This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 116

2015 Belarus Team Selection Test, 1
Tags: frobenius , additive combinatorics , additive number theory , additive representation , induction , number theory
Let $n \ge 2$ be an integer, and let $A_n$ be the set \[A_n = \{2^n - 2^k\mid k \in \mathbb{Z},\, 0 \le k < n\}.\] Determine the largest positive integer that cannot be written as the sum of one or more (not necessarily distinct) elements of $A_n$ . [i]Proposed by Serbia[/i]

1993 IMO Shortlist, 2
Tags: combinatorics , additive combinatorics , additive number theory
Let $n,k \in \mathbb{Z}^{+}$ with $k \leq n$ and let $S$ be a set containing $n$ distinct real numbers. Let $T$ be a set of all real numbers of the form $x_1 + x_2 + \ldots + x_k$ where $x_1, x_2, \ldots, x_k$ are distinct elements of $S.$ Prove that $T$ contains at least $k(n-k)+1$ distinct elements.

PEN P Problems, 25
Tags: number theory , additive number theory , relatively prime
Let $a$ and $b$ be positive integers with $\gcd(a, b)=1$. Show that every integer greater than $ab-a-b$ can be expressed in the form $ax+by$, where $x, y \in \mathbb{N}_{0}$.

1992 IMO Shortlist, 15
Tags: perfect power , additive combinatorics , additive number theory , number theory
Does there exist a set $ M$ with the following properties? [i](i)[/i] The set $ M$ consists of 1992 natural numbers. [i](ii)[/i] Every element in $ M$ and the sum of any number of elements have the form $ m^k$ $ (m, k \in \mathbb{N}, k \geq 2).$

2002 India IMO Training Camp, 16
Tags: additive number theory , sum , number theory , modular arithmetic
Is it possible to find $100$ positive integers not exceeding $25,000$, such that all pairwise sums of them are different?

1969 IMO Shortlist, 13
Tags: number theory , modular arithmetic , divisibility , additive number theory
$(CZS 2)$ Let $p$ be a prime odd number. Is it possible to find $p-1$ natural numbers $n + 1, n + 2, . . . , n + p -1$ such that the sum of the squares of these numbers is divisible by the sum of these numbers?

1977 IMO Shortlist, 3
Tags: additive number theory , remainder , divisibility , number theory
Let $a,b$ be two natural numbers. When we divide $a^2+b^2$ by $a+b$, we the the remainder $r$ and the quotient $q.$ Determine all pairs $(a, b)$ for which $q^2 + r = 1977.$

2012 ELMO Shortlist, 3
Tags: number theory , additive number theory , extremal combinatorics , perfect power
Let $s(k)$ be the number of ways to express $k$ as the sum of distinct $2012^{th}$ powers, where order does not matter. Show that for every real number $c$ there exists an integer $n$ such that $s(n)>cn$. [i]Alex Zhu.[/i]

1993 IMO Shortlist, 4
Tags: subset , additive number theory , divisibility , induction , number theory
Show that for any finite set $S$ of distinct positive integers, we can find a set $T \supseteq S$ such that every member of $T$ divides the sum of all the members of $T$. [b]Original Statement:[/b] A finite set of (distinct) positive integers is called a [b]DS-set[/b] if each of the integers divides the sum of them all. Prove that every finite set of positive integers is a subset of some [b]DS-set[/b].

PEN P Problems, 1
Tags: additive number theory
Show that any integer can be expressed as a sum of two squares and a cube.

1983 IMO, 3
Tags: frobenius , additive number theory , additive combinatorics , number theory , modular arithmetic
Let $a,b$ and $c$ be positive integers, no two of which have a common divisor greater than $1$. Show that $2abc-ab-bc-ca$ is the largest integer which cannot be expressed in the form $xbc+yca+zab$, where $x,y,z$ are non-negative integers.

2015 Peru IMO TST, 11
Tags: frobenius , additive combinatorics , additive number theory , additive representation , number theory , induction
Let $n \ge 2$ be an integer, and let $A_n$ be the set \[A_n = \{2^n - 2^k\mid k \in \mathbb{Z},\, 0 \le k < n\}.\] Determine the largest positive integer that cannot be written as the sum of one or more (not necessarily distinct) elements of $A_n$ . [i]Proposed by Serbia[/i]

1997 Pre-Preparation Course Examination, 1
Tags: combinatorics , additive combinatorics , additive number theory , extremal combinatorics , subset
Let $ k,m,n$ be integers such that $ 1 < n \leq m \minus{} 1 \leq k.$ Determine the maximum size of a subset $ S$ of the set $ \{1,2,3, \ldots, k\minus{}1,k\}$ such that no $ n$ distinct elements of $ S$ add up to $ m.$

1992 IMO Longlists, 45
Tags: function , additive number theory , additive combinatorics , counting , number theory
Let $n$ be a positive integer. Prove that the number of ways to express $n$ as a sum of distinct positive integers (up to order) and the number of ways to express $n$ as a sum of odd positive integers (up to order) are the same.

1977 Germany Team Selection Test, 3
Tags: sequence , additive number theory , modular arithmetic , number theory
Let $a_{1}, \ldots, a_{n}$ be an infinite sequence of strictly positive integers, so that $a_{k} < a_{k+1}$ for any $k.$ Prove that there exists an infinity of terms $ a_{m},$ which can be written like $a_m = x \cdot a_p + y \cdot a_q$ with $x,y$ strictly positive integers and $p \neq q.$

PEN P Problems, 32
Tags: additive number theory
A composite positive integer is a product $ab$ with $a$ and $b$ not necessarily distinct integers in $\{2,3,4,\dots\}$. Show that every composite positive integer is expressible as $xy+xz+yz+1$, with $x,y,z$ positive integers.

PEN P Problems, 8
Tags: algorithm , additive number theory
Prove that any positive integer can be represented as an aggregate of different powers of $3$, the terms in the aggregate being combined by the signs $+$ and $-$ appropriately chosen.

PEN P Problems, 6
Tags: pigeonhole principle , additive number theory , floor function , inequalities , function
Show that every integer greater than $1$ can be written as a sum of two square-free integers.

1969 IMO Longlists, 13
Tags: number theory , divisibility , additive number theory , modular arithmetic
$(CZS 2)$ Let $p$ be a prime odd number. Is it possible to find $p-1$ natural numbers $n + 1, n + 2, . . . , n + p -1$ such that the sum of the squares of these numbers is divisible by the sum of these numbers?

PEN P Problems, 34
Tags: additive number theory
If $n$ is a positive integer which can be expressed in the form $n=a^{2}+b^{2}+c^{2}$, where $a, b, c$ are positive integers, prove that for each positive integer $k$, $n^{2k}$ can be expressed in the form $A^2 +B^2 +C^2$, where $A, B, C$ are positive integers.

1992 IMO Longlists, 64
Tags: algorithm , additive number theory , additive combinatorics , algebra
For any positive integer $n$ consider all representations $n = a_1 + \cdots+ a_k$, where $a_1 > a_2 > \cdots > a_k > 0$ are integers such that for all $i \in \{1, 2, \cdots , k - 1\}$, the number $a_i$ is divisible by $a_{i+1}$. Find the longest such representation of the number $1992.$

1983 IMO Longlists, 27
Tags: frobenius , additive number theory , additive combinatorics , number theory , modular arithmetic
Let $a,b$ and $c$ be positive integers, no two of which have a common divisor greater than $1$. Show that $2abc-ab-bc-ca$ is the largest integer which cannot be expressed in the form $xbc+yca+zab$, where $x,y,z$ are non-negative integers.

1999 IMO Shortlist, 4
Tags: counting , additive combinatorics , additive number theory , modular arithmetic , combinatorics
Let $A$ be a set of $N$ residues $\pmod{N^{2}}$. Prove that there exists a set $B$ of of $N$ residues $\pmod{N^{2}}$ such that $A + B = \{a+b|a \in A, b \in B\}$ contains at least half of all the residues $\pmod{N^{2}}$.

1979 IMO Longlists, 50
Tags: number theory , additive number theory , additive combinatorics , system of equations
Let $m$ positive integers $a_1, \dots , a_m$ be given. Prove that there exist fewer than $2^m$ positive integers $b_1, \dots , b_n$ such that all sums of distinct $b_k$’s are distinct and all $a_i \ (i \leq m)$ occur among them.

1966 IMO Shortlist, 11
Tags: number theory , factorial , additive number theory
Does there exist an integer $z$ that can be written in two different ways as $z = x! + y!$, where $x, y$ are natural numbers with $x \le y$ ?