Found problems: 15460
1958 AMC 12/AHSME, 32
With $ \$1000$ a rancher is to buy steers at $ \$25$ each and cows at $ \$26$ each. If the number of steers $ s$ and the number of cows $ c$ are both positive integers, then:
$ \textbf{(A)}\ \text{this problem has no solution}\qquad\\
\textbf{(B)}\ \text{there are two solutions with }{s}\text{ exceeding }{c}\qquad \\
\textbf{(C)}\ \text{there are two solutions with }{c}\text{ exceeding }{s}\qquad \\
\textbf{(D)}\ \text{there is one solution with }{s}\text{ exceeding }{c}\qquad \\
\textbf{(E)}\ \text{there is one solution with }{c}\text{ exceeding }{s}$
2019 Malaysia National Olympiad, 4
Let $A=\{1,2,...,100\}$ and $f(k), k\in N$ be the size of the largest subset of $A$ such that no two elements differ by $k$. How many solutions are there to $f(k)=50$?
2004 Iran MO (2nd round), 4
$\mathbb{N}$ is the set of positive integers. Determine all functions $f:\mathbb{N}\to\mathbb{N}$ such that for every pair $(m,n)\in\mathbb{N}^2$ we have that:
\[f(m)+f(n) \ | \ m+n .\]
2023 Dutch IMO TST, 4
Find all positive integers $n$, such that $\sigma(n) =\tau(n) \lceil {\sqrt{n}} \rceil$.
2008 Gheorghe Vranceanu, 4
Find the largest natural number $ k $ which has the property that there is a partition of the natural numbers $ \bigcup_{1\le j\le k} V_j, $ an index $ i\in\{ 1,\ldots ,k \} $ and three natural numbers $ a,b,c\in V_i, $ satisfying $ a+2b=4c. $
2008 IMC, 1
Let $ n, k$ be positive integers and suppose that the polynomial $ x^{2k}\minus{}x^k\plus{}1$ divides $ x^{2n}\plus{}x^n\plus{}1$. Prove that $ x^{2k}\plus{}x^k\plus{}1$ divides $ x^{2n}\plus{}x^n\plus{}1$.
2019 Iran MO (2nd Round), 3
$x_1,x_2,...,x_n>1$ are natural numbers and $n \geq 3$
Prove that : $(x_1x_2...x_n)^2 \ne x_1^3 + x_2^3 +...+x_n^3$
2014 Cuba MO, 5
The number 2013 is written on a blackboard. Two players participate, alternating in turns, in the next game. A movement consists in changing the number that is on the board for the difference of this number and one of its divisors. The player who writes a zero loses. Which of the two players can guarantee victory?
2010 China Team Selection Test, 2
Given integer $a_1\geq 2$. For integer $n\geq 2$, define $a_n$ to be the smallest positive integer which is not coprime to $a_{n-1}$ and not equal to $a_1,a_2,\cdots, a_{n-1}$. Prove that every positive integer except 1 appears in this sequence $\{a_n\}$.
2015 China Northern MO, 7
Use $[x]$ to represent the greatest integer no more than a real number $x$. Let
$$S_n=\left[1+\frac12 +\frac13+...+\frac{1}{n}\right], \,\, (n =1,2,..,)$$ Prove that there are infinitely many $n$ such that $C_n^{S_n}$ is an even number.
[b]PS.[/b] [i]Attached is the original wording which forgets left [/i] [b][ [/b][i]. I hope it is ok where I put it.[/i]
2012 Romanian Masters In Mathematics, 4
Prove that there are infinitely many positive integers $n$ such that $2^{2^n+1}+1$ is divisible by $n$ but $2^n+1$ is not.
[i](Russia) Valery Senderov[/i]
2004 India IMO Training Camp, 4
Let $f$ be a bijection of the set of all natural numbers on to itself. Prove that there exists positive integers $a < a+d < a+ 2d$ such that $f(a) < f(a+d) <f(a+2d)$
1996 Greece National Olympiad, 3
Prove that among $81$ natural numbers whose prime divisors are in the set $\{2, 3, 5\}$ there exist four numbers whose product is the fourth power of an integer.
2015 Rioplatense Mathematical Olympiad, Level 3, 2
Let $a , b , c$ positive integers, coprime. For each whole number $n \ge 1$, we denote by $s ( n )$ the number of elements in the set $\{ a , b , c \}$ that divide $n$. We consider $k_1< k_2< k_3<...$ .the sequence of all positive integers that are divisible by some element of $\{ a , b , c \}$. Finally we define the characteristic sequence of $( a , b , c )$ like the succession $ s ( k_1) , s ( k_2) , s ( k_3) , .... $ .
Prove that if the characteristic sequences of $( a , b , c )$ and $( a', b', c')$ are equal, then $a = a', b = b'$ and $c=c'$
1992 IMO Longlists, 46
Prove that the sequence $5, 12, 19, 26, 33,\cdots $ contains no term of the form $2^n -1.$
2006 Iran MO (2nd round), 1
[b]a.)[/b] Let $m>1$ be a positive integer. Prove there exist finite number of positive integers $n$ such that $m+n|mn+1$.
[b]b.)[/b] For positive integers $m,n>2$, prove that there exists a sequence $a_0,a_1,\cdots,a_k$ from positive integers greater than $2$ that $a_0=m$, $a_k=n$ and $a_i+a_{i+1}|a_ia_{i+1}+1$ for $i=0,1,\cdots,k-1$.
2012 USAMO, 4
Find all functions $f:\mathbb{Z}^+ \rightarrow \mathbb{Z}^+$ (where $\mathbb{Z}^+$ is the set of positive integers) such that $f(n!) = f(n)!$ for all positive integers $n$ and such that $m-n$ divides $f(m) - f(n)$ for all distinct positive integers $m, n$.
2023 ISI Entrance UGB, 4
Let $n_1, n_2, \cdots , n_{51}$ be distinct natural numbers each of which has exactly $2023$ positive integer factors. For instance, $2^{2022}$ has exactly $2023$ positive integer factors $1,2, 2^{2}, 2^{3}, \cdots 2^{2021}, 2^{2022}$. Assume that no prime larger than $11$ divides any of the $n_{i}$'s. Show that there must be some perfect cube among the $n_{i}$'s.
1997 All-Russian Olympiad, 1
Find all integer solutions of the equation $(x^2 - y^2)^2 = 1 + 16y$.
[i]M. Sonkin[/i]
2011 IMO, 1
Given any set $A = \{a_1, a_2, a_3, a_4\}$ of four distinct positive integers, we denote the sum $a_1 +a_2 +a_3 +a_4$ by $s_A$. Let $n_A$ denote the number of pairs $(i, j)$ with $1 \leq i < j \leq 4$ for which $a_i +a_j$ divides $s_A$. Find all sets $A$ of four distinct positive integers which achieve the largest possible value of $n_A$.
[i]Proposed by Fernando Campos, Mexico[/i]
2012 USA Team Selection Test, 4
Find all positive integers $a,n\ge1$ such that for all primes $p$ dividing $a^n-1$, there exists a positive integer $m<n$ such that $p\mid a^m-1$.
2009 Peru Iberoamerican Team Selection Test, P1
A set $P$ has the following property: “For any positive integer $k$, if $p$ is a prime factor of $k^3+6$, then $p$ belongs to $P$ ”. Prove that $P$ is infinite.
2008 Dutch IMO TST, 4
Let $n$ be positive integer such that $\sqrt{1 + 12n^2}$ is an integer.
Prove that $2 + 2\sqrt{1 + 12n^2}$ is the square of an integer.
2022 BMT, Tie 3
Let $A$ be the product of all positive integers less than $1000$ whose ones or hundreds digit is $7$. Compute the remainder when $A/101$ is divided by $101$.
2023 Romania Team Selection Test, P1
Let $m$ and $n$ be positive integers, where $m < 2^n.$ Determine the smallest possible number of not necessarily pairwise distinct powers of two that add up to $m\cdot(2^n- 1).$
[i]The Problem Selection Committee[/i]