Olympiad problems

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 $2015$.

2012 EGMO, 2

Tags: combinatorics , combinatorial number theory , number theory

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]

2001 Baltic Way, 17

Tags: number theory , combinatorial number theory

Let $n$ be a positive integer. Prove that at least $2^{n-1}+n$ numbers can be chosen from the set $\{1, 2, 3,\ldots ,2^n\}$ such that for any two different chosen numbers $x$ and $y$, $x+y$ is not a divisor of $x\cdot y$.

2022 Bulgarian Spring Math Competition, Problem 11.4

Tags: combinatorial number theory , subset , number theory , combinatorics , probabilistic method

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 $A$.

1994 IMO Shortlist, 3

Tags: modular arithmetic , prime number , combinatorial number theory , combinatorics , number theory

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$.

2001 IMO Shortlist, 4

Tags: combinatorics , combinatorial number theory , partition , coloring , greedy algorithm

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.

1994 IMO, 6

Tags: modular arithmetic , number theory , prime number , combinatorial number theory , combinatorics

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$.

2018 EGMO, 6

Tags: number theory , combinatorial number theory

[list=a] [*]Prove that for every real number $t$ such that $0 < t < \tfrac{1}{2}$ there exists a positive integer $n$ with the following property: for every set $S$ of $n$ positive integers there exist two different elements $x$ and $y$ of $S$, and a non-negative integer $m$ (i.e. $m \ge 0 $), such that \[ |x-my|\leq ty.\] [*]Determine whether for every real number $t$ such that $0 < t < \tfrac{1}{2} $ there exists an infinite set $S$ of positive integers such that \[|x-my| > ty\] for every pair of different elements $x$ and $y$ of $S$ and every positive integer $m$ (i.e. $m > 0$).

2014 Contests, 3

Tags: quadratic , number theory , combinatorial number theory , divisor

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 $a + b = n$.

2019 South East Mathematical Olympiad, 5

Tags: combinatorics , combinatorial number theory

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.