Found problems: 15460
Oliforum Contest I 2008, 1
Consider the sequence of integer such that:
$ a_1 = 2$
$ a_2 = 5$
$ a_{n + 1} = (2 - n^2)a_n + (2 + n^2)a_{n - 1}, \forall n\ge 2$
Find all triplies $ (x,y,z) \in \mathbb{N}^3$ such that $ a_xa_y = a_z$.
2017 Princeton University Math Competition, B2
Let $S = \{1, 22, 333, \dots , 999999999\}$. For how many pairs of integers $(a, b)$ where $a, b \in S$ and $a < b$ is it the case that $a$ divides $b$?
2013 Harvard-MIT Mathematics Tournament, 30
How many positive integers $k$ are there such that \[\dfrac k{2013}(a+b)=lcm(a,b)\] has a solution in positive integers $(a,b)$?
1938 Moscow Mathematical Olympiad, 042
How many positive integers smaller than $1000$ and not divisible by $5$ and by $7$ are there?
1998 Tournament Of Towns, 1
(a) Prove that for any two positive integers a and b the equation $lcm (a, a + 5) = lcm (b, b + 5)$ implies $a = b$.
(b) Is it possible that $lcm (a, b) = lcm (a + c, b + c)$ for positive integers $a, b$ and $c$?
(A Shapovalov)
PS. part (a) for Juniors, both part for Seniors
2022 Thailand TSTST, 3
Let $n > 1$ be a given integer. Prove that infinitely many terms of the sequence $(a_k )_{k\ge 1}$, defined by \[a_k=\left\lfloor\frac{n^k}{k}\right\rfloor,\] are odd. (For a real number $x$, $\lfloor x\rfloor$ denotes the largest integer not exceeding $x$.)
[i]Proposed by Hong Kong[/i]
2012 Mathcenter Contest + Longlist, 7
The arithmetic function $\nu$ is defined by $$\nu (n) = \begin{cases}0, \,\,\,\,\, n=1 \\ k, \,\,\,\,\, n=p_1^{a_1} p_2^{a_2} ... p_k^{a_k}\end{cases}$$, where $n=p_1^{a_1} p_2^{a_2} ... p_k^{a_k}$ represents the prime factorization of the number. Prove that for any naturals $m,n$, $$\tau (n^m) = \sum_{d | n} m^{\nu (d)}.$$ [i](PP-nine)[/i]
2016 CHMMC (Fall), 10
For a positive integer $n$, let $p(n)$ denote the number of prime divisors of $n$, counting multiplicity (i.e. $p(12)=3$). A sequence $a_n$ is defined such that $a_0 = 2$ and for $n > 0$, $a_n = 8^{p(a_{n-1})} + 2$. Compute $$\sum_{n=0}^{\infty} \frac{a_n}{2^n}$$
1996 Bulgaria National Olympiad, 1
Sequence $\{a_n\}$ it define $a_1=1$ and
\[a_{n+1}=\frac{a_n}{n}+\frac{n}{a_n}\] for all $n\ge 1$\\
Prove that $\lfloor a_n^2\rfloor=n$ for all $n\ge 4.$
2017 Iran Team Selection Test, 6
Let $k>1$ be an integer. The sequence $a_1,a_2, \cdots$ is defined as:
$a_1=1, a_2=k$ and for all $n>1$ we have: $a_{n+1}-(k+1)a_n+a_{n-1}=0$
Find all positive integers $n$ such that $a_n$ is a power of $k$.
[i]Proposed by Amirhossein Pooya[/i]
2012 Purple Comet Problems, 6
Volume $A$ equals one fourth of the sum of the volumes $B$ and $C$, while volume $B$ equals one sixth of the sum of the volumes $A$ and $C$. There are relatively prime positive integers $m$ and $n$ so that the ratio of volume $C$ to the sum of the other two volumes is $\frac{m}{n}$. Find $m+n$.
2019 European Mathematical Cup, 1
For positive integers $a$ and $b$, let $M(a,b)$ denote their greatest common divisor. Determine all pairs of positive integers $(m,n)$ such that for any two positive integers $x$ and $y$ such that $x\mid m$ and $y\mid n$,
$$M(x+y,mn)>1.$$
[i]Proposed by Ivan Novak[/i]
2019 VJIMC, 1
Let $\{a_n \}_{n=0}^{\infty}$ be a sequence given recrusively such that $a_0=1$ and $$a_{n+1}=\frac{7a_n+\sqrt{45a_n^2-36}}{2}$$ for $n\geq 0$
Show that :
a) $a_n$ is a positive integer.
b) $a_n a_{n+1}-1$ is a square of an integer.
[i]Proposed by Stefan Gyurki (Matej Bel University, Banska Bystrica).[/i]
2022 Kosovo National Mathematical Olympiad, 4
Find all prime numbers $p$ and $q$ such that $pq-p-q+3$ is a perfect square.
2002 France Team Selection Test, 3
Let $p\ge 3$ be a prime number. Show that there exist $p$ positive integers $a_1,a_2,\ldots ,a_p$ not exceeding $2p^2$ such that the $\frac{p(p-1)}{2}$ sums $a_i+a_j\ (i<j)$ are all distinct.
1997 Brazil Team Selection Test, Problem 3
Find all positive integers $x>1, y$ and primes $p,q$ such that $p^{x}=2^{y}+q^{x}$
2013 Online Math Open Problems, 17
Determine the number of ordered pairs of positive integers $(x,y)$ with $y < x \le 100$ such that $x^2-y^2$ and $x^3 - y^3$ are relatively prime. (Two numbers are [i]relatively prime[/i] if they have no common factor other than $1$.)
[i]Ray Li[/i]
1999 Brazil Team Selection Test, Problem 4
Let Q+ and Z denote the set of positive rationals and the set of inte-
gers, respectively. Find all functions f : Q+ → Z satisfying the following
conditions:
(i) f(1999) = 1;
(ii) f(ab) = f(a) + f(b) for all a, b ∈ Q+;
(iii) f(a + b) ≥ min{f(a), f(b)} for all a, b ∈ Q+.
2013 Online Math Open Problems, 20
A positive integer $n$ is called [i]mythical[/i] if every divisor of $n$ is two less than a prime. Find the unique mythical number with the largest number of divisors.
[i]Proposed by Evan Chen[/i]
1997 Flanders Math Olympiad, 1
Write the number 1997 as the sum of positive integers for which the product is maximal, and prove there's no better solution.
1984 Bulgaria National Olympiad, Problem 1
Solve the equation $5^x7^y+4=3^z$ in nonnegative integers.
2023 China Team Selection Test, P4
Given $m,n\in\mathbb N_+,$ define
$$S(m,n)=\left\{(a,b)\in\mathbb N_+^2\mid 1\leq a\leq m,1\leq b\leq n,\gcd (a,b)=1\right\}.$$
Prove that: for $\forall d,r\in\mathbb N_+,$ there exists $m,n\in\mathbb N_+,m,n\geq d$ and $\left|S(m,n)\right|\equiv r\pmod d.$
2018 ITAMO, 4
$4.$ Let $N$ be an integer greater than $1$.Denote by $x$ the smallest positive integer with the following property:there exists a positive integer $y$ strictly less than $x-1$ , such that $x$ divides $N+y$.Prove that x is either $p^n$ or $2p$ , where $p$ is a prime number and $n$ is a positive integer
2018 Regional Olympiad of Mexico Northeast, 3
Find the smallest natural number $n$ for which there exists a natural number $x$ such that
$$(x+1)^3 + (x + 2)^3 + (x + 3)^3 + (x + 4)^3 = (x + n)^3.$$
2023 Girls in Mathematics Tournament, 1
Define $(a_n)$ a sequence, where $a_1= 12, a_2= 24$ and for $n\geq 3$, we have: $$a_n= a_{n-2}+14$$
a) Is $2023$ in the sequence?
b) Show that there are no perfect squares in the sequence.