Olympiad problems

2012 AMC 12/AHSME, 24

Define the function $f_1$ on the positive integers by setting $f_1(1)=1$ and if $n=p_1^{e_1}p_2^{e_2}...p_k^{e_k}$ is the prime factorization of $n>1$, then \[f_1(n)=(p_1+1)^{e_1-1}(p_2+1)^{e_2-1}\cdots (p_k+1)^{e_k-1}.\] For every $m \ge 2$, let $f_m(n)=f_1(f_{m-1}(n))$. For how many $N$ in the range $1 \le N \le 400$ is the sequence $(f_1(N), f_2(N), f_3(N),...)$ unbounded? [b]Note:[/b] a sequence of positive numbers is unbounded if for every integer $B$, there is a member of the sequence greater than $B$. $ \textbf{(A)}\ 15 \qquad\textbf{(B)}\ 16 \qquad\textbf{(C)}\ 17 \qquad\textbf{(D)}\ 18\qquad\textbf{(E)}\ 19 $

2015 China Northern MO, 3

Tags: number theory , prime factorization

If $n=p_1^{a_1},p_2^{a_2}...p_s^{a_s}$ then $\phi (n)=n \left(1- \frac{1}{p_1}\right)\left(1 - \frac{1}{p_2}\right)...\left(1- \frac{1}{p_s}\right)$. Find the smallest positive integer $n$ such that $\phi (n)=\frac{2^5}{47}n.$

2009 Spain Mathematical Olympiad, 4

Tags: prime factorization , number theory

Find all the integer pairs $ (x,y)$ such that: \[ x^2\minus{}y^4\equal{}2009\]

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 ($p$).

2013 AMC 12/AHSME, 15

Tags: factorial , induction , number theory , prime factorization

The number $2013$ is expressed in the form \[2013=\frac{a_1!a_2!\cdots a_m!}{b_1!b_2!\cdots b_n!},\] where $a_1\ge a_2\ge\cdots\ge a_m$ and $b_1\ge b_2\ge\cdots\ge b_n$ are positive integers and $a_1+b_1$ is as small as possible. What is $|a_1-b_1|$? ${ \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D}}\ 4\qquad\textbf{(E)}\ 5 $

1997 IMO Shortlist, 17

Tags: number theory , equation , prime factorization

Find all pairs $ (a,b)$ of positive integers that satisfy the equation: $ a^{b^2} \equal{} b^a$.

2010 Math Prize For Girls Problems, 8

Tags: prime factorization , number theory

When Meena turned 16 years old, her parents gave her a cake with $n$ candles, where $n$ has exactly 16 different positive integer divisors. What is the smallest possible value of $n$?

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 $

2005 AIME Problems, 12

Tags: number theory , prime factorization

For positive integers $n$, let $\tau (n)$ denote the number of positive integer divisors of $n$, including $1$ and $n$. For example, $\tau (1)=1$ and $\tau(6) =4$. Define $S(n)$ by \[S(n)=\tau(1)+ \tau(2) + ... + \tau(n).\] Let $a$ denote the number of positive integers $n \leq 2005$ with $S(n)$ odd, and let $b$ denote the number of positive integers $n \leq 2005$ with $S(n)$ even. Find $|a-b|$.

2013 Harvard-MIT Mathematics Tournament, 11

Tags: hmmt , number theory , prime factorization , prime number , binomial coefficient , pascal triangle

Compute the prime factorization of $1007021035035021007001$. (You should write your answer in the form $p_1^{e_1}p_2^{e_2}\ldots p_k^{e_k}$ where $p_1,\ldots,p_k$ are distinct prime numbers and $e_1,\ldots,e_k$ are positive integers.)

2005 AIME Problems, 2

Tags: arithmetic sequence , number theory , prime factorization

For each positive integer $k$, let $S_k$ denote the increasing arithmetic sequence of integers whose first term is $1$ and whose common difference is $k$. For example, $S_3$ is the sequence $1,4,7,10,...$. For how many values of $k$ does $S_k$ contain the term $2005$?

1998 Poland - First Round, 5

Tags: number theory , prime factorization

Find all pairs of positive integers $ x,y$ satisfying the equation \[ y^x \equal{} x^{50}\]

1991 AIME Problems, 5

Tags: number theory , prime number , prime factorization

Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $ 20!$ be the resulting product?

2023 Brazil Team Selection Test, 3

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

Let $Q$ be a set of prime numbers, not necessarily finite. For a positive integer $n$ consider its prime factorization: define $p(n)$ to be the sum of all the exponents and $q(n)$ to be the sum of the exponents corresponding only to primes in $Q$. A positive integer $n$ is called [i]special[/i] if $p(n)+p(n+1)$ and $q(n)+q(n+1)$ are both even integers. Prove that there is a constant $c>0$ independent of the set $Q$ such that for any positive integer $N>100$, the number of special integers in $[1,N]$ is at least $cN$. (For example, if $Q=\{3,7\}$, then $p(42)=3$, $q(42)=2$, $p(63)=3$, $q(63)=3$, $p(2022)=3$, $q(2022)=1$.)

2008 Paraguay Mathematical Olympiad, 1

Tags: prime factorization , number theory

How many positive integers $n < 500$ exist such that its prime factors are exclusively $2$, $7$, $11$, or a combination of these?

2013 USAMTS Problems, 4

Tags: inequalities , modular arithmetic , relatively prime , prime factorization , arithmetic sequence , number theory

Bunbury the bunny is hopping on the positive integers. First, he is told a positive integer $n$. Then Bunbury chooses positive integers $a,d$ and hops on all of the spaces $a,a+d,a+2d,\dots,a+2013d$. However, Bunbury must make these choices so that the number of every space that he hops on is less than $n$ and relatively prime to $n$. A positive integer $n$ is called [i]bunny-unfriendly[/i] if, when given that $n$, Bunbury is unable to find positive integers $a,d$ that allow him to perform the hops he wants. Find the maximum bunny-unfriendly integer, or prove that no such maximum exists.

2013 Harvard-MIT Mathematics Tournament, 7

Tags: hmmt , number theory , prime factorization

Find the number of positive divisors $d$ of $15!=15\cdot 14\cdot\cdots\cdot 2\cdot 1$ such that $\gcd(d,60)=5$.

2014 NIMO Problems, 1

Tags: function , geometry , 3d geometry , modular arithmetic , number theory , prime factorization , logarithm

Let $\eta(m)$ be the product of all positive integers that divide $m$, including $1$ and $m$. If $\eta(\eta(\eta(10))) = 10^n$, compute $n$. [i]Proposed by Kevin Sun[/i]

2006 AIME Problems, 4

Tags: factorial , number theory , prime factorization

Let $N$ be the number of consecutive 0's at the right end of the decimal representation of the product $1!\times2!\times3!\times4!\cdots99!\times100!.$ Find the remainder when $N$ is divided by 1000.

2012 Bulgaria National Olympiad, 2

Tags: geometric sequence , number theory , prime factorization , combinatorics

Prove that the natural numbers can be divided into two groups in a way that both conditions are fulfilled: 1) For every prime number $p$ and every natural number $n$, the numbers $p^n,p^{n+1}$ and $p^{n+2}$ do not have the same colour. 2) There does not exist an infinite geometric sequence of natural numbers of the same colour.

2010 Purple Comet Problems, 2

Tags: number theory , prime factorization

The prime factorization of $12 = 2 \cdot 2 \cdot 3$ has three prime factors. Find the number of prime factors in the factorization of $12! = 12 \cdot 11 \cdot 10 \cdot 9 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1.$

PEN P Problems, 43

Tags: number theory , prime factorization , additive number theory

A positive integer $n$ is abundant if the sum of its proper divisors exceeds $n$. Show that every integer greater than $89 \times 315$ is the sum of two abundant numbers.

2013 NIMO Problems, 8

Tags: euler , number theory , prime factorization

For a finite set $X$ define \[ S(X) = \sum_{x \in X} x \text{ and } P(x) = \prod_{x \in X} x. \] Let $A$ and $B$ be two finite sets of positive integers such that $\left\lvert A \right\rvert = \left\lvert B \right\rvert$, $P(A) = P(B)$ and $S(A) \neq S(B)$. Suppose for any $n \in A \cup B$ and prime $p$ dividing $n$, we have $p^{36} \mid n$ and $p^{37} \nmid n$. Prove that \[ \left\lvert S(A) - S(B) \right\rvert > 1.9 \cdot 10^{6}. \][i]Proposed by Evan Chen[/i]

2003 AMC 12-AHSME, 18

Tags: number theory , prime factorization

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$

2015 AMC 12/AHSME, 18

Tags: function , number theory , prime factorization

For every composite positive integer $n$, define $r(n)$ to be the sum of the factors in the prime factorization of $n$. For example, $r(50)=12$ because the prime factorization of $50$ is $ 2 \cdot 5^2 $, and $ 2 + 5 + 5 = 12 $. What is the range of the function $r$, $ \{ r(n) : n \ \text{is a composite positive integer} \} $? [b](A)[/b] the set of positive integers [b](B)[/b] the set of composite positive integers [b](C)[/b] the set of even positive integers [b](D)[/b] the set of integers greater than 3 [b](E)[/b] the set of integers greater than 4