Found problems: 15460
1991 AMC 12/AHSME, 14
If $x$ is the cube of a positive integer and $d$ is the number of positive integers that are divisors of $x$, then $d$ could be
$ \textbf{(A)}\ 200\qquad\textbf{(B)}\ 201\qquad\textbf{(C)}\ 202\qquad\textbf{(D)}\ 203\qquad\textbf{(E)}\ 204 $
2010 Ukraine Team Selection Test, 12
Is there a positive integer $n$ for which the following holds:
for an arbitrary rational $r$ there exists an integer $b$ and non-zero integers $a _1, a_2, ..., a_n$ such that $r=b+\frac{1}{a_1}+\frac{1}{a_2}+...+\frac{1}{a_n}$ ?
2022 Switzerland - Final Round, 5
For an integer $a \ge 2$, denote by $\delta_(a) $ the second largest divisor of $a$. Let $(a_n)_{n\ge 1}$ be a sequence
of integers such that $a_1 \ge 2$ and $$a_{n+1} = a_n + \delta_(a_n)$$
for all $n \ge 1$. Prove that there exists a positive integer $k$ such that $a_k$ is divisible by $3^{2022}$.
2017 Federal Competition For Advanced Students, P2, 3
Let $(a_n)_{n\ge 0}$ be the sequence of rational numbers with $a_0 = 2016$ and $a_{n+1} = a_n + \frac{2}{a_n}$ for all $n \ge 0$.
Show that the sequence does not contain a square of a rational number.
Proposed by Theresia Eisenkölbl
2011 Rioplatense Mathematical Olympiad, Level 3, 1
Given a positive integer $n$, an operation consists of replacing $n$ with either $2n-1$, $3n-2$ or $5n-4$. A number $b$ is said to be a [i]follower[/i] of number $a$ if $b$ can be obtained from $a$ using this operation multiple times. Find all positive integers $a < 2011$ that have a common follower with $2011$.
2009 Tournament Of Towns, 3
Alex is going to make a set of cubical blocks of the same size and to write a digit on each of their faces so that it would be possible to form every $30$-digit integer with these blocks. What is the minimal number of blocks in a set with this property? (The digits $6$ and $9$ do not turn one into another.)
2008 India Regional Mathematical Olympiad, 5
Let $N$ be a ten digit positive integer divisible by $7$. Suppose the first and the last digit of $N$ are interchanged and the resulting number (not necessarily ten digit) is also divisible by $7$ then we say that $N$ is a good integer. How many ten digit good integers are there?
2006 Estonia Math Open Junior Contests, 4
Does there exist a natural number with the sum of digits of its $ kth$ power being
equal to $ k$, if a) $ k \equal{} 2004$; b) $ k \equal{} 2006?$
2013 Saudi Arabia BMO TST, 4
Find all positive integers $n < 589$ for which $589$ divides $n^2 + n + 1$.
1985 USAMO, 1
Determine whether or not there are any positive integral solutions of the simultaneous equations
\begin{align*}x_1^2+x_2^2+\cdots+x_{1985}^2&=y^3,\\
x_1^3+x_2^3+\cdots+x_{1985}^3&=z^2\end{align*}
with distinct integers $x_1$, $x_2$, $\ldots$, $x_{1985}$.
KoMaL A Problems 2019/2020, A. 769
Find all triples $(a,b,c)$ of distinct positive integers so that there exists a subset $S$ of the positive integers for which for all positive integers $n$ exactly one element of the triple $(an,bn,cn)$ is in $S$.
Proposed by Carl Schildkraut, MIT
2012 Czech-Polish-Slovak Junior Match, 5
Find all triplets $(a, k, m)$ of positive integers that satisfy the equation $k + a^k = m + 2a^m$.
2003 Abels Math Contest (Norwegian MO), 2a
Find all pairs $(x, y)$ of integers numbers such that $y^3+5=x(y^2+2)$
2006 All-Russian Olympiad, 5
Let $a_1$, $a_2$, ..., $a_{10}$ be positive integers such that $a_1<a_2<...<a_{10}$. For every $k$, denote by $b_k$ the greatest divisor of $a_k$ such that $b_k<a_k$. Assume that $b_1>b_2>...>b_{10}$. Show that $a_{10}>500$.
1948 Kurschak Competition, 1
Knowing that $23$ October $1948$ was a Saturday, which is more frequent for New Year’s Day, Sunday or Monday?
2013 Cuba MO, 4
A subset of the set $\{1, 2, 3, ..., 30\}$ is called [i]delicious [/i ]if it doesn't contain elements a and b such that $a = 3b$. A [i]delicious[/i] subset It is called [i]super delicious[/i] if, in addition to being delicious, it is verified that no [i]delicious[/i] subset has more elements than it has. Determine the number of [i]super delicious[/i] subsets
1987 IMO Shortlist, 8
(a) Let $\gcd(m, k) = 1$. Prove that there exist integers $a_1, a_2, . . . , a_m$ and $b_1, b_2, . . . , b_k$ such that each product $a_ib_j$ ($i = 1, 2, \cdots ,m; \ j = 1, 2, \cdots, k$) gives a different residue when divided by $mk.$
(b) Let $\gcd(m, k) > 1$. Prove that for any integers $a_1, a_2, . . . , a_m$ and $b_1, b_2, . . . , b_k$ there must be two products $a_ib_j$ and $a_sb_t$ ($(i, j) \neq (s, t)$) that give the same residue when divided by $mk.$
[i]Proposed by Hungary.[/i]
2006 MOP Homework, 2
Determine all pairs of positive integers $(m,n)$ such that m is but divisible by every integer from $1$ to $n$ (inclusive), but not divisible by $n + 1, n + 2$, and $n + 3$.
2002 Baltic Way, 20
Does there exist an infinite non-constant arithmetic progression, each term of which is of the form $a^b$, where $a$ and $b$ are positive integers with $b\ge 2$?
1965 All Russian Mathematical Olympiad, 059
A bus ticket is considered to be lucky if the sum of the first three digits equals to the sum of the last three ($6$ digits in Russian buses). Prove that the sum of all the lucky numbers is divisible by $13$.
1998 Cono Sur Olympiad, 3
Prove that, least $30$% of the natural numbers $n$ between $1$ and $1000000$ the first digit of $2^n$ is $1$.
2023 Balkan MO Shortlist, N2
Find all positive integers, such that there exist positive integers $a, b, c$, satisfying $\gcd(a, b, c)=1$ and $n=\gcd(ab+c, ac-b)=a+b+c$.
1976 Miklós Schweitzer, 3
Let $ H$ denote the set of those natural numbers for which $ \tau(n)$ divides $ n$, where $ \tau(n)$ is the number of divisors of $ n$. Show that
a) $ n! \in H$ for all sufficiently large $ n$,
b)$ H$ has density $ 0$.
[i]P. Erdos[/i]
2020 Malaysia IMONST 1, 9
What is the smallest positive multiple of $225$ that can be written using
digits $0$ and $1$ only?
2011 APMO, 1
Let $a,b,c$ be positive integers. Prove that it is impossible to have all of the three numbers $a^2+b+c,b^2+c+a,c^2+a+b$ to be perfect squares.