i]
2014 EGMO, 3
We denote the number of positive divisors of a positive integer $m$ by $d(m)$ and the number of distinct prime divisors of $m$ by $\omega(m)$. Let $k$ be a positive integer. Prove that there exist infinitely many positive integers $n$ such that $\omega(n) = k$ and $d(n)$ does not divide $d(a^2+b^2)$ for any positive integers $a, b$ satisfying $a + b = n$.
2018 Czech and Slovak Olympiad III A, 6
Determine the least positive integer $n$ with the following property – for every 3-coloring of numbers $1,2,\ldots,n$ there are two (different) numbers $a,b$ of the same color such that $|a-b|$ is a perfect square.
1994 IMO, 6
Show that there exists a set $ A$ of positive integers with the following property: for any infinite set $ S$ of primes, there exist [i]two[/i] positive integers $ m$ in $ A$ and $ n$ not in $ A$, each of which is a product of $ k$ distinct elements of $ S$ for some $ k \geq 2$.
2020/2021 Tournament of Towns, P7
Let $p{}$ and $q{}$ be two coprime positive integers. A frog hops along the integer line so that on every hop it moves either $p{}$ units to the right or $q{}$ units to the left. Eventually, the frog returns to the initial point. Prove that for every positive integer $d{}$ with $d < p + q$ there are two numbers visited by the frog which differ just by $d{}$.
[i]Nikolay Belukhov[/i]
2022 Nigerian MO round 3, Problem 3
A unit square is removed from the corner of an $n \times n$ grid, where $n \geq 2$. Prove that the remainder can be covered by copies of the figures of $3$ or $5$ unit squares depicted in the drawing below.
[asy]
import geometry;
draw((-1.5,0)--(-3.5,0)--(-3.5,2)--(-2.5,2)--(-2.5,1)--(-1.5,1)--cycle);
draw((-3.5,1)--(-2.5,1)--(-2.5,0));
draw((0.5,0)--(0.5,3)--(1.5,3)--(1.5,1)--(3.5,1)--(3.5,0)--cycle);
draw((1.5,0)--(1.5,1));
draw((2.5,0)--(2.5,1));
draw((0.5,1)--(1.5,1));
draw((0.5,2)--(1.5,2));
[/asy]
[b]Note:[/b] Every square must be covered once and figures must not go over the bounds of the grid.
2019 Czech and Slovak Olympiad III A, 6
Assume we can fill a table $n\times n$ with all numbers $1,2,\ldots,n^2-1,n^2$ in such way that arithmetic means of numbers in every row and every column is an integer. Determine all such positive integers $n$.
2016 Bosnia And Herzegovina - Regional Olympiad, 2
Find all elements $n \in A = \{2,3,...,2016\} \subset \mathbb{N}$ such that:
every number $m \in A$ smaller than $n$, and coprime with $n$, must be a prime number
2015 PAMO, Problem 3
Let $a_1,a_2,...,a_{11}$ be integers. Prove that there are numbers $b_1,b_2,...,b_{11}$, each $b_i$ equal $-1,0$ or $1$, but not all being $0$, such that the number
$$N=a_1b_1+a_2b_2+...+a_{11}b_{11}$$
is divisible by $2015$.
2007 IMO Shortlist, 7
For a prime $ p$ and a given integer $ n$ let $ \nu_p(n)$ denote the exponent of $ p$ in the prime factorisation of $ n!$. Given $ d \in \mathbb{N}$ and $ \{p_1,p_2,\ldots,p_k\}$ a set of $ k$ primes, show that there are infinitely many positive integers $ n$ such that $ d\mid \nu_{p_i}(n)$ for all $ 1 \leq i \leq k$.
[i]Author: Tejaswi Navilarekkallu, India[/i]
2016 Bosnia And Herzegovina - Regional Olympiad, 4
Let $A$ be a set of $65$ integers with pairwise different remainders modulo $2016$. Prove that exists a subset $B=\{a,b,c,d\}$ of set $A$ such that $a+b-c-d$ is divisible with $2016$
2019 South East Mathematical Olympiad, 5
Let $S=\{1928,1929,1930,\cdots,1949\}.$ We call one of $S$’s subset $M$ is a [i]red[/i] subset, if the sum of any two different elements of $M$ isn’t divided by $4.$ Let $x,y$ be the number of the [i]red[/i] subsets of $S$ with $4$ and $5$ elements,respectively. Determine which of $x,y$ is greater and prove that.
2015 Azerbaijan IMO TST, 1
We say that $A$$=${$a_1,a_2,a_3\cdots a_n$} consisting $n>2$ distinct positive integers is $good$ if for every $i=1,2,3\cdots n$ the number ${a_i}^{2015}$ is divisible by the product of all numbers in $A$ except $a_i$. Find all integers $n>2$ such that exists a $good$ set consisting of $n$ positive integers.
2013 EGMO, 3
Let $n$ be a positive integer.
(a) Prove that there exists a set $S$ of $6n$ pairwise different positive integers, such that the least common multiple of any two elements of $S$ is no larger than $32n^2$.
(b) Prove that every set $T$ of $6n$ pairwise different positive integers contains two elements the least common multiple of which is larger than $9n^2$.
2021 Romanian Master of Mathematics Shortlist, N2
We call a set of positive integers [i]suitable [/i] if none of its elements is coprime to the sum of all
elements of that set. Given a real number $\varepsilon \in (0,1)$, prove that, for all large enough positive
integers $N$, there exists a suitable set of size at least $\varepsilon N$, each element of which is at most $N$.
2019 Iran Team Selection Test, 2
Hesam chose $10$ distinct positive integers and he gave all pairwise $\gcd$'s and pairwise ${\text lcm}$'s (a total of $90$ numbers) to Masoud. Can Masoud always find the first $10$ numbers, just by knowing these $90$ numbers?
[i]Proposed by Morteza Saghafian [/i]
Kvant 2021, M2653
Let $p{}$ and $q{}$ be two coprime positive integers. A frog hops along the integer line so that on every hop it moves either $p{}$ units to the right or $q{}$ units to the left. Eventually, the frog returns to the initial point. Prove that for every positive integer $d{}$ with $d < p + q$ there are two numbers visited by the frog which differ just by $d{}$.
[i]Nikolay Belukhov[/i]
2000 IMO Shortlist, 6
A nonempty set $ A$ of real numbers is called a $ B_3$-set if the conditions $ a_1, a_2, a_3, a_4, a_5, a_6 \in A$ and $ a_1 \plus{} a_2 \plus{} a_3 \equal{} a_4 \plus{} a_5 \plus{} a_6$ imply that the sequences $ (a_1, a_2, a_3)$ and $ (a_4, a_5, a_6)$ are identical up to a permutation. Let $A = \{a_0 = 0 < a_1 < a_2 < \cdots \}$, $B = \{b_0 = 0 < b_1 < b_2 < \cdots \}$ be infinite sequences of real numbers with $ D(A) \equal{} D(B),$ where, for a set $ X$ of real numbers, $ D(X)$ denotes the difference set $ \{|x\minus{}y|\mid x, y \in X \}.$ Prove that if $ A$ is a $ B_3$-set, then $ A \equal{} B.$
1989 Czech And Slovak Olympiad IIIA, 6
Consider a finite sequence $a_1, a_2,...,a_n$ whose terms are natural numbers at most equal to $n$. Determine the maximum number of terms of such a sequence, if you know that every two of its neighboring terms are different and at the same time there is no quartet of terms in it such that $a_p = a_r \ne a_q = a_s$ for $p < q < r < s$.