Olympiad problems

2011 Costa Rica - Final Round, 2

Find the biggest positive integer $n$ such that $n$ is $167$ times the amount of it's positive divisors.

2014 IMS, 11

Tags: number theory proposed , number theory

Let the equation $a^2 + b^2 + 1=abc$ have answer in $\mathbb{N}$.Prove that $c=3$.

2010 Danube Mathematical Olympiad, 4

Tags: number theory proposed , number theory

Let $p$ be a prime number of the form $4k+3$. Prove that there are no integers $w,x,y,z$ whose product is not divisible by $p$, such that: \[ w^{2p}+x^{2p}+y^{2p}=z^{2p}. \]

1974 Miklós Schweitzer, 3

Tags: search , number theory proposed , number theory

Prove that a necessary and sufficient for the existence of a set $ S \subset \{1,2,...,n \}$ with the property that the integers $ 0,1,...,n\minus{}1$ all have an odd number of representations in the form $ x\minus{}y, x,y \in S$, is that $ (2n\minus{}1)$ has a multiple of the form $ 2.4^k\minus{}1$ [i]L. Lovasz, J. Pelikan[/i]

2002 Romania National Olympiad, 3

Tags: modular arithmetic , number theory proposed , number theory

Let $k$ and $n$ be positive integers with $n>2$. Show that the equation: \[x^n-y^n=2^k\] has no positive integer solutions.

2012 China Team Selection Test, 1

Tags: inequalities , induction , number theory proposed , number theory

Given an integer $n\ge 2$. Prove that there only exist a finite number of n-tuples of positive integers $(a_1,a_2,\ldots,a_n)$ which simultaneously satisfy the following three conditions: [list] [*] $a_1>a_2>\ldots>a_n$; [*] $\gcd (a_1,a_2,\ldots,a_n)=1$; [*] $a_1=\sum_{i=1}^{n}\gcd (a_i,a_{i+1})$,where $a_{n+1}=a_1$.[/list]

2003 Tournament Of Towns, 1

Tags: arithmetic sequence , number theory , relatively prime , prime numbers , number theory proposed

An increasing arithmetic progression consists of one hundred positive integers. Is it possible that every two of them are relatively prime?

1996 Turkey Team Selection Test, 2

Tags: linear algebra , matrix , number theory proposed , number theory

Find the maximum number of pairwise disjoint sets of the form $S_{a,b} = \{n^{2}+an+b | n \in \mathbb{Z}\}$, $a, b \in \mathbb{Z}$.

2010 Contests, 1

Tags: number theory proposed , number theory

Find all primes $p,q$ such that $p^3-q^7=p-q$.

2014 ELMO Shortlist, 7

Tags: number theory , number theory proposed , Hi

Find all triples $(a,b,c)$ of positive integers such that if $n$ is not divisible by any prime less than $2014$, then $n+c$ divides $a^n+b^n+n$. [i]Proposed by Evan Chen[/i]

2012 All-Russian Olympiad, 3

Tags: number theory proposed , number theory

Initially, ten consecutive natural numbers are written on the board. In one turn, you may pick any two numbers from the board (call them $a$ and $b$) and replace them with the numbers $a^2-2011b^2$ and $ab$. After several turns, there were no initial numbers left on the board. Could there, at this point, be again, ten consecutive natural numbers?

2016 India National Olympiad, P3

Tags: number theory , number theory proposed , function

Let $\mathbb{N}$ denote the set of natural numbers. Define a function $T:\mathbb{N}\rightarrow\mathbb{N}$ by $T(2k)=k$ and $T(2k+1)=2k+2$. We write $T^2(n)=T(T(n))$ and in general $T^k(n)=T^{k-1}(T(n))$ for any $k>1$. (i) Show that for each $n\in\mathbb{N}$, there exists $k$ such that $T^k(n)=1$. (ii) For $k\in\mathbb{N}$, let $c_k$ denote the number of elements in the set $\{n: T^k(n)=1\}$. Prove that $c_{k+2}=c_{k+1}+c_k$, for $k\ge 1$.

2024 All-Russian Olympiad, 1

Tags: number theory , number theory proposed

Let $p$ and $q$ be different prime numbers. We are given an infinite decreasing arithmetic progression in which each of the numbers $p^{23}, p^{24}, q^{23}$ and $q^{24}$ occurs. Show that the numbers $p$ and $q$ also occur in this progression. [i]Proposed by A. Kuznetsov[/i]

2012 Indonesia TST, 4

Tags: number theory proposed , number theory

Find all odd prime $p$ such that $1+k(p-1)$ is prime for all integer $k$ where $1 \le k \le \dfrac{p-1}{2}$.

2008 Iran MO (3rd Round), 8

Tags: inequalities , number theory proposed , number theory

In an old script found in ruins of Perspolis is written: [code] This script has been finished in a year whose 13th power is 258145266804692077858261512663 You should know that if you are skilled in Arithmetics you will know the year this script is finished easily.[/code] Find the year the script is finished. Give a reason for your answer.

2005 Bulgaria National Olympiad, 6

Tags: algebra , polynomial , symmetry , modular arithmetic , number theory proposed , number theory

Let $a,b$ and $c$ be positive integers such that $ab$ divides $c(c^{2}-c+1)$ and $a+b$ is divisible by $c^{2}+1$. Prove that the sets $\{a,b\}$ and $\{c,c^{2}-c+1\}$ coincide.

2010 Iran MO (3rd Round), 4

Tags: function , number theory proposed , number theory

sppose that $\sigma_k:\mathbb N \longrightarrow \mathbb R$ is a function such that $\sigma_k(n)=\sum_{d|n}d^k$. $\rho_k:\mathbb N \longrightarrow \mathbb R$ is a function such that $\rho_k\ast \sigma_k=\delta$. find a formula for $\rho_k$.($\frac{100}{6}$ points)

2010 Korea Junior Math Olympiad, 1

Tags: modular arithmetic , logarithms , number theory proposed , number theory

Prove that $ 7^{2^{20}} + 7^{2^{19}} + 1 $ has at least $ 21 $ distinct prime divisors.

2006 Cono Sur Olympiad, 5

Tags: floor function , number theory proposed , number theory

Find all positive integer number $n$ such that $[\sqrt{n}]-2$ divides $n-4$ and $[\sqrt{n}]+2$ divides $n+4$. Note: $[r]$ denotes the integer part of $r$.

2013 Saint Petersburg Mathematical Olympiad, 7

Tags: modular arithmetic , number theory proposed , number theory

Given is a natural number $a$ with $54$ digits, each digit equal to $0$ or $1$. Prove the remainder of $a$ when divide by $ 33\cdot 34\cdots 39 $ is larger than $100000$. [hide](It's mean: $a \equiv r \pmod{33\cdot 34\cdots 39 }$ with $ 0<r<33\cdot 34\cdots 39 $ then prove that $r>100000$ )[/hide] M. Antipov

2010 China National Olympiad, 3

Tags: algebra , polynomial , number theory proposed , number theory

Suppose $a_1,a_2,a_3,b_1,b_2,b_3$ are distinct positive integers such that \[(n \plus{} 1)a_1^n \plus{} na_2^n \plus{} (n \minus{} 1)a_3^n|(n \plus{} 1)b_1^n \plus{} nb_2^n \plus{} (n \minus{} 1)b_3^n\] holds for all positive integers $n$. Prove that there exists $k\in N$ such that $ b_i \equal{} ka_i$ for $ i \equal{} 1,2,3$.

2014 Contests, 1

Tags: number theory , greatest common divisor , number theory proposed

For $x, y$ positive integers, $x^2-4y+1$ is a multiple of $(x-2y)(1-2y)$. Prove that $|x-2y|$ is a square number.

2007 All-Russian Olympiad Regional Round, 8.6

Tags: number theory proposed , number theory

A number $ B$ is obtained from a positive integer number $ A$ by permuting its decimal digits. The number $ A\minus{}B\equal{}11...1$ ($ n$ of $ 1's$). Find the smallest possible positive value of $ n$.

2009 India IMO Training Camp, 3

Tags: AMC , USA(J)MO , USAMO , number theory proposed , number theory

Let $ a,b$ be two distinct odd natural numbers.Define a Sequence $ { < a_n > }_{n\ge 0}$ like following: $ a_1 \equal{} a \\ a_2 \equal{} b \\ a_n \equal{} \text{largest odd divisor of }(a_{n \minus{} 1} \plus{} a_{n \minus{} 2})$. Prove that there exists a natural number $ N$ such that $ a_n \equal{} gcd(a,b) \forall n\ge N$.

Russian TST 2015, P3

Tags: number theory proposed , number theory , Operation

Fix positive integers $n$ and $k\ge 2$. A list of $n$ integers is written in a row on a blackboard. You can choose a contiguous block of integers, and I will either add $1$ to all of them or subtract $1$ from all of them. You can repeat this step as often as you like, possibly adapting your selections based on what I do. Prove that after a finite number of steps, you can reach a state where at least $n-k+2$ of the numbers on the blackboard are all simultaneously divisible by $k$.