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

2019 IMO Shortlist, N7
Tags: number theory , additive number theory
Prove that there is a constant $c>0$ and infinitely many positive integers $n$ with the following property: there are infinitely many positive integers that cannot be expressed as the sum of fewer than $cn\log(n)$ pairwise coprime $n$th powers. [i]Canada[/i]

PEN P Problems, 11
Tags: inequalities , additive number theory
For each positive integer $n$, let $f(n)$ denote the number of ways of representing $n$ as a sum of powers of 2 with nonnegative integer exponents. Representations which differ only in the ordering of their summands are considered to be the same. For instance, $f(4)=4$, because the number $4$ can be represented in the following four ways: \[4, 2+2, 2+1+1, 1+1+1+1.\] Prove that, for any integer $n \geq 3$, \[2^{n^{2}/4}< f(2^{n}) < 2^{n^{2}/2}.\]

PEN P Problems, 41
Tags: additive number theory
The famous conjecture of Goldbach is the assertion that every even integer greater than $2$ is the sum of two primes. Except $2$, $4$, and $6$, every even integer is a sum of two positive composite integers: $n=4+(n-4)$. What is the largest positive even integer that is not a sum of two odd composite integers?

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]

2014 IMO Shortlist, N1
Tags: number theory , frobenius , additive combinatorics , additive number theory , additive representation , 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]

1998 IMO Shortlist, 8
Tags: number theory , integer sequence , additive combinatorics , additive number theory
Let $a_{0},a_{1},a_{2},\ldots $ be an increasing sequence of nonnegative integers such that every nonnegative integer can be expressed uniquely in the form $a_{i}+2a_{j}+4a_{k}$, where $i,j$ and $k$ are not necessarily distinct. Determine $a_{1998}$.

1983 IMO Longlists, 51
Tags: number theory , additive number theory , additive combinatorics
Decide whether there exists a set $M$ of positive integers satisfying the following conditions: (i) For any natural number $m>1$ there exist $a, b \in M$ such that $a+b = m.$ (ii) If $a, b, c, d \in M$, $a, b, c, d > 10$ and $a + b = c + d$, then $a = c$ or $a = d.$

2021 China Team Selection Test, 3
Tags: number theory , additive number theory
Given positive integer $n$. Prove that for any integers $a_1,a_2,\cdots,a_n,$ at least $\lceil \tfrac{n(n-6)}{19} \rceil$ numbers from the set $\{ 1,2, \cdots, \tfrac{n(n-1)}{2} \}$ cannot be represented as $a_i-a_j (1 \le i, j \le n)$.

1969 IMO Shortlist, 13
Tags: modular arithmetic , number theory , 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?

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]

2010 Contests, 2
Tags: number theory , perfect power , additive number theory
For a positive integer $k,$ call an integer a $pure$ $k-th$ $power$ if it can be represented as $m^k$ for some integer $m.$ Show that for every positive integer $n,$ there exists $n$ distinct positive integers such that their sum is a pure $2009-$th power and their product is a pure $2010-$th power.

1969 IMO Shortlist, 63
Tags: quadratic , number theory , additive number theory
$(SWE 6)$ Prove that there are infinitely many positive integers that cannot be expressed as the sum of squares of three positive integers.

2016 Bosnia and Herzegovina Team Selection Test, 4
Tags: number theory , additive number theory , coprime
Determine the largest positive integer $n$ which cannot be written as the sum of three numbers bigger than $1$ which are pairwise coprime.

1979 IMO Shortlist, 18
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.

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

PEN P Problems, 39
Tags: modular arithmetic , additive number theory
In how many ways can $2^{n}$ be expressed as the sum of four squares of natural numbers?