Olympiad problems

PEN K Problems, 2

Find all surjective functions $f: \mathbb{N}\to \mathbb{N}$ such that for all $m,n\in \mathbb{N}$: \[m \vert n \Longleftrightarrow f(m) \vert f(n).\]

2005 IMO Shortlist, 5

Tags: number theory , prime number , prime factorization

Denote by $d(n)$ the number of divisors of the positive integer $n$. A positive integer $n$ is called highly divisible if $d(n) > d(m)$ for all positive integers $m < n$. Two highly divisible integers $m$ and $n$ with $m < n$ are called consecutive if there exists no highly divisible integer $s$ satisfying $m < s < n$. (a) Show that there are only finitely many pairs of consecutive highly divisible integers of the form $(a, b)$ with $a\mid b$. (b) Show that for every prime number $p$ there exist infinitely many positive highly divisible integers $r$ such that $pr$ is also highly divisible.

PEN J Problems, 6

Tags: prime factorization , modular arithmetic , divisor function , quadratic , relatively prime , number theory

Show that if $m$ and $n$ are relatively prime positive integers, then $\phi( 5^m -1) \neq 5^{n}-1$.

MathLinks Contest 7th, 2.2

Tags: number theory , prime factorization , modular arithmetic , vector

For a prime $ p$ an a positive integer $ n$, denote by $ \nu_p(n)$ the exponent of $ p$ in the prime factorization of $ n!$. Given a positive integer $ d$ and a finite set $ \{p_1,p_2,\ldots, p_k\}$ of primes, show that there are infinitely many positive integers $ n$ such that $ \nu_{p_i}(n) \equiv 0 \pmod d$, for all $ 1\leq i \leq k$.

2011 Turkey Team Selection Test, 3

Tags: floor function , limit , greatest common divisor , prime factorization , number theory , induction

Let $t(n)$ be the sum of the digits in the binary representation of a positive integer $n,$ and let $k \geq 2$ be an integer. [b]a.[/b] Show that there exists a sequence $(a_i)_{i=1}^{\infty}$ of integers such that $a_m \geq 3$ is an odd integer and $t(a_1a_2 \cdots a_m)=k$ for all $m \geq 1.$ [b]b.[/b] Show that there is an integer $N$ such that $t(3 \cdot 5 \cdots (2m+1))>k$ for all integers $m \geq N.$

2010 AMC 12/AHSME, 3

Tags: prime factorization , number theory

A ticket to a school play costs $ x$ dollars, where $ x$ is a whole number. A group of 9th graders buys tickets costing a total of $ \$48$, and a group of 10th graders buys tickets costing a total of $ \$64$. How many values of $ x$ are possible? $ \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5$

Oliforum Contest II 2009, 1

Tags: function , modular arithmetic , prime factorization , number theory

Let $ \sigma(\cdot): \mathbb{N}_0 \to \mathbb{N}_0$ be the function from every positive integer $ n$ to the sum of divisors $ \sum_{d \mid n}{d}$ (i.e. $ \sigma(6) \equal{} 6 \plus{} 3 \plus{} 2 \plus{} 1$ and $ \sigma(8) \equal{} 8 \plus{} 4 \plus{} 2 \plus{} 1$). Find all primes $ p$ such that $ p \mid \sigma(p \minus{} 1)$. [i](Salvatore Tringali)[/i]

2016 Romania National Olympiad, 3

Tags: number theory , prime factorization

Find all the positive integers $p$ with the property that the sum of the first $p$ positive integers is a four-digit positive integer whose decomposition into prime factors is of the form $2^m3^n(m + n)$, where $m, n \in N^*$.

1998 South africa National Olympiad, 1

Tags: prime factorization , rational number , irrational number , logarithm , number theory

Is $\log_{10}{8}$ rational?

2011 Junior Balkan Team Selection Tests - Romania, 1

Tags: number theory , prime factorization , consecutive , prime

It is said that a positive integer $n > 1$ has the property ($p$) if in its prime factorization $n = p_1^{a_1} \cdot ... \cdot p_j^{a_j}$ at least one of the prime factors $p_1, ... , p_j$ has the exponent equal to $2$. a) Find the largest number $k$ for which there exist $k$ consecutive positive integers that do not have the property ($p$). b) Prove that there is an infinite number of positive integers $n$ such that $n, n + 1$ and $n + 2$ have the property ($p$).

1963 AMC 12/AHSME, 8

Tags: prime factorization , number theory

The smallest positive integer $x$ for which $1260x=N^3$, where $N$ is an integer, is: $\textbf{(A)}\ 1050 \qquad \textbf{(B)}\ 1260 \qquad \textbf{(C)}\ 1260^2 \qquad \textbf{(D)}\ 7350 \qquad \textbf{(E)}\ 44100$

2001 Bundeswettbewerb Mathematik, 4

Tags: number theory , prime factorization , prime number

Prove: For each positive integer is the number of divisors whose decimal representations ends with a 1 or 9 not less than the number of divisors whose decimal representations ends with 3 or 7.

1977 IMO Longlists, 27

Tags: number theory , prime number , prime factorization , dirichlet s theorem

Let $n$ be a given number greater than 2. We consider the set $V_n$ of all the integers of the form $1 + kn$ with $k = 1, 2, \ldots$ A number $m$ from $V_n$ is called indecomposable in $V_n$ if there are not two numbers $p$ and $q$ from $V_n$ so that $m = pq.$ Prove that there exist a number $r \in V_n$ that can be expressed as the product of elements indecomposable in $V_n$ in more than one way. (Expressions which differ only in order of the elements of $V_n$ will be considered the same.)

2003 AMC 12-AHSME, 18

Tags: prime factorization , number theory

Let $ x$ and $ y$ be positive integers such that $ 7x^5 \equal{} 11y^{13}$. The minimum possible value of $ x$ has a prime factorization $ a^cb^d$. What is $ a \plus{} b \plus{} c \plus{} d$? $ \textbf{(A)}\ 30 \qquad \textbf{(B)}\ 31 \qquad \textbf{(C)}\ 32 \qquad \textbf{(D)}\ 33 \qquad \textbf{(E)}\ 34$

2008 ITest, 29

Tags: number theory , linear algebra , prime factorization , matrix

Find the number of ordered triplets $(a,b,c)$ of positive integers such that $abc=2008$ (the product of $a$, $b$, and $c$ is $2008$).

2013 AMC 12/AHSME, 17

Tags: number theory , prime factorization

A group of $ 12 $ pirates agree to divide a treasure chest of gold coins among themselves as follows. The $ k^\text{th} $ pirate to take a share takes $ \frac{k}{12} $ of the coins that remain in the chest. The number of coins initially in the chest is the smallest number for which this arrangement will allow each pirate to receive a positive whole number of coins. How many coins does the $ 12^{\text{th}} $ pirate receive? $ \textbf{(A)} \ 720 \qquad \textbf{(B)} \ 1296 \qquad \textbf{(C)} \ 1728 \qquad \textbf{(D)} \ 1925 \qquad \textbf{(E)} \ 3850 $

2012 AMC 8, 18

Tags: number theory , prime factorization

What is the smallest positive integer that is neither prime nor square and that has no prime factor less than 50? $\textbf{(A)}\hspace{.05in}3127 \qquad \textbf{(B)}\hspace{.05in}3133 \qquad \textbf{(C)}\hspace{.05in}3137 \qquad \textbf{(D)}\hspace{.05in}3139 \qquad \textbf{(E)}\hspace{.05in}3149 $

2013 AMC 10, 21

Tags: number theory , prime factorization

A group of $ 12 $ pirates agree to divide a treasure chest of gold coins among themselves as follows. The $ k^\text{th} $ pirate to take a share takes $ \frac{k}{12} $ of the coins that remain in the chest. The number of coins initially in the chest is the smallest number for which this arrangement will allow each pirate to receive a positive whole number of coins. How many coins does the $ 12^{\text{th}} $ pirate receive? $ \textbf{(A)} \ 720 \qquad \textbf{(B)} \ 1296 \qquad \textbf{(C)} \ 1728 \qquad \textbf{(D)} \ 1925 \qquad \textbf{(E)} \ 3850 $

2011 Federal Competition For Advanced Students, Part 1, 1

Tags: calculus , modular arithmetic , integration , quadratic , prime factorization , number theory

Determine all integer triplets $(x,y,z)$ such that \[x^4+x^2=7^zy^2\mbox{.}\]

2018 PUMaC Number Theory A, 2

Tags: prime factorization , number theory

For a positive integer $n$, let $f(n)$ be the number of (not necessarily distinct) primes in the prime factorization of $k$. For example, $f(1) = 0, f(2) = 1, $ and $f(4) = f(6) = 2$. let $g(n)$ be the number of positive integers $k \leq n$ such that $f(k) \geq f(j)$ for all $j \leq n$. Find $g(1) + g(2) + \ldots + g(100)$.

1992 Dutch Mathematical Olympiad, 1

Tags: probability , prime factorization , number theory

Four dice are thrown. What is the probability that the product of the number equals $ 36?$

2004 Indonesia MO, 1

Tags: geometry , number theory , 3d geometry , prime factorization

Determine the number of positive odd and even factor of $ 5^6\minus{}1$.

2004 China Team Selection Test, 1

Tags: number theory , prime factorization

Let $ m_1$, $ m_2$, $ \cdots$, $ m_r$ (may not distinct) and $ n_1$, $ n_2$ $ \cdots$, $ n_s$ (may not distinct) be two groups of positive integers such that for any positive integer $ d$ larger than $ 1$, the numbers of which can be divided by $ d$ in group $ m_1$, $ m_2$, $ \cdots$, $ m_r$ (including repeated numbers) are no less than that in group $ n_1$, $ n_2$ $ \cdots$, $ n_s$ (including repeated numbers). Prove that $ \displaystyle \frac{m_1 \cdot m_2 \cdots m_r}{n_1 \cdot n_2 \cdots n_s}$ is integer.

2010 AMC 10, 8

Tags: number theory , prime factorization

A ticket to a school play costs $ x$ dollars, where $ x$ is a whole number. A group of 9th graders buys tickets costing a total of $ \$48$, and a group of 10th graders buys tickets costing a total of $ \$64$. How many values of $ x$ are possible? $ \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5$

PEN A Problems, 69

Tags: modular arithmetic , prime factorization , divisibility theory , number theory

Prove that if the odd prime $p$ divides $a^{b}-1$, where $a$ and $b$ are positive integers, then $p$ appears to the same power in the prime factorization of $b(a^{d}-1)$, where $d=\gcd(b,p-1)$.