Olympiad problems

2005 MOP Homework, 1

Let $a0$, $a1$, ..., $a_n$ be integers, not all zero, and all at least $-1$. Given that $a_0+2a_1+2^2a_2+...+2^na_n =0$, prove that $a_0+a_1+...+a_n>0$.

2010 China Team Selection Test, 3

Tags: number theory unsolved , number theory

For integers $n>1$, define $f(n)$ to be the sum of all postive divisors of $n$ that are less than $n$. Prove that for any positive integer $k$, there exists a positive integer $n>1$ such that $n<f(n)<f^2(n)<\cdots<f^k(n)$, where $f^i(n)=f(f^{i-1}(n))$ for $i>1$ and $f^1(n)=f(n)$.

2013 Dutch IMO TST, 1

Tags: factorial , number theory unsolved , number theory

Show that $\sum_{n=0}^{2013}\frac{4026!}{(n!(2013-n)!)^2}$ is a perfect square.

2007 Brazil National Olympiad, 2

Tags: quadratics , modular arithmetic , induction , number theory unsolved , number theory

Find the number of integers $ c$ such that $ \minus{}2007 \leq c \leq 2007$ and there exists an integer $ x$ such that $ x^2 \plus{} c$ is a multiple of $ 2^{2007}$.

2010 Postal Coaching, 2

Tags: arithmetic sequence , number theory unsolved , number theory

Call a triple $(a, b, c)$ of positive integers a [b]nice[/b] triple if $a, b, c$ forms a non-decreasing arithmetic progression, $gcd(b, a) = gcd(b, c) = 1$ and the product $abc$ is a perfect square. Prove that given a nice triple, there exists some other nice triple having at least one element common with the given triple.

2005 Postal Coaching, 27

Tags: induction , number theory unsolved , number theory

Let $k$ be an even positive integer and define a sequence $<x_n>$ by \[ x_1= 1 , x_{n+1} = k^{x_n} +1. \] Show that $x_n ^2$ divides $x_{n-1}x_{n+1}$ for each $n \geq 2.$

1995 Romania Team Selection Test, 2

Tags: number theory unsolved , number theory

For each positive integer $ n$,define $ f(n)\equal{}lcm(1,2,...,n)$. (a)Prove that for every $ k$ there exist $ k$ consecutive positive integers on which $ f$ is constant. (b)Find the maximum possible cardinality of a set of consecutive positive integers on which $ f$ is strictly increasing and find all sets for which this maximum is attained.

2011 Mongolia Team Selection Test, 1

Tags: modular arithmetic , number theory unsolved , number theory

Let $v(n)$ be the order of $2$ in $n!$. Prove that for any positive integers $a$ and $m$ there exists $n$ ($n>1$) such that $v(n) \equiv a (\mod m)$. I have a book with Mongolian problems from this year, and this problem appeared in it. Perhaps I am terribly misinterpreting this problem, but it seems like it is wrong. Any ideas?

2014 Romania Team Selection Test, 2

Tags: number theory unsolved , number theory

For every positive integer $n$, let $\sigma(n)$ denote the sum of all positive divisors of $n$ ($1$ and $n$, inclusive). Show that a positive integer $n$, which has at most two distinct prime factors, satisfies the condition $\sigma(n)=2n-2$ if and only if $n=2^k(2^{k+1}+1)$, where $k$ is a non-negative integer and $2^{k+1}+1$ is prime.

2009 Argentina Iberoamerican TST, 2

Tags: number theory unsolved , number theory

Let $ a$ and $ k$ be positive integers. Let $ a_i$ be the sequence defined by $ a_1 \equal{} a$ and $ a_{n \plus{} 1} \equal{} a_n \plus{} k\pi(a_n)$ where $ \pi(x)$ is the product of the digits of $ x$ (written in base ten) Prove that we can choose $ a$ and $ k$ such that the infinite sequence $ a_i$ contains exactly $ 100$ distinct terms

2010 South africa National Olympiad, 3

Tags: algorithm , number theory unsolved , number theory

Determine all positive integers $n$ such that $5^n - 1$ can be written as a product of an even number of consecutive integers.

2005 India National Olympiad, 4

Tags: number theory unsolved , number theory

All possible $6$-digit numbers, in each of which the digits occur in nonincreasing order (from left to right, e.g. $877550$) are written as a sequence in increasing order. Find the $2005$-th number in this sequence.

1999 Bulgaria National Olympiad, 2

Tags: modular arithmetic , number theory unsolved , number theory

Let $\{a_n\}$ be a sequence of integers satisfying $(n-1)a_{n+1}=(n+1)a_n-2(n-1) \forall n\ge 1$. If $2000|a_{1999}$, find the smallest $n\ge 2$ such that $2000|a_n$.

2007 Pan African, 1

Tags: modular arithmetic , number theory unsolved , number theory

Find all natural numbers $N$ consisting of exactly $1112$ digits (in decimal notation) such that: (a) The sum of the digits of $N$ is divisible by $2000$; (b) The sum of the digits of $N+1$ is divisible by $2000$; (c) $1$ is a digit of $N$.

2004 Rioplatense Mathematical Olympiad, Level 3, 2

Tags: modular arithmetic , number theory unsolved , number theory

Find the smallest integer $n$ such that each subset of $\{1,2,\ldots, 2004\}$ with $n$ elements has two distinct elements $a$ and $b$ for which $a^2-b^2$ is a multiple of $2004$.

2008 Philippine MO, 2

Tags: number theory unsolved , number theory

Find the largest integer $n$ for which $\frac{n^{2007}+n^{2006}+\cdots+n^2+n+1}{n+2007}$ is an integer.

2001 Bundeswettbewerb Mathematik, 4

Tags: number theory , prime factorization , prime numbers , number theory unsolved

Prove: For each positive integer is the number of divisors whose decimal representations ends with a 1 or 9 not less than the number of divisors whose decimal representations ends with 3 or 7.

1999 APMO, 4

Tags: inequalities , number theory unsolved , number theory

Determine all pairs $(a,b)$ of integers with the property that the numbers $a^2+4b$ and $b^2+4a$ are both perfect squares.

2007 Pre-Preparation Course Examination, 11

Tags: number theory unsolved , number theory

Let $p \geq 3$ be a prime and $a_1,a_2,\cdots , a_{p-2}$ be a sequence of positive integers such that for every $k \in \{1,2,\cdots,p-2\}$ neither $a_k$ nor $a_k^k-1$ is divisible by $p$. Prove that product of some of members of this sequence is equivalent to $2$ modulo $p$.

2006 Lithuania Team Selection Test, 2

Tags: inequalities , quadratics , number theory unsolved , number theory

Solve in integers $x$ and $y$ the equation $x^3-y^3=2xy+8$.

2005 International Zhautykov Olympiad, 1

Tags: algebra , polynomial , quadratics , modular arithmetic , number theory unsolved , number theory

Prove that the equation $ x^{5} \plus{} 31 \equal{} y^{2}$ has no integer solution.

2005 Korea National Olympiad, 7

Tags: number theory unsolved , number theory

For a positive integer $n$, let $f(n)$ be the number of factors of $n^2+n+1$. Show that there are infinitely many integers $n$ which satisfy $f(n) \geq f(n+1)$.

1997 China National Olympiad, 2

Tags: modular arithmetic , number theory unsolved , number theory

Let $A=\{1,2,3,\cdots ,17\}$. A mapping $f:A\rightarrow A$ is defined as follows: $f^{[1]}(x)=f(x), f^{[k+1]}(x)=f(f^{[k]}(x))$ for $k\in\mathbb{N}$. Suppose that $f$ is bijective and that there exists a natural number $M$ such that: i) when $m<M$ and $1\le i\le 16$, we have $f^{[m]}(i+1)- f^{[m]}(i) \not=\pm 1\pmod{17}$ and $f^{[m]}(1)- f^{[m]}(17) \not=\pm 1\pmod{17}$; ii) when $1\le i\le 16$, we have $f^{[M]}(i+1)- f^{[M]}(i)=\pm 1 \pmod{17}$ and $f^{[M]}(1)- f^{[M]}(17)=\pm 1\pmod{17}$. Find the maximal value of $M$.

2004 Czech-Polish-Slovak Match, 2

Tags: limit , number theory unsolved , number theory

Show that for each natural number $k$ there exist only finitely many triples $(p, q, r)$ of distinct primes for which $p$ divides $qr-k$, $q$ divides $pr-k$, and $r$ divides $pq - k$.

1984 IMO Longlists, 7

Tags: number theory , least common multiple , floor function , number theory unsolved

Prove that for any natural number $n$, the number $\dbinom{2n}{n}$ divides the least common multiple of the numbers $1, 2,\cdots, 2n -1, 2n$.