Olympiad problems

1969 IMO Shortlist, 7

$(BUL 1)$ Prove that the equation $\sqrt{x^3 + y^3 + z^3}=1969$ has no integral solutions.

Let $S_{n}=\{1,n,n^{2},n^{3}, \cdots \}$, where $n$ is an integer greater than $1$. Find the smallest number $k=k(n)$ such that there is a number which may be expressed as a sum of $k$ (possibly repeated) elements in $S_{n}$ in more than one way. (Rearrangements are considered the same.)

PEN P Problems, 31

Tags: quadratic , additive number theory

A finite sequence of integers $a_{0}, a_{1}, \cdots, a_{n}$ is called quadratic if for each $i \in \{1,2,\cdots,n \}$ we have the equality $\vert a_{i}-a_{i-1} \vert = i^2$. [list=a] [*] Prove that for any two integers $b$ and $c$, there exists a natural number $n$ and a quadratic sequence with $a_{0}=b$ and $a_{n}=c$. [*] Find the smallest natural number $n$ for which there exists a quadratic sequence with $a_{0}=0$ and $a_{n}=1996$. [/list]

1995 IMO Shortlist, 7

Tags: combinatorics , partition , additive combinatorics , additive number theory

Does there exist an integer $ n > 1$ which satisfies the following condition? The set of positive integers can be partitioned into $ n$ nonempty subsets, such that an arbitrary sum of $ n \minus{} 1$ integers, one taken from each of any $ n \minus{} 1$ of the subsets, lies in the remaining subset.

PEN P Problems, 30

Tags: additive number theory

Let $a_{1}, a_{2}, a_{3}, \cdots$ 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 $a_{1998}$.

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, 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 $4k+2 \; (k \in \mathbb{Z})$.

PEN P Problems, 23

Tags: modular arithmetic , additive number theory

Show that there are infinitely many positive integers which cannot be expressed as the sum of squares.

1999 IMO Shortlist, 4

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

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 $\pmod{N^{2}}$.

PEN P Problems, 2

Tags: geometry , 3d geometry , modular arithmetic , additive number theory

Show that each integer $n$ can be written as the sum of five perfect cubes (not necessarily positive).

1983 IMO, 3

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.

PEN P Problems, 25

Tags: number theory , relatively prime , additive number theory

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

PEN P Problems, 18

Tags: modular arithmetic , number theory , additive number theory

Let $p$ be a prime with $p \equiv 1 \pmod{4}$. Let $a$ be the unique integer such that \[p=a^{2}+b^{2}, \; a \equiv-1 \pmod{4}, \; b \equiv 0 \; \pmod{2}\] Prove that \[\sum^{p-1}_{i=0}\left( \frac{i^{3}+6i^{2}+i }{p}\right) = 2 \left( \frac{2}{p}\right),\] where $\left(\frac{k}{p}\right)$ denotes the Legendre Symbol.