Olympiad problems

2010 Singapore MO Open, 3

Suppose that $a_1,...,a_{15}$ are prime numbers forming an arithmetic progression with common difference $d > 0$ if $a_1 > 15$ show that $d > 30000$

2010 Balkan MO Shortlist, N3

Tags: function , number theory , greatest common divisor , relatively prime , totient function , number theory proposed

For each integer $n$ ($n \ge 2$), let $f(n)$ denote the sum of all positive integers that are at most $n$ and not relatively prime to $n$. Prove that $f(n+p) \neq f(n)$ for each such $n$ and every prime $p$.

2010 ELMO Shortlist, 5

Tags: modular arithmetic , number theory proposed , number theory

Find the set $S$ of primes such that $p \in S$ if and only if there exists an integer $x$ such that $x^{2010} + x^{2009} + \cdots + 1 \equiv p^{2010} \pmod{p^{2011}}$. [i]Brian Hamrick.[/i]

2015 Kazakhstan National Olympiad, 2

Tags: number theory , Diophantine equation , number theory proposed , Kazakhstan

Solve in positive integers $x^yy^x=(x+y)^z$

2005 Cono Sur Olympiad, 2

Tags: number theory proposed , number theory

We say that a number of 20 digits is [i]special[/i] if its impossible to represent it as an product of a number of 10 digits by a number of 11 digits. Find the maximum quantity of consecutive numbers that are specials.

2014 ELMO Shortlist, 8

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

Let $\mathbb N$ denote the set of positive integers. Find all functions $f: \mathbb{N} \to \mathbb{N}$ such that: (i) The greatest common divisor of the sequence $f(1), f(2), \dots$ is $1$. (ii) For all sufficiently large integers $n$, we have $f(n) \neq 1$ and \[ f(a)^n \mid f(a+b)^{a^{n-1}} - f(b)^{a^{n-1}} \] for all positive integers $a$ and $b$. [i]Proposed by Yang Liu[/i]

2012 Macedonia National Olympiad, 1

Tags: parameterization , number theory proposed , number theory

Solve the equation $~$ $x^4+2y^4+4z^4+8t^4=16xyzt$ $~$ in the set of integer numbers.

2007 Singapore Team Selection Test, 1

Tags: number theory proposed , number theory

Find all pairs of nonnegative integers $ (x, y)$ satisfying $ (14y)^x \plus{} y^{x\plus{}y} \equal{} 2007$.

2014 ELMO Shortlist, 11

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

Let $p$ be a prime satisfying $p^2\mid 2^{p-1}-1$, and let $n$ be a positive integer. Define \[ f(x) = \frac{(x-1)^{p^n}-(x^{p^n}-1)}{p(x-1)}. \] Find the largest positive integer $N$ such that there exist polynomials $g(x)$, $h(x)$ with integer coefficients and an integer $r$ satisfying $f(x) = (x-r)^N g(x) + p \cdot h(x)$. [i]Proposed by Victor Wang[/i]

2007 Indonesia TST, 3

Tags: number theory proposed , number theory

Let $ a_1,a_2,a_3,\dots$ be infinite sequence of positive integers satisfying the following conditon: for each prime number $ p$, there are only finite number of positive integers $ i$ such that $ p|a_i$. Prove that that sequence contains a sub-sequence $ a_{i_1},a_{i_2},a_{i_3},\dots$, with $ 1 \le i_1<i_2<i_3<\dots$, such that for each $ m \ne n$, $ \gcd(a_{i_m},a_{i_n})\equal{}1$.

2007 Romania Team Selection Test, 3

Tags: induction , number theory proposed , number theory

Find all subsets $A$ of $\left\{ 1, 2, 3, 4, \ldots \right\}$, with $|A| \geq 2$, such that for all $x,y \in A, \, x \neq y$, we have that $\frac{x+y}{\gcd (x,y)}\in A$. [i]Dan Schwarz[/i]

2008 Iran MO (3rd Round), 4

Tags: number theory proposed , number theory

Let $ u$ be an odd number. Prove that $ \frac{3^{3u}\minus{}1}{3^u\minus{}1}$ can be written as sum of two squares.

2014 ELMO Shortlist, 6

Tags: modular arithmetic , number theory proposed , number theory

Show that the numerator of \[ \frac{2^{p-1}}{p+1} - \left(\sum_{k = 0}^{p-1}\frac{\binom{p-1}{k}}{(1-kp)^2}\right) \] is a multiple of $p^3$ for any odd prime $p$. [i]Proposed by Yang Liu[/i]

1999 Taiwan National Olympiad, 2

Tags: number theory proposed , number theory

Let $a_{1},a_{2},...,a_{1999}$ be a sequence of nonnegative integers such that for any $i,j$ with $i+j\leq 1999$ , $a_{i}+a_{j}\leq a_{i+j}\leq a_{i}+a_{j}+1$. Prove that there exists a real number $x$ such that $a_{n}=[nx]\forall n$.

2013 China Team Selection Test, 1

Tags: modular arithmetic , geometry , geometric transformation , real analysis , number theory proposed , number theory

Let $p$ be a prime number and $a, k$ be positive integers such that $p^a<k<2p^a$. Prove that there exists a positive integer $n$ such that \[n<p^{2a}, C_n^k\equiv n\equiv k\pmod {p^a}.\]

1983 IMO Longlists, 13

Tags: modular arithmetic , number theory proposed , number theory

Let $p$ be a prime number and $a_1, a_2, \ldots, a_{(p+1)/2}$ different natural numbers less than or equal to $p.$ Prove that for each natural number $r$ less than or equal to $p$, there exist two numbers (perhaps equal) $a_i$ and $a_j$ such that \[p \equiv a_i a_j \pmod r.\]

1997 Canada National Olympiad, 1

Tags: number theory , least common multiple , number theory proposed

Determine the number of pairs of positive integers $x,y$ such that $x\le y$, $\gcd (x,y)=5!$ and $\text{lcm}(x,y)=50!$.

2011 All-Russian Olympiad, 3

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

For positive integers $a>b>1$, define \[x_n = \frac {a^n-1}{b^n-1}\] Find the least $d$ such that for any $a,b$, the sequence $x_n$ does not contain $d$ consecutive prime numbers. [i]V. Senderov[/i]

2009 Vietnam Team Selection Test, 3

Tags: quadratics , Diophantine equation , number theory , number theory proposed

Let a, b be positive integers. a, b and a.b are not perfect squares. Prove that at most one of following equations $ ax^2 \minus{} by^2 \equal{} 1$ and $ ax^2 \minus{} by^2 \equal{} \minus{} 1$ has solutions in positive integers.

1992 Hungary-Israel Binational, 2

Tags: number theory proposed , number theory

A set $S$ consists of $1992$ positive integers among whose units digits all $10$ digits occur. Show that there is such a set $S$ having no nonempty subset $S_{1}$ whose sum of elements is divisible by $2000$.

2005 Junior Balkan Team Selection Tests - Romania, 2

Tags: inequalities , number theory proposed , number theory

Find the largest positive integer $n>10$ such that the residue of $n$ when divided by each perfect square between $2$ and $\dfrac n2$ is an odd number.

2006 All-Russian Olympiad, 2

Tags: number theory proposed , number theory

The sum and the product of two purely periodic decimal fractions $a$ and $b$ are purely periodic decimal fractions of period length $T$. Show that the lengths of the periods of the fractions $a$ and $b$ are not greater than $T$. [i]Note.[/i] A [i]purely periodic decimal fraction[/i] is a periodic decimal fraction without a non-periodic starting part.

2024 Czech-Polish-Slovak Junior Match, 5

Tags: number theory , number theory proposed , Digits , palindromes , Powers of 2

Is there a positive integer $n$ such that when we write the decimal digits of $2^n$ in opposite order, we get another integer power of $2$?

2014 ELMO Shortlist, 2

Tags: modular arithmetic , number theory proposed , number theory

Define the Fibanocci sequence recursively by $F_1=1$, $F_2=1$ and $F_{i+2} = F_i + F_{i+1}$ for all $i$. Prove that for all integers $b,c>1$, there exists an integer $n$ such that the sum of the digits of $F_n$ when written in base $b$ is greater than $c$. [i]Proposed by Ryan Alweiss[/i]

2010 ELMO Problems, 2

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

Let $r$ and $s$ be positive integers. Define $a_0 = 0$, $a_1 = 1$, and $a_n = ra_{n-1} + sa_{n-2}$ for $n \geq 2$. Let $f_n = a_1a_2\cdots a_n$. Prove that $\displaystyle\frac{f_n}{f_kf_{n-k}}$ is an integer for all integers $n$ and $k$ such that $0 < k < n$. [i]Evan O' Dorney.[/i]