Olympiad problems

2009 Indonesia MO, 2

For any real $ x$, let $ \lfloor x\rfloor$ be the largest integer that is not more than $ x$. Given a sequence of positive integers $ a_1,a_2,a_3,\ldots$ such that $ a_1>1$ and \[ \left\lfloor\frac{a_1\plus{}1}{a_2}\right\rfloor\equal{}\left\lfloor\frac{a_2\plus{}1}{a_3}\right\rfloor\equal{}\left\lfloor\frac{a_3\plus{}1}{a_4}\right\rfloor\equal{}\cdots\] Prove that \[ \left\lfloor\frac{a_n\plus{}1}{a_{n\plus{}1}}\right\rfloor\leq1\] holds for every positive integer $ n$.

2002 All-Russian Olympiad, 1

Tags: number theory unsolved , number theory

Determine the smallest natural number which can be represented both as the sum of $2002$ positive integers with the same sum of decimal digits, and as the sum of $2003$ integers with the same sum of decimal digits.

2007 Turkey MO (2nd round), 1

Tags: modular arithmetic , number theory unsolved , number theory

Let $k>1$ be an integer, $p=6k+1$ be a prime number and $m=2^{p}-1$ . Prove that $\frac{2^{m-1}-1}{127m}$ is an integer.

2009 Croatia Team Selection Test, 4

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

Prove that there are infinite many positive integers $ n$ such that $ n^2\plus{}1\mid n!$, and infinite many of those for which $ n^2\plus{}1 \nmid n!$.

1990 Federal Competition For Advanced Students, P2, 1

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

Determine the number of integers $ n$ with $ 1 \le n \le N\equal{}1990^{1990}$ such that $ n^2\minus{}1$ and $ N$ are coprime.

2011 Akdeniz University MO, 1

Tags: number theory , number theory unsolved

Let $a$ be a positive number, and we show decimal part of the $a$ with $\left\{a\right\}$.For a positive number $x$ with $\sqrt 2< x <\sqrt 3$ such that, $\left\{\frac{1}{x}\right\}$=$\left\{x^2\right\}$.Find value of the $$x(x^7-21)$$

2011 Indonesia TST, 1

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

Find all real number $x$ which could be represented as $x = \frac{a_0}{a_1a_2 . . . a_n} + \frac{a_1}{a_2a_3 . . . a_n} + \frac{a_2}{a_3a_4 . . . a_n} + . . . + \frac{a_{n-2}}{a_{n-1}a_n} + \frac{a_{n-1}}{a_n}$ , with $n, a_1, a_2, . . . . , a_n$ are positive integers and $1 = a_0 \leq a_1 < a_2 < . . . < a_n$

2008 Baltic Way, 7

Tags: symmetry , number theory unsolved , number theory

How many pairs $ (m,n)$ of positive integers with $ m < n$ fulfill the equation $ \frac {3}{2008} \equal{} \frac 1m \plus{} \frac 1n$?

2014 India IMO Training Camp, 1

Tags: number theory unsolved , number theory , euler totient function

Let $p$ be an odd prime and $r$ an odd natural number.Show that $pr+1$ does not divide $p^p-1$

2008 Grigore Moisil Intercounty, 2

Tags: modular arithmetic , number theory unsolved , number theory

Determine the natural numbers a, b, c s.t. : $ \frac{3a+2b}{6a}=\frac{8b+c}{10b}=\frac{3a+2c}{3c} $ and $ a^{2}+b^{2}+c^{2}=975 $ The challenge here is to come up with as basic solution as possible.

2009 India Regional Mathematical Olympiad, 2

Tags: modular arithmetic , number theory unsolved , number theory

Show that there is no integer $ a$ such that $ a^2 \minus{} 3a \minus{} 19$ is divisible by $ 289$.

1972 IMO Longlists, 23

Tags: number theory unsolved , number theory

Does there exist a $2n$-digit number $\overline{a_{2n}a_{2n-1}\cdots a_1}$(for an arbitrary $n$) for which the following equality holds: \[\overline{a_{2n}\cdots a_1}= (\overline{a_n \cdots a_1})^2?\]

1990 Polish MO Finals, 2

Tags: limit , number theory unsolved , number theory

Suppose that $(a_n)$ is a sequence of positive integers such that $\lim\limits_{n\to \infty} \dfrac{n}{a_n}=0$ Prove that there exists $k$ such that there are at least $1990$ perfect squares between $a_1 + a_2 + ... + a_k$ and $a_1 + a_2 + ... + a_{k+1}$.

1992 Vietnam National Olympiad, 2

Tags: number theory unsolved , number theory

For any positive integer $a$, denote $f(a)=|\{b\in\mathbb{N}| b|a$ $\text{and}$ $b\mod{10}\in\{1,9\}\}|$ and $g(a)=|\{b\in\mathbb{N}| b|a$ $\text{and}$ $b\mod{10}\in\{3,7\}\}|$. Prove that $f(a)\geq g(a)\forall a\in\mathbb{N}$.

1993 Romania Team Selection Test, 3

Tags: search , number theory unsolved , number theory

Let $ p\geq 5$ be a prime number.Prove that for any partition of the set $ P\equal{}\{1,2,3,...,p\minus{}1\}$ in $ 3$ subsets there exists numbers $ x,y,z$ each belonging to a distinct subset,such that $ x\plus{}y\equiv z (mod p)$

1999 Brazil Team Selection Test, Problem 1

Tags: number theory unsolved , number theory

For a positive integer n, let $w(n)$ denote the number of distinct prime divisors of n. Determine the least positive integer k such that $2^{w(n)} \leq k \sqrt[4]{n}$ for all positive integers n.

2014 India IMO Training Camp, 1

Tags: number theory unsolved , number theory , euler totient function

Let $p$ be an odd prime and $r$ an odd natural number.Show that $pr+1$ does not divide $p^p-1$

2012 Uzbekistan National Olympiad, 2

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

For any positive integers $n$ and $m$ satisfying the equation $n^3+(n+1)^3+(n+2)^3=m^3$, prove that $4\mid n+1$.

1991 IMTS, 5

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

Prove that there are infinitely many positive integers $n$ such that $n \times n \times n$ can not be filled completely with 2 x 2 x 2 and 3 x 3 x 3 solid cubes.

2012 Indonesia TST, 4

Tags: number theory unsolved , number theory

Find all quadruplets of positive integers $(m,n,k,l)$ such that $3^m = 2^k + 7^n$ and $m^k = 1 + k + k^2 + k^3 + \ldots + k^l$.

1999 All-Russian Olympiad, 5

Tags: number theory unsolved , number theory

The sum of the (decimal) digits of a natural number $n$ equals $100$, and the sum of digits of $44n$ equals $800$. Determine the sum of digits of $3n$.

2001 India IMO Training Camp, 1

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

For any positive integer $n$, show that there exists a polynomial $P(x)$ of degree $n$ with integer coefficients such that $P(0),P(1), \ldots, P(n)$ are all distinct powers of $2$.

2008 Rioplatense Mathematical Olympiad, Level 3, 3

Tags: number theory unsolved , number theory

Find all integers $k\ge 2$ such that for all integers $n\ge 2$, $n$ does not divide the greatest odd divisor of $k^n+1$.

2014 Portugal MO, 4

Tags: number theory unsolved , number theory

Determine all natural numbers $x$, $y$ and $z$, such that $x\leq y\leq z$ and \[\left(1+\frac1x\right)\left(1+\frac1y\right)\left(1+\frac1z\right) = 3\text{.}\]

2003 Brazil National Olympiad, 1

Tags: quadratics , Gauss , number theory unsolved , number theory

Find the smallest positive prime that divides $n^2 + 5n + 23$ for some integer $n$.