Olympiad problems

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.

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

1990 IMO Shortlist, 1

Tags: number theory , Additive combinatorics , Additive Number Theory , counting , IMO Shortlist

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?

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?

1969 IMO Shortlist, 25

Tags: number theory , Divisibility , Frobenius , Additive Number Theory , IMO Shortlist , IMO Longlist

$(GBR 2)$ 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 $N(k).$

1969 IMO Shortlist, 13

Tags: modular arithmetic , number theory , Divisibility , Additive Number Theory , IMO Shortlist , IMO Longlist

$(CZS 2)$ 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?

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

1996 IMO Shortlist, 3

Tags: combinatorics , Additive combinatorics , Additive Number Theory , Extremal combinatorics , Subsets , IMO Shortlist

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.$

1989 IMO Longlists, 50

Tags: number theory , system of equations , Diophantine equation , Additive Number Theory , IMO Shortlist

Let $ a, b, c, d,m, n \in \mathbb{Z}^\plus{}$ such that \[ a^2\plus{}b^2\plus{}c^2\plus{}d^2 \equal{} 1989,\] \[ a\plus{}b\plus{}c\plus{}d \equal{} m^2,\] and the largest of $ a, b, c, d$ is $ n^2.$ Determine, with proof, the values of $m$ and $ n.$

PEN P Problems, 26

Tags: Additive Number Theory

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 $x, y, z \in \mathbb{N}_{0}$

1992 IMO Longlists, 23

Tags: number theory , egyptian fractions , IMO Shortlist , IMO Longlist , Additive Number Theory

An [i]Egyptian number[/i] is a positive integer that can be expressed as a sum of positive integers, not necessarily distinct, such that the sum of their reciprocals is $1$. For example, $32 = 2 + 3 + 9 + 18$ is Egyptian because $\frac 12 +\frac 13 +\frac 19 +\frac{1}{18}=1$ . Prove that all integers greater than $23$ are [i]Egyptian[/i].

PEN P Problems, 6

Tags: function , inequalities , pigeonhole principle , floor function , Additive Number Theory

Show that every integer greater than $1$ can be written as a sum of two square-free integers.

1966 IMO Shortlist, 11

Tags: number theory , factorial , Additive Number Theory , IMO Shortlist , IMO Longlist

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 $x \le y$ ?

1969 IMO Longlists, 13

Tags: modular arithmetic , number theory , Divisibility , Additive Number Theory , IMO Shortlist , IMO Longlist

$(CZS 2)$ 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, 13

Tags: induction , Additive Number Theory

Let $a_{1}=1$, $a_{2}=2$, $a_{3}$, $a_{4}$, $\cdots$ be the sequence of positive integers of the form $2^{\alpha}3^{\beta}$, where $\alpha$ and $\beta$ are nonnegative integers. Prove that every positive integer is expressible in the form \[a_{i_{1}}+a_{i_{2}}+\cdots+a_{i_{n}},\] where no summand is a multiple of any other.

PEN P Problems, 40

Tags: number theory , prime numbers , 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]