Olympiad problems

2010 Postal Coaching, 4

Prove that the following statement is true for two natural nos. $m,n$ if and only $v(m) = v(n)$ where $v(k)$ is the highest power of $2$ dividing $k$. $\exists$ a set $A$ of positive integers such that $(i)$ $x,y \in \mathbb{N}, |x-y| = m \implies x \in A $ or $y \in A$ $(ii)$ $x,y \in \mathbb{N}, |x-y| = n \implies x \not\in A $ or $y \not\in A$

2009 South africa National Olympiad, 1

Tags: number theory unsolved , number theory

Determine the smallest integer $n > 1$ with the property that $n^2(n - 1)$ is divisible by 2009.

2009 Croatia Team Selection Test, 4

Tags: number theory unsolved , number theory

Determine all natural $ n$ for which there exists natural $ m$ divisible by all natural numbers from 1 to $ n$ but not divisible by any of the numbers $ n \plus{} 1$, $ n \plus{} 2$, $ n \plus{} 3$.

1985 IMO Longlists, 76

Tags: quadratics , number theory unsolved , number theory

Are there integers $m$ and $n$ such that \[5m^2 - 6mn + 7n^2 = 1985 \ ?\]

2008 JBMO Shortlist, 7

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

Determine the minimum value of prime $p> 3$ for which there is no natural number $n> 0$ such that $2^n+3^n\equiv 0\pmod{p} $.

2011 Postal Coaching, 4

Tags: number theory unsolved , number theory

Let $a, b, c$ be positive integers for which \[ac = b^2 + b + 1\] Prove that the equation \[ax^2 - (2b + 1)xy + cy^2 = 1\] has an integer solution.

2014 China Girls Math Olympiad, 8

Tags: number theory , relatively prime , number theory unsolved

Let $n$ be a positive integer, and set $S$ be the set of all integers in $\{1,2,\dots,n\}$ which are relatively prime to $n$. Set $S_1 = S \cap \left(0, \frac n3 \right]$, $S_2 = S \cap \left( \frac n3, \frac {2n}3 \right]$, $S_3 = S \cap \left( \frac{2n}{3}, n \right]$. If the cardinality of $S$ is a multiple of $3$, prove that $S_1$, $S_2$, $S_3$ have the same cardinality.

2002 China Team Selection Test, 3

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

The positive integers $ \alpha, \beta, \gamma$ are the roots of a polynomial $ f(x)$ with degree $ 4$ and the coefficient of the first term is $ 1$. If there exists an integer such that $ f(\minus{}1)\equal{}f^2(s)$. Prove that $ \alpha\beta$ is not a perfect square.

2004 Postal Coaching, 8

Tags: number theory unsolved , number theory

Solve for integers $a,b,c$ \[ (a+b+c)^3 + \frac{1}{2} (b+c)(c+a)(a+b) = 1 - abc \]

1993 Cono Sur Olympiad, 3

Tags: ceiling function , number theory unsolved , number theory

Find the number of elements that a set $B$ can have, contained in $(1, 2, ... , n)$, according to the following property: For any elements $a$ and $b$ on $B$ ($a \ne b$), $(a-b) \not| (a+b)$.

2005 MOP Homework, 3

Tags: number theory unsolved , number theory

Suppose that $p$ and $q$ are distinct primes and $S$ is a subset of $\{1, 2, ..., p-1\}$. Let $N(S)$ denote the number of ordered $q$-tuples $(x_1,x_2,...,x_q)$ with $x_i \in S$, $1 \le i \le q$, such that $x_1+x_2+...+x_q \cong 0 (mod p)$.

2011 Puerto Rico Team Selection Test, 1

Tags: number theory unsolved , number theory

The product of 22 integers is 1. Show that their sum can not be 0.

2014 Iran Team Selection Test, 4

Tags: quadratics , number theory unsolved , number theory

$n$ is a natural number. We shall call a permutation $a_1,\dots,a_n$ of $1,\dots,n$ a quadratic(cubic) permutation if $\forall 1\leq i \leq n-1$ we have $a_ia_{i+1}+1$ is a perfect square(cube). $(a)$ Prove that for infinitely many natural numbers $n$ there exists a quadratic permutation. $(b)$ Prove that for no natural number $n$ exists a cubic permutation.

2001 Tournament Of Towns, 2

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

The decimal expression of the natural number $a$ consists of $n$ digits, while that of $a^3$ consists of $m$ digits. Can $n + m$ be equal to 2001?

2007 Croatia Team Selection Test, 8

Tags: number theory unsolved , number theory

Positive integers $x>1$ and $y$ satisfy an equation $2x^2-1=y^{15}$. Prove that 5 divides $x$.

2005 Polish MO Finals, 1

Tags: LaTeX , inequalities , function , number theory unsolved , number theory

Find all triplets $(x,y,n)$ of positive integers which satisfy: \[ (x-y)^n=xy \]

2006 IberoAmerican, 1

Tags: number theory unsolved , number theory

Find all pairs $(a,\, b)$ of positive integers such that $2a-1$ and $2b+1$ are coprime and $a+b$ divides $4ab+1.$

2005 MOP Homework, 4

Tags: number theory unsolved , number theory

Find all prime numbers $p$ and $q$ such that $3p^4+5q^4+15=13p^2q^2$.

2010 IFYM, Sozopol, 5

Tags: number theory , greatest common divisor , Diophantine equation , number theory unsolved

Let n is a natural number,for which $\sqrt{1+12n^2}$ is a whole number.Prove that $2+2\sqrt{1+12n^2}$ is perfect square.

2004 Postal Coaching, 18

Tags: number theory unsolved , number theory

Let $0 = a_1 < a_2 < a_3 < \cdots < a_n < 1$ and $0 = b_1 < b_2 < b_3 \cdots < b_m < 1$ be real numbers such that for no $a_j$ and $b_k$ the relation $a_j + b_k = 1$ is satisfied. Prove that if the $mn$ numbers ${\ a_j + b_k : 1 \leq j \leq n , 1 \leq k \leq m \}}$ are reduced modulo $1$, then at least $m+n -1$ residues will be distinct.

2007 Estonia Math Open Junior Contests, 4

Tags: calculus , integration , ratio , geometry , perimeter , number theory unsolved , number theory

Call a scalene triangle K [i]disguisable[/i] if there exists a triangle K′ similar to K with two shorter sides precisely as long as the two longer sides of K, respectively. Call a disguisable triangle [i]integral[/i] if the lengths of all its sides are integers. (a) Find the side lengths of the integral disguisable triangle with the smallest possible perimeter. (b) Let K be an arbitrary integral disguisable triangle for which no smaller integral disguisable triangle similar to it exists. Prove that at least two side lengths of K are perfect squares.

2005 MOP Homework, 1

Tags: group theory , abstract algebra , number theory unsolved , number theory

We call a natural number 3-partite if the set of its divisors can be partitioned into 3 subsets each with the same sum. Show that there exist infinitely many 3-partite numbers.

1993 Kurschak Competition, 1

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

Let $a$ and $b$ be positive integers. Prove that the numbers $an^2+b$ and $a(n+1)^2+b$ are both perfect squares only for finitely many integers $n$.

2005 China Team Selection Test, 2

Tags: number theory unsolved , number theory

Given prime number $p$. $a_1,a_2 \cdots a_k$ ($k \geq 3$) are integers not divible by $p$ and have different residuals when divided by $p$. Let \[ S_n= \{ n \mid 1 \leq n \leq p-1, (na_1)_p < \cdots < (na_k)_p \} \] Here $(b)_p$ denotes the residual when integer $b$ is divided by $p$. Prove that $|S|< \frac{2p}{k+1}$.

2010 Postal Coaching, 5

Tags: search , number theory unsolved , number theory

Let $a, b, c$ be integers such that \[\frac ab+\frac bc+\frac ca= 3\] Prove that $abc$ is a cube of an integer.