Olympiad problems

2012 India IMO Training Camp, 2

Show that there exist infinitely many pairs $(a, b)$ of positive integers with the property that $a+b$ divides $ab+1$, $a-b$ divides $ab-1$, $b>1$ and $a>b\sqrt{3}-1$

2009 IMAR Test, 2

Tags: geometry , combinatorial geometry , greatest common divisor , combinatorics , number theory

Of the vertices of a cube, $7$ of them have assigned the value $0$, and the eighth the value $1$. A [i]move[/i] is selecting an edge and increasing the numbers at its ends by an integer value $k > 0$. Prove that after any finite number of [i]moves[/i], the g.c.d. of the $8$ numbers at vertices is equal to $1$. Russian M.O.

2014 Brazil Team Selection Test, 1

Tags: greatest common divisor , gcd , number theory

For $m$ and $n$ positive integers that are prime to each other, determine the possible values of $$\gcd (5^m + 7^m, 5^n + 7^n)$$

2008 Bosnia And Herzegovina - Regional Olympiad, 3

Tags: number theory , inequalities , abstract algebra , greatest common divisor , prime number

Prove that equation $ p^{4}\plus{}q^{4}\equal{}r^{4}$ does not have solution in set of prime numbers.

2011 NIMO Summer Contest, 11

Tags: number theory , least common multiple , greatest common divisor

How many ordered pairs of positive integers $(m, n)$ satisfy the system \begin{align*} \gcd (m^3, n^2) & = 2^2 \cdot 3^2, \\ \text{LCM} [m^2, n^3] & = 2^4 \cdot 3^4 \cdot 5^6, \end{align*} where $\gcd(a, b)$ and $\text{LCM}[a, b]$ denote the greatest common divisor and least common multiple of $a$ and $b$, respectively?

2008 Putnam, B5

Tags: derivative , integration , greatest common divisor , function , number theory , calculus

Find all continuously differentiable functions $ f: \mathbb{R}\to\mathbb{R}$ such that for every rational number $ q,$ the number $ f(q)$ is rational and has the same denominator as $ q.$ (The denominator of a rational number $ q$ is the unique positive integer $ b$ such that $ q\equal{}a/b$ for some integer $ a$ with $ \gcd(a,b)\equal{}1.$) (Note: $ \gcd$ means greatest common divisor.)

1983 Vietnam National Olympiad, 1

Tags: modular arithmetic , greatest common divisor , number theory

Are there positive integers $a, b$ with $b \ge 2$ such that $2^a + 1$ is divisible by $2^b - 1$?

1956 Putnam, B6

Tags: series , limit , greatest common divisor

Given $T_1 =2, T_{n+1}= T_{n}^{2} -T_n +1$ for $n>0.$ Prove: (i) If $m \ne n,$ $T_m$ and $T_n$ have no common factor greater than $1.$ (ii) $\sum_{i=1}^{\infty} \frac{1}{T_i }=1.$