Olympiad problems

2014 Contests, 2

For some positive integers $m,n$, the system $x+y^2 = m$ and $x^2+y = n$ has exactly one integral solution $(x,y)$. Determine all possible values of $m-n$.

2009 BMO TST, 3

Tags: function , modular arithmetic , Euler , number theory unsolved , number theory

For the give functions in $\mathbb{N}$: [b](a)[/b] Euler's $\phi$ function ($\phi(n)$- the number of natural numbers smaller than $n$ and coprime with $n$); [b](b)[/b] the $\sigma$ function such that the $\sigma(n)$ is the sum of natural divisors of $n$. solve the equation $\phi(\sigma(2^x))=2^x$.

2008 Tuymaada Olympiad, 2

Tags: AMC , USA(J)MO , USAMO , number theory , prime numbers , number theory unsolved

Is it possible to arrange on a circle all composite positive integers not exceeding $ 10^6$, so that no two neighbouring numbers are coprime? [i]Author: L. Emelyanov[/i] [hide="Tuymaada 2008, Junior League, First Day, Problem 2."]Prove that all composite positive integers not exceeding $ 10^6$ may be arranged on a circle so that no two neighbouring numbers are coprime. [/hide]

2014 Contests, 2

Tags: number theory unsolved , number theory

Solve the following equation in $\mathbb{Z}$: \[3^{2a + 1}b^2 + 1 = 2^c\]

2002 India IMO Training Camp, 14

Tags: modular arithmetic , number theory unsolved , number theory

Let $p$ be an odd prime and let $a$ be an integer not divisible by $p$. Show that there are $p^2+1$ triples of integers $(x,y,z)$ with $0 \le x,y,z < p$ and such that $(x+y+z)^2 \equiv axyz \pmod p$

1971 IMO Longlists, 14

Tags: geometry , 3D geometry , Diophantine equation , number theory unsolved , number theory

Note that $8^3 - 7^3 = 169 = 13^2$ and $13 = 2^2 + 3^2.$ Prove that if the difference between two consecutive cubes is a square, then it is the square of the sum of two consecutive squares.

2009 Germany Team Selection Test, 3

Tags: invariant , number theory unsolved , number theory

Initially, on a board there a positive integer. If board contains the number $x,$ then we may additionally write the numbers $2x+1$ and $\frac{x}{x+2}.$ At some point 2008 is written on the board. Prove, that this number was there from the beginning.

2006 Federal Competition For Advanced Students, Part 2, 1

Tags: function , modular arithmetic , floor function , number theory , relatively prime , number theory unsolved

Let $ N$ be a positive integer. How many non-negative integers $ n \le N$ are there that have an integer multiple, that only uses the digits $ 2$ and $ 6$ in decimal representation?

2002 India IMO Training Camp, 14

Tags: modular arithmetic , number theory unsolved , number theory

Let $p$ be an odd prime and let $a$ be an integer not divisible by $p$. Show that there are $p^2+1$ triples of integers $(x,y,z)$ with $0 \le x,y,z < p$ and such that $(x+y+z)^2 \equiv axyz \pmod p$

1999 China Team Selection Test, 2

Tags: number theory , prime numbers , number theory unsolved

Find all prime numbers $p$ which satisfy the following condition: For any prime $q < p$, if $p = kq + r, 0 \leq r < q$, there does not exist an integer $q > 1$ such that $a^{2} \mid r$.

2005 MOP Homework, 4

Tags: modular arithmetic , number theory unsolved , number theory

Prove that there does not exist an integer $n>1$ such that $n$ divides $3^n-2^n$.

2010 Singapore MO Open, 5

Tags: modular arithmetic , number theory , number theory unsolved

A prime number $p$ and integers $x, y, z$ with $0 < x < y < z < p$ are given. Show that if the numbers $x^3, y^3, z^3$ give the same remainder when divided by $p$, then $x^2 + y^2 + z^2$ is divisible by $x + y + z.$

2000 France Team Selection Test, 2

Tags: function , number theory unsolved , number theory

A function from the positive integers to the positive integers satisfies these properties 1. $f(ab)=f(a)f(b)$ for any two coprime positive integers $a,b$. 2. $f(p+q)=f(p)+f(q)$ for any two primes $p,q$. Prove that $f(2)=2, f(3)=3, f(1999)=1999$.

2023 Myanmar IMO Training, 5

Tags: function , floor function , modular arithmetic , number theory , relatively prime , number theory unsolved

For a real number $x$, let $\lfloor x\rfloor$ stand for the largest integer that is less than or equal to $x$. Prove that \[ \left\lfloor{(n-1)!\over n(n+1)}\right\rfloor \] is even for every positive integer $n$.

2011 China Team Selection Test, 2

Tags: induction , number theory unsolved , number theory

Let $\{b_n\}_{n\geq 1}^{\infty}$ be a sequence of positive integers. The sequence $\{a_n\}_{n\geq 1}^{\infty}$ is defined as follows: $a_1$ is a fixed positive integer and \[a_{n+1}=a_n^{b_n}+1 ,\qquad \forall n\geq 1.\] Find all positive integers $m\geq 3$ with the following property: If the sequence $\{a_n\mod m\}_{n\geq 1 }^{\infty}$ is eventually periodic, then there exist positive integers $q,u,v$ with $2\leq q\leq m-1$, such that the sequence $\{b_{v+ut}\mod q\}_{t\geq 1}^{\infty}$ is purely periodic.

2012 Tuymaada Olympiad, 2

Tags: number theory unsolved , number theory

Solve in positive integers the following equation: \[{1\over n^2}-{3\over 2n^3}={1\over m^2}\] [i]Proposed by A. Golovanov[/i]

2008 Germany Team Selection Test, 3

Tags: number theory unsolved , number theory

Prove there is an integer $ k$ for which $ k^3 \minus{} 36 k^2 \plus{} 51 k \minus{} 97$ is a multiple of $ 3^{2008.}$

2012 South East Mathematical Olympiad, 1

Tags: modular arithmetic , number theory unsolved , number theory

A nonnegative integer $m$ is called a “six-composited number” if $m$ and the sum of its digits are both multiples of $6$. How many “six-composited numbers” that are less than $2012$ are there?

1997 Kurschak Competition, 1

Tags: geometry , linear algebra , matrix , number theory unsolved , number theory

Let $p>2$ be a prime number and let $L=\{0,1,\dots,p-1\}^2$. Prove that we can find $p$ points in $L$ with no three of them collinear.

2007 All-Russian Olympiad, 2

Tags: number theory unsolved , number theory

$100$ fractions are written on a board, their numerators are numbers from $1$ to $100$ (each once) and denominators are also numbers from $1$ to $100$ (also each once). It appears that the sum of these fractions equals to $a/2$ for some odd $a$. Prove that it is possible to interchange numerators of two fractions so that sum becomes a fraction with odd denominator. [i]N. Agakhanov, I. Bogdanov [/i]

2005 Postal Coaching, 24

Tags: calculus , integration , number theory , number theory unsolved

Find all nonnegative integers $x,y$ such that \[ 2 \cdot 3^{x} +1 = 7 \cdot 5^{y}. \]

1994 Cono Sur Olympiad, 1

Tags: number theory unsolved , number theory

Pedro and Cecilia play the following game: Pedro chooses a positive integer number $a$ and Cecilia wins if she finds a positive integrer number $b$, prime with $a$, such that, in the factorization of $a^3+b^3$ will appear three different prime numbers. Prove that Cecilia can always win.

2004 Croatia Team Selection Test, 1

Tags: quadratics , algebra , number theory unsolved , number theory

Find all pairs $(x,y)$ of positive integers such that $x(x+y)=y^2+1.$

2006 China Team Selection Test, 3

Tags: number theory unsolved , number theory

Given positive integers $m$ and $n$ so there is a chessboard with $mn$ $1 \times 1$ grids. Colour the grids into red and blue (Grids that have a common side are not the same colour and the grid in the left corner at the bottom is red). Now the diagnol that goes from the left corner at the bottom to the top right corner is coloured into red and blue segments (Every segment has the same colour with the grid that contains it). Find the sum of the length of all the red segments.

2003 South africa National Olympiad, 5

Tags: algebra , polynomial , number theory unsolved , number theory

Prove that the sum of the squares of two consecutive positive integers cannot be equal to a sum of the fourth powers of two consecutive positive integers.