Olympiad problems

PEN P Problems, 3

Prove that infinitely many positive integers cannot be written in the form \[{x_{1}}^{3}+{x_{2}}^{5}+{x_{3}}^{7}+{x_{4}}^{9}+{x_{5}}^{11},\] where $x_{1}, x_{2}, x_{3}, x_{4}, x_{5}\in \mathbb{N}$.

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.

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 $a_i-a_j (1 \le i, j \le n)$.

2000 IMO Shortlist, 6

Tags: additive number theory , sum of squares , perfect square , number theory

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

PEN P Problems, 27

Tags: floor function , additive number theory

Determine, with proof, the largest number which is the product of positive integers whose sum is $1976$.

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.

1992 IMO Longlists, 64

Tags: algorithm , algebra , additive number theory , additive combinatorics

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 $1992.$

PEN P Problems, 22

Tags: additive number theory

Show that an integer can be expressed as the difference of two squares if and only if it is not of the form $4k+2 \; (k \in \mathbb{Z})$.

2002 India IMO Training Camp, 16

Tags: sum , modular arithmetic , number theory , additive number theory

Is it possible to find $100$ positive integers not exceeding $25,000$, such that all pairwise sums of them are different?

2017 Baltic Way, 3

Tags: fibonacci , additive number theory , extremal combinatorics , algebra

Positive integers $x_1,...,x_m$ (not necessarily distinct) are written on a blackboard. It is known that each of the numbers $F_1,...,F_{2018}$ can be represented as a sum of one or more of the numbers on the blackboard. What is the smallest possible value of $m$? (Here $F_1,...,F_{2018}$ are the first $2018$ Fibonacci numbers: $F_1=F_2=1, F_{k+1}=F_k+F_{k-1}$ for $k>1$.)

PEN P Problems, 43

Tags: number theory , prime factorization , additive number theory

A positive integer $n$ is abundant if the sum of its proper divisors exceeds $n$. Show that every integer greater than $89 \times 315$ is the sum of two abundant numbers.

2020 Taiwan TST Round 3, 1

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]

1990 IMO Longlists, 3

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?

2001 Romania Team Selection Test, 3

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

Let $ p$ and $ q$ be relatively prime positive integers. A subset $ S$ of $ \{0, 1, 2, \ldots \}$ is called [b]ideal[/b] if $ 0 \in S$ and for each element $ n \in S,$ the integers $ n \plus{} p$ and $ n \plus{} q$ belong to $ S.$ Determine the number of ideal subsets of $ \{0, 1, 2, \ldots \}.$

PEN P Problems, 19

Tags: relatively prime , additive number theory , 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 $n$.

PEN P Problems, 15

Tags: quadratic , algebra , diophantine equation , additive number theory

Find all integers $m>1$ such that $m^3$ is a sum of $m$ squares of consecutive integers.

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 $x_{1}$, $x_{2}$, $\cdots$, $x_{n}$.

1993 IMO Shortlist, 4

Tags: induction , number theory , subset , additive number theory , divisibility

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].

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?

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?

2015 Belarus Team Selection Test, 1

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]

1977 Germany Team Selection Test, 3

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 $p \neq q.$

1992 IMO Longlists, 60

Tags: number theory , perfect power , additive combinatorics , additive 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).$

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.

1992 IMO Shortlist, 15

Tags: number theory , perfect power , additive combinatorics , additive 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).$