Olympiad problems

2014 Putnam, 1

Prove that every nonzero coefficient of the Taylor series of $(1-x+x^2)e^x$ about $x=0$ is a rational number whose numerator (in lowest terms) is either $1$ or a prime number.

PEN J Problems, 6

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

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

2013 AMC 10, 20

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 $

2016 CMIMC, 4

Tags: function , number theory , prime factorization

For some positive integer $n$, consider the usual prime factorization \[n = \displaystyle \prod_{i=1}^{k} p_{i}^{e_{i}}=p_1^{e_1}p_2^{e_2}\ldots p_k^{e_k},\] where $k$ is the number of primes factors of $n$ and $p_{i}$ are the prime factors of $n$. Define $Q(n), R(n)$ by \[ Q(n) = \prod_{i=1}^{k} p_{i}^{p_{i}} \text{ and } R(n) = \prod_{i=1}^{k} e_{i}^{e_{i}}. \] For how many $1 \leq n \leq 70$ does $R(n)$ divide $Q(n)$?

1977 IMO Shortlist, 10

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.)

2014 Putnam, 1

PEN J Problems, 6

2013 AMC 10, 20

2016 CMIMC, 4

1977 IMO Shortlist, 10

1963 AMC 12/AHSME, 8

2019 Malaysia National Olympiad, 6

2000 AIME Problems, 4