Olympiad problems

2014 Germany Team Selection Test, 3

Let $a_1 \leq a_2 \leq \cdots$ be a non-decreasing sequence of positive integers. A positive integer $n$ is called [i]good[/i] if there is an index $i$ such that $n=\dfrac{i}{a_i}$. Prove that if $2013$ is [i]good[/i], then so is $20$.

2014 Contests, 4

Tags: limit , real analysis , number theory proposed , number theory , Analytic Number Theory

Let $a>1$ be a positive integer and $f\in \mathbb{Z}[x]$ with positive leading coefficient. Let $S$ be the set of integers $n$ such that \[n \mid a^{f(n)}-1.\] Prove that $S$ has density $0$; that is, prove that $\lim_{n\rightarrow \infty} \frac{|S\cap \{1,...,n\}|}{n}=0$.

1990 IMO Longlists, 39

Tags: number theory proposed , number theory

Let $a, b, c$ be integers. Prove that there exist integers $p_1, q_1, r_1, p_2, q_2$ and $r_2$, satisfying $a = q_1r_2 - q_2r_1, b = r_1p_2 - r_2p_1$ and $c = p_1q_2 - p_2q_1.$

2013 China Team Selection Test, 1

Tags: symmetry , number theory proposed , number theory

For a positive integer $N>1$ with unique factorization $N=p_1^{\alpha_1}p_2^{\alpha_2}\dotsb p_k^{\alpha_k}$, we define \[\Omega(N)=\alpha_1+\alpha_2+\dotsb+\alpha_k.\] Let $a_1,a_2,\dotsc, a_n$ be positive integers and $p(x)=(x+a_1)(x+a_2)\dotsb (x+a_n)$ such that for all positive integers $k$, $\Omega(P(k))$ is even. Show that $n$ is an even number.

2012 All-Russian Olympiad, 1

Tags: modular arithmetic , number theory proposed , number theory

Let $a_1,\ldots ,a_{10}$ be distinct positive integers, all at least $3$ and with sum $678$. Does there exist a positive integer $n$ such that the sum of the $20$ remainders of $n$ after division by $a_1,a_2,\ldots ,a_{10},2a_1,2a_2,\ldots ,2a_{10}$ is $2012$?

2013 Bulgaria National Olympiad, 1

Tags: modular arithmetic , number theory , Diophantine equation , number theory proposed

Find all prime numbers $p,q$, for which $p^{q+1}+q^{p+1}$ is a perfect square. [i]Proposed by P. Boyvalenkov[/i]

2013 Iran Team Selection Test, 15

Tags: number theory proposed , number theory

a) Does there exist a sequence $a_1<a_2<\dots$ of positive integers, such that there is a positive integer $N$ that $\forall m>N$, $a_m$ has exactly $d(m)-1$ divisors among $a_i$s? b) Does there exist a sequence $a_1<a_2<\dots$ of positive integers, such that there is a positive integer $N$ that $\forall m>N$, $a_m$ has exactly $d(m)+1$ divisors among $a_i$s?

2009 Indonesia TST, 4

Tags: number theory proposed , number theory

Prove that there exist infinitely many positive integers $ n$ such that $ n!$ is not divisible by $ n^2\plus{}1$.

2007 IberoAmerican, 5

Tags: group theory , abstract algebra , inequalities , geometry , geometric transformation , number theory proposed , number theory

Let's say a positive integer $ n$ is [i]atresvido[/i] if the set of its divisors (including 1 and $ n$) can be split in in 3 subsets such that the sum of the elements of each is the same. Determine the least number of divisors an atresvido number can have.

1997 Baltic Way, 8

Tags: number theory proposed , number theory

If we add $1996$ to $1997$, we first add the unit digits $6$ and $7$. Obtaining $13$, we write down $3$ and “carry” $1$ to the next column. Thus we make a carry. Continuing, we see that we are to make three carries in total. Does there exist a positive integer $k$ such that adding $1996\cdot k$ to $1997\cdot k$ no carry arises during the whole calculation?

2003 ITAMO, 1

Tags: modular arithmetic , number theory proposed , number theory

Find all three digit numbers $n$ which are equal to the number formed by three last digit of $n^2$.

2012 China Western Mathematical Olympiad, 4

Tags: number theory proposed , number theory

Find all prime number $p$, such that there exist an infinite number of positive integer $n$ satisfying the following condition: $p|n^{ n+1}+(n+1)^n.$ (September 29, 2012, Hohhot)

2012 China Team Selection Test, 1

Tags: logarithms , floor function , function , number theory proposed , number theory

Given an integer $n\ge 4$. $S=\{1,2,\ldots,n\}$. $A,B$ are two subsets of $S$ such that for every pair of $(a,b),a\in A,b\in B, ab+1$ is a perfect square. Prove that \[\min \{|A|,|B|\}\le\log _2n.\]

2016 Tuymaada Olympiad, 4

Tags: number theory proposed , Diophantine Equations , number theory , Tuymaada 2016

For each positive integer $k$ find the number of solutions in nonnegative integers $x,y,z$ with $x\le y \le z$ of the equation $$8^k=x^3+y^3+z^3-3xyz$$

2002 Baltic Way, 19

Tags: modular arithmetic , number theory proposed , number theory

Let $n$ be a positive integer. Prove that the equation \[x+y+\frac{1}{x}+\frac{1}{y}=3n\] does not have solutions in positive rational numbers.

2010 Romania Team Selection Test, 5

Tags: floor function , number theory , number theory proposed

Let $a$ and $n$ be two positive integer numbers such that the (positive) prime factors of $a$ be all greater than $n$. Prove that $n!$ divides $(a - 1)(a^2 - 1)\cdots (a^{n-1} - 1)$. [i]AMM Magazine[/i]

2006 Germany Team Selection Test, 1

Tags: Euler , number theory proposed , number theory

For any positive integer $n$, let $w\left(n\right)$ denote the number of different prime divisors of the number $n$. (For instance, $w\left(12\right)=2$.) Show that there exist infinitely many positive integers $n$ such that $w\left(n\right)<w\left(n+1\right)<w\left(n+2\right)$.

1997 Bulgaria National Olympiad, 3

Tags: number theory proposed , number theory

Let $n$ and $m$ be natural numbers such that $m+ i=a_ib_i^2$ for $i=1,2, \cdots n$ where $a_i$ and $b_i$ are natural numbers and $a_i$ is not divisible by a square of a prime number. Find all $n$ for which there exists an $m$ such that $\sum_{i=1}^{n}a_i=12$

2012 China National Olympiad, 2

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

Consider a square-free even integer $n$ and a prime $p$, such that 1) $(n,p)=1$; 2) $p\le 2\sqrt{n}$; 3) There exists an integer $k$ such that $p|n+k^2$. Prove that there exists pairwise distinct positive integers $a,b,c$ such that $n=ab+bc+ca$. [i]Proposed by Hongbing Yu[/i]

1998 Turkey Team Selection Test, 3

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

Let $f(x_{1}, x_{2}, . . . , x_{n})$ be a polynomial with integer coeﬃcients of degree less than $n$. Prove that if $N$ is the number of $n$-tuples $(x_{1}, . . . , x_{n})$ with $0 \leq x_{i} < 13$ and $f(x_{1}, . . . , x_{n}) = 0 (mod 13)$, then $N$ is divisible by 13.

2014 Contests, 3

Tags: number theory proposed , number theory

Let $a_1 \leq a_2 \leq \cdots$ be a non-decreasing sequence of positive integers. A positive integer $n$ is called [i]good[/i] if there is an index $i$ such that $n=\dfrac{i}{a_i}$. Prove that if $2013$ is [i]good[/i], then so is $20$.

2007 Junior Balkan MO, 4

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

Prove that if $ p$ is a prime number, then $ 7p+3^{p}-4$ is not a perfect square.

2007 Baltic Way, 16

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

Let $a$ and $b$ be rational numbers such that $s=a+b=a^2+b^2$. Prove that $s$ can be written as a fraction where the denominator is relatively prime to $6$.

2013 India IMO Training Camp, 1

Tags: modular arithmetic , Diophantine equation , number theory proposed , number theory

A positive integer $a$ is called a [i]double number[/i] if it has an even number of digits (in base 10) and its base 10 representation has the form $a = a_1a_2 \cdots a_k a_1 a_2 \cdots a_k$ with $0 \le a_i \le 9$ for $1 \le i \le k$, and $a_1 \ne 0$. For example, $283283$ is a double number. Determine whether or not there are infinitely many double numbers $a$ such that $a + 1$ is a square and $a + 1$ is not a power of $10$.

2005 Brazil National Olympiad, 1

Tags: number theory proposed , number theory

A natural number is a [i]palindrome[/i] when one obtains the same number when writing its digits in reverse order. For example, $481184$, $131$ and $2$ are palindromes. Determine all pairs $(m,n)$ of positive integers such that $\underbrace{111\ldots 1}_{m\ {\rm ones}}\times\underbrace{111\ldots 1}_{n\ {\rm ones}}$ is a palindrome.