Found problems: 71
2011 IMAR Test, 4
Given an integer number $n \ge 3$, show that the number of lists of jointly coprime positive integer numbers that sum to $n$ is divisible by $3$.
(For instance, if $n = 4$, there are six such lists: $(3, 1), (1, 3), (2, 1, 1), (1, 2, 1), (1, 1, 2)$ and $(1, 1, 1, 1)$.)
2022 Durer Math Competition (First Round), 4
We want to partition the integers $1, 2, 3, . . . , 100$ into several groups such that within each group either any two numbers are coprime or any two are not coprime. At least how many groups are needed for such a partition?
[i]We call two integers coprime if they have no common divisor greater than $1$.[/i]
2012 Singapore Junior Math Olympiad, 5
Suppose $S = \{a_1, a_2,..., a_{15}\}$ is a set of $1 5$ distinct positive integers chosen from $2 , 3, ... , 2012$ such that every two of them are coprime. Prove that $S$ contains a prime number.
(Note: Two positive integers $m, n$ are coprime if their only common factor is 1)
2012 Danube Mathematical Competition, 2
Consider the natural number prime $p, p> 5$. From the decimal number $\frac1p$, randomly remove $2012$ numbers, after the comma. Show that the remaining number can be represented as $\frac{a}{b}$ , where $a$ and $b$ are coprime numbers , and $b$ is multiple of $p$.
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 $i \neq j$.
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.
2015 Saudi Arabia BMO TST, 4
Let $n \ge 2$ be an integer and $p_1 < p_2 < ... < p_n$ prime numbers. Prove that there exists an integer $k$ relatively prime with $p_1p_2... p_n$ and such that $gcd (k + p_1p_2...p_i, p_1p_2...p_n) = 1$ for all $i = 1, 2,..., n - 1$.
Malik Talbi
VI Soros Olympiad 1999 - 2000 (Russia), 11.7
Prove that there are arithmetic progressions of arbitrary length, consisting of different pairwise coprime natural numbers.
2015 Bosnia And Herzegovina - Regional Olympiad, 2
For positive integer $n$, find all pairs of coprime integers $p$ and $q$ such that $p+q^2=(n^2+1)p^2+q$
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.
2019 Junior Balkan Team Selection Tests - Romania, 2
Let $n$ be a positive integer and $A$ a set containing $8n + 1$ positive integers co-prime with $6$ and less than $30n$. Prove that there exist $a, b \in A$ two different numbers such that $a$ divides $b$.
2011 Korea Junior Math Olympiad, 6
For a positive integer $n$, define the set $S_n$ as $S_n =\{(a, b)|a, b \in N, lcm[a, b] = n\}$ . Let $f(n)$ be the sum of $\phi (a)\phi (b)$ for all $(a, b) \in S_n$. If a prime $p$ relatively prime to $n$ is a divisor of $f(n)$, prove that there exists a prime $q|n$ such that $p|q^2 - 1$.
2012 Greece JBMO TST, 2
Find all pairs of coprime positive integers $(p,q)$ such that $p^2+2q^2+334=[p^2,q^2]$ where $[p^2,q^2]$ is the leact common multiple of $p^2,q^2$ .
2024 Dutch IMO TST, 1
For a positive integer $n$, let $\alpha(n)$ be the arithmetic mean of the divisors of $n$, and let $\beta(n)$ be the arithmetic mean of the numbers $k \le n$ with $\text{gcd}(k,n)=1$.
Determine all positive integers $n$ with $\alpha(n)=\beta(n)$.
2011 Regional Olympiad of Mexico Center Zone, 3
We have $n$ positive integers greater than $1$ and less than $10000$ such that neither of them is prime but any two of them are relative prime. Find the maximum value of $n $.
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 $1$.
1993 Abels Math Contest (Norwegian MO), 3
The Fermat-numbers are defined by $F_n = 2^{2^n}+1$ for $n\in N$.
(a) Prove that $F_n = F_{n-1}F_{n-2}....F_1F_0 +2$ for $n > 0$.
(b) Prove that any two different Fermat numbers are coprime
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.
2016 Bosnia and Herzegovina Team Selection Test, 4
Determine the largest positive integer $n$ which cannot be written as the sum of three numbers bigger than $1$ which are pairwise coprime.
2018 Rioplatense Mathematical Olympiad, Level 3, 3
Determine all the triples $\{a, b, c \}$ of positive integers coprime (not necessarily pairwise prime) such that $a + b + c$ simultaneously divides the three numbers $a^{12} + b^{12}+ c^{12}$, $ a^{23} + b^{23} + c^{23} $ and $ a^{11004} + b^{11004} + c^{11004}$
2024 Brazil Team Selection Test, 1
Given an integer $n > 1$, let $1 = a_1 < a_2 < \cdots < a_t = n - 1$ be all positive integers less than $n$ that are coprime to $n$. Find all $n$ such that there is no $i \in \{1, 2, \ldots , t - 1\}$ satisfying $3 | a_i + a_{i+1}$.