Found problems: 15460
2010 CHMMC Fall, 3
Talithia throws a party on the fifth Saturday of every month that has five Saturdays. That is, if a month has five Saturdays, Talithia has a party on the fifth Saturday of that month, and if a month has four Saturdays, then Talithia does not have a party that month. Given that January $1$, $2010$ was a Friday, compute the number of parties Talithia will have in $2010$.
2018 Harvard-MIT Mathematics Tournament, 3
There are two prime numbers $p$ so that $5p$ can be expressed in the form $\left\lfloor \dfrac{n^2}{5}\right\rfloor$ for some positive integer $n.$ What is the sum of these two prime numbers?
2014 ELMO Shortlist, 10
Find all positive integer bases $b \ge 9$ so that the number
\[ \frac{{\overbrace{11 \cdots 1}^{n-1 \ 1's}0\overbrace{77 \cdots 7}^{n-1\ 7's}8\overbrace{11 \cdots 1}^{n \ 1's}}_b}{3} \]
is a perfect cube in base 10 for all sufficiently large positive integers $n$.
[i]Proposed by Yang Liu[/i]
2019 Bosnia and Herzegovina Junior BMO TST, 4
$4.$ Let there be a variable positive integer whose last two digits are $3's$. Prove that this number is divisible by a prime greater than $7$.
2009 Thailand Mathematical Olympiad, 1
Let $a$ and $b$ be integers and $p$ a prime. For each positive integer k, define$ A_k = \{n \in Z^+ |p^k$ divides $a^n - b^n\}$. Show that if $A_1$ is nonempty then $A_k$ is nonempty for all positive integers $k$
2024 Kyiv City MO Round 1, Problem 1
Find the number of positive integers for which the product of digits and the sum of digits are the same and equal to $8$.
LMT Team Rounds 2021+, 13
Find the sum of $$\frac{\sigma(n) \cdot d(n)}{ \phi (n)}$$ over all positive $n$ that divide $ 60$.
Note: The function $d(i)$ outputs the number of divisors of $i$, $\sigma (i)$ outputs the sum of the factors of $i$, and $\phi (i)$ outputs the number of positive integers less than or equal to $i$ that are relatively prime to $i$.
2017 Irish Math Olympiad, 5
The sequence $a = (a_0, a_1,a_2,...)$ is defined by $a_0 = 0, a_1 =2$ and $$a_{n+2} = 2a_{n+1} + 41a_n$$Prove that $a_{2016}$ is divisible by $2017.$
2019 China Second Round Olympiad, 2
Find all the positive integers $n$ such that:
$(1)$ $n$ has at least $4$ positive divisors.
$(2)$ if all positive divisors of $n$ are $d_1,d_2,\cdots ,d_k,$ then $d_2-d_1,d_3-d_2,\cdots ,d_k-d_{k-1}$ form a geometric sequence.
2025 Abelkonkurransen Finale, 2b
Which positive integers $a$ have the property that \(n!-a\) is a perfect square for infinitely many positive integers \(n\)?
2023 Vietnam Team Selection Test, 4
Given are two coprime positive integers $a, b$ with $b$ odd and $a>2$. The sequence $(x_n)$ is defined by $x_0=2, x_1=a$ and $x_{n+2}=ax_{n+1}+bx_n$ for $n \geq 1$. Prove that:
$a)$ If $a$ is even then there do not exist positive integers $m, n, p$ such that $\frac{x_m} {x_nx_p}$ is a positive integer.
$b)$ If $a$ is odd then there do not exist positive integers $m, n, p$ such that $mnp$ is even and $\frac{x_m} {x_nx_p}$ is a perfect square.
2012 IMAC Arhimede, 4
Solve the following equations in the set of natural numbers:
a) $(5+11\sqrt2)^p=(11+5\sqrt2)^q$
b) $1005^x+2011^y=1006^z$
1991 Federal Competition For Advanced Students, 1
Suppose that $ a,b,$ and $ \sqrt[3]{a}\plus{}\sqrt[3]{b}$ are rational numbers. Prove that $ \sqrt[3]{a}$ and $ \sqrt[3]{b}$ are also rational.
2005 Taiwan TST Round 3, 2
Find all primes $p$ such that the number of distinct positive factors of $p^2+2543$ is less than 16.
2018 Purple Comet Problems, 23
Let $a, b$, and $c$ be integers simultaneously satisfying the equations $4abc + a + b + c = 2018$ and $ab + bc + ca = -507$. Find $|a| + |b|+ |c|$.
2013 Israel National Olympiad, 2
Let $A=\{n\in\mathbb{Z}\mid 0<n<2013\}$. A subset $B\subseteq A$ is called [b]reduced[/b] if for any two numbers $x,y\in B$, we must have $x\cdot y \notin B$. For example, any subset containing the numbers $3,5,15$ cannot be reduced, and same for a subset containing $4,16$.
[list=a]
[*] Find the maximal size of a reduced subset of $A$.
[*] How many reduced subsets are there with that maximal size?
[/list]
2006 Switzerland Team Selection Test, 1
Let $n$ be natural number and $1=d_1<d_2<\ldots <d_k=n$ be the positive divisors of $n$.
Find all $n$ such that $2n = d_5^2+ d_6^2 -1$.
2002 Iran MO (3rd Round), 20
$a_{0}=2,a_{1}=1$ and for $n\geq 1$ we know that : $a_{n+1}=a_{n}+a_{n-1}$
$m$ is an even number and $p$ is prime number such that $p$ divides $a_{m}-2$. Prove that $p$ divides $a_{m+1}-1$.
2014 Contests, 4
Let $x_1,x_2,\dots,x_{2014}$ be integers among which no two are congurent modulo $2014$. Let $y_1,y_2,\dots,y_{2014}$ be integers among which no two are congurent modulo $2014$. Prove that one can rearrange $y_1,y_2,\dots,y_{2014}$ to $z_1,z_2,\dots,z_{2014}$, so that among \[x_1+z_1,x_2+z_2,\dots,x_{2014}+z_{2014}\] no two are congurent modulo $4028$.
2008 Brazil Team Selection Test, 4
Find all odd integers $n$ for which $\frac{2^{\phi (n)}-1}{n}$ is a perfect square.
1983 Tournament Of Towns, (035) O4
The natural numbers $M$ and $K$ are represented by different permutations of the same digits. Prove that
(a) The sum of the digits of $2M$ equals the sum of the digits of $2K$.
(b) The sum of the digits of $M/2$ equals the sum of the digits of $K/2$ ($M, K$ both even).
(c) The sum of the digits of $5M$ equals the sum of the digits of $5 K$.
(AD Lisitskiy)
2024 Serbia National Math Olympiad, 6
Find all non-constant polynomials $P(x)$ with integer coefficients and positive leading coefficient, such that $P^{2mn}(m^2)+n^2$ is a perfect square for all positive integers $m, n$.
1994 AIME Problems, 1
The increasing sequence $3, 15, 24, 48, \ldots$ consists of those positive multiples of 3 that are one less than a perfect square. What is the remainder when the 1994th term of the sequence is divided by 1000?
2012 Abels Math Contest (Norwegian MO) Final, 1b
Every integer is painted white or black, so that if $m$ is white then $m + 20$ is also white, and if $k$ is black then $k + 35$ is also black. For which $n$ can exactly $n$ of the numbers $1, 2, ..., 50$ be white?
2010 Silk Road, 2
Let $N = 2010!+1$. Prove that
a) $N$ is not divisible by $4021$;
b) $N$ is not divisible by $2027,2029,2039$;
c)$ N$ has a prime divisor greater than $2050$.