Olympiad problems

2019 IMO Shortlist, N7

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]

1996 IMO Shortlist, 3

Tags: additive combinatorics , additive number theory , extremal combinatorics , subset , combinatorics

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.$

1969 IMO Longlists, 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?

1977 IMO, 2

Tags: number theory , additive number theory , remainder , divisibility

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 $q^2 + r = 1977.$

PEN P Problems, 20

Tags: additive number theory

If an integer $n$ is such that $7n$ is the form $a^2 +3b^2$, prove that $n$ is also of that form.

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?

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.$

1966 IMO Longlists, 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 $x \le y$ ?

PEN P Problems, 33

Tags: induction , number theory , relatively prime , additive number theory

Let $a_{1}, a_{2}, \cdots, a_{k}$ be relatively prime positive integers. Determine the largest integer which cannot be expressed in the form \[x_{1}a_{2}a_{3}\cdots a_{k}+x_{2}a_{1}a_{3}\cdots a_{k}+\cdots+x_{k}a_{1}a_{2}\cdots a_{k-1}\] for some nonnegative integers $x_{1}, x_{2}, \cdots, x_{k}$.

1969 IMO Shortlist, 25

Tags: number theory , divisibility , frobenius , additive number theory

$(GBR 2)$ Let $a, b, x, y$ be positive integers such that $a$ and $b$ have no common divisor greater than $1$. Prove that the largest number not expressible in the form $ax + by$ is $ab - a - b$. If $N(k)$ is the largest number not expressible in the form $ax + by$ in only $k$ ways, find $N(k).$

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.

PEN P Problems, 29

Tags: additive number theory

Show that the set of positive integers which cannot be represented as a sum of distinct perfect squares is finite.

1992 IMO Longlists, 45

Tags: function , number theory , additive number theory , additive combinatorics , counting

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.

2010 APMO, 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.

2015 Peru IMO TST, 11

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]

PEN P Problems, 42

Tags: pigeonhole principle , inequalities , additive number theory

Prove that for each positive integer $K$ there exist infinitely many even positive integers which can be written in more than $K$ ways as the sum of two odd primes.

PEN P Problems, 19

Tags: number theory , relatively prime , additive number theory

Let $n$ be an integer of the form $a^2 + b^2$, where $a$ and $b$ are relatively prime integers and such that if $p$ is a prime, $p \leq \sqrt{n}$, then $p$ divides $ab$. Determine all such $n$.

1990 IMO Longlists, 16

Tags: number theory , additive number theory , additive combinatorics

We call an integer $k \geq 1$ having property $P$, if there exists at least one integer $m \geq 1$ which cannot be expressed in the form $m = \varepsilon_1 z_1^k + \varepsilon_2 z_2^k + \cdots + \varepsilon_{2k} z_{2k}^k $ , where $z_i$ are nonnegative integer and $\varepsilon _i = 1$ or $-1$, $i = 1, 2, \ldots, 2k$. Prove that there are infinitely many integers $k$ having the property $P.$

1983 IMO Shortlist, 18

Tags: modular arithmetic , number theory , frobenius , additive number theory , additive combinatorics

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.

1990 IMO Shortlist, 1

Tags: number theory , additive combinatorics , additive number theory , counting

The integer $ 9$ can be written as a sum of two consecutive integers: $ 9 \equal{} 4\plus{}5.$ Moreover, it can be written as a sum of (more than one) consecutive positive integers in exactly two ways: $ 9 \equal{} 4\plus{}5 \equal{} 2\plus{}3\plus{}4.$ Is there an integer that can be written as a sum of $ 1990$ consecutive integers and that can be written as a sum of (more than one) consecutive positive integers in exactly $ 1990$ ways?

1992 IMO Longlists, 77

Tags: ramsey theory , additive number theory , additive combinatorics , extremal combinatorics , combinatorics

Show that if $994$ integers are chosen from $1, 2,\cdots , 1992$ and one of the chosen integers is less than $64$, then there exist two among the chosen integers such that one of them is a factor of the other.

1975 IMO Shortlist, 11

Tags: modular arithmetic , number theory , sequence , additive 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, 40

Tags: number theory , prime number , additive number theory

Show that [list=a][*] infinitely many perfect squares are a sum of a perfect square and a prime number, [*] infinitely many perfect squares are not a sum of a perfect square and a prime number. [/list]

PEN P Problems, 38

Tags: additive number theory

Find the smallest possible $n$ for which there exist integers $x_{1}$, $x_{2}$, $\cdots$, $x_{n}$ such that each integer between $1000$ and $2000$ (inclusive) can be written as the sum (without repetition), of one or more of the integers $x_{1}$, $x_{2}$, $\cdots$, $x_{n}$.

PEN P Problems, 4

Tags: additive number theory

Determine all positive integers that are expressible in the form \[a^{2}+b^{2}+c^{2}+c,\] where $a$, $b$, $c$ are integers.