Found problems: 71
2018 Dutch BxMO TST, 2
Let $\vartriangle ABC$ be a triangle of which the side lengths are positive integers which are pairwise coprime. The tangent in $A$ to the circumcircle intersects line $BC$ in $D$. Prove that $BD$ is not an integer.
2004 All-Russian Olympiad Regional Round, 10.5
Equation $$x^n + a_1x^{n-1} + a_2x^{n-2} +...+ a_{n-1}x + a_n = 0$$ with integer non-zero coefficients $a_1$, $a_2$, $...$ , $a_n$ has $n$ different integer roots. Prove that if any two roots are relatively prime, then the numbers $a_{n-1}$ and $a_n$ are coprime.
2023 Indonesia TST, N
Given an integer $a>1$. Prove that there exists a sequence of positive integers
\[ n_1, n_2, n_3, \ldots \]
Such that
\[ \gcd(a^{n_i+1} + a^{n_i} - 1, \ a^{n_j + 1} + a^{n_j} - 1) =1 \] For every $i \neq j$.
2016 Saudi Arabia Pre-TST, 1.4
Let $p$ be a given prime. For each prime $r$, we defind the function as following $F(r) =\frac{(p^{rp} - 1) (p - 1)}{(p^r - 1) (p^p - 1)}$.
1. Show that $F(r)$ is a positive integer for any prime $r \ne p$.
2. Show that $F(r)$ and $F(s)$ are coprime for any primes $r$ and $s$ such that $r \ne p, s \ne p$ and $r \ne s$.
3. Fix a prime $r \ne p$. Show that there is a prime divisor $q$ of $F(r)$ such that $p| q - 1$ but $p^2 \nmid q - 1$.
1988 Mexico National Olympiad, 5
If $a$ and $b$ are coprime positive integers and $n$ an integer, prove that the greatest common divisor of $a^2+b^2-nab$ and $a+b$ divides $n+2$.
1992 All Soviet Union Mathematical Olympiad, 567
Show that if $15$ numbers lie between $2$ and $1992$ and each pair is coprime, then at least one is prime.
1991 Romania Team Selection Test, 2
The sequence ($a_n$) is defined by $a_1 = a_2 = 1$ and $a_{n+2 }= a_{n+1} +a_n +k$, where $k$ is a positive integer.
Find the least $k$ for which $a_{1991}$ and $1991$ are not coprime.
2017 Ecuador Juniors, 5
Two positive integers are coprime if their greatest common divisor is $1$. Let $C$ be the set of all divisors of the number $8775$ that are greater than $ 1$. A set of $k$ consecutive positive integers satisfies that each of them is coprime with some element of $C$. Determine the largest possible value of $K$.
2016 Saudi Arabia IMO TST, 1
Call a positive integer $N \ge 2$ [i]special [/i] if for every k such that $2 \le k \le N, N$ can be expressed as a sum of $k$ positive integers that are relatively prime to $N$ (although not necessarily relatively prime to each other). Find all special positive integers.
2017 Singapore Junior Math Olympiad, 5
Let $a, b, c$ be nonzero integers, with $1$ as their only positive common divisor, such that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}= 0$. Find the number of such triples $(a, b, c)$ with $50 \ge |a| \ge |b| \ge |c| 1$.
2014 IFYM, Sozopol, 5
Let $f(x)$ be a polynomial with integer coefficients, for which there exist $a,b\in \mathbb{Z}$ ($a\neq b$), such that $f(a)$ and $f(b)$ are coprime. Prove that there exist infinitely many values for $x$, such that each $f(x)$ is coprime with any other.
2012 IMAR Test, 2
Given an integer $n \ge 2$, evaluate $\Sigma \frac{1}{pq}$ ,where the summation is over all coprime integers $p$ and $q$ such that $1 \le p < q \le n$ and $p + q > n$.
2018 Dutch BxMO TST, 2
Let $\vartriangle ABC$ be a triangle of which the side lengths are positive integers which are pairwise coprime. The tangent in $A$ to the circumcircle intersects line $BC$ in $D$. Prove that $BD$ is not an integer.
2013 Danube Mathematical Competition, 2
Let $a, b, c, n$ be four integers, where n$\ge 2$, and let $p$ be a prime dividing both $a^2+ab+b^2$ and $a^n+b^n+c^n$, but not $a+b+c$. for instance, $a \equiv b \equiv -1 (mod \,\, 3), c \equiv 1 (mod \,\, 3), n$ a positive even integer, and $p = 3$ or $a = 4, b = 7, c = -13, n = 5$, and $p = 31$ satisfy these conditions. Show that $n$ and $p - 1$ are not coprime.
1940 Eotvos Mathematical Competition, 2
Let $m$ and $n$ be distinct positive integers. Prove that $2^{2^m} + 1$ and $2^{2^n} + 1$ have no common divisor greater than $1$.
VI Soros Olympiad 1999 - 2000 (Russia), 11.7
Prove that there are arithmetic progressions of arbitrary length, consisting of different pairwise coprime natural numbers.
1979 Poland - Second Round, 5
Prove that among every ten consecutive natural numbers there is one that is coprime to each of the other nine.
2016 Balkan MO Shortlist, N1
Find all natural numbers $n$ for which $1^{\phi (n)} + 2^{\phi (n)} +... + n^{\phi (n)}$ is coprime with $n$.
2016 Silk Road, 3
Given natural numbers $a,b$ and function $f: \mathbb{N} \to \mathbb{N} $ such that for any natural number $n, f\left( n+a \right)$ is divided by $f\left( {\left[ {\sqrt n } \right] + b} \right)$. Prove that for any natural $n$ exist $n$ pairwise distinct and pairwise relatively prime natural numbers ${{a}_{1}}$, ${{a}_{2}}$, $\ldots$, ${{a}_{n}}$ such that the number $f\left( {{a}_{i+1}} \right)$ is divided by $f\left( {{a}_{i}} \right)$ for each $i=1,2, \dots ,n-1$ .
(Here $[x]$ is the integer part of number $x$, that is, the largest integer not exceeding $x$.)
1999 Bundeswettbewerb Mathematik, 2
The sequences $(a_n)$ and $(b_n)$ are defined by $a_1 = b_1 = 1$ and $a_{n+1} = a_n +b_n, b_{n+1} = a_nb_n$ for $n = 1,2,...$ Show that every two distinct terms of the sequence $(a_n)$ are coprime
VMEO III 2006, 11.4
Given an integer $a>1$. Let $p_1 < p_2 <...< p_k$ be all prime divisors of $a$. For each positive integer $n$ we define:
$C_0(n) = a^{2n}, C_1(n) =\frac{a^{2n}}{p^2_1}, .... , C_k(n) =\frac{a^{2_n}}{p^2_k}$
$A = a^2 + 1$
$T(n) = A^{C_0(n)} - 1$
$M(n) = LCM(a^{2n+2}, A^{C_1(n)} - 1, ..., A^{C_k(n)} - 1)$
$A_n =\frac{T(n)}{M(n)}$
Prove that the sequence $A_1, A_2, ... $ satisfies the properties:
(i) Every number in the sequence is an integer greater than $1$ and has only prime divisors of the form $am + 1$.
(ii) Any two different numbers in the sequence are coprime.