This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 121

PEN A Problems, 58
Tags: divisibility theory
Let $k\ge 14$ be an integer, and let $p_k$ be the largest prime number which is strictly less than $k$. You may assume that $p_k\ge \tfrac{3k}{4}$. Let $n$ be a composite integer. Prove that [list=a] [*] if $n=2p_k$, then $n$ does not divide $(n-k)!$, [*] if $n>2p_k$, then $n$ divides $(n-k)!$. [/list]

PEN A Problems, 22
Tags: algebra , polynomial , binomial theorem , divisibility theory
Prove that the number \[\sum_{k=0}^{n}\binom{2n+1}{2k+1}2^{3k}\] is not divisible by $5$ for any integer $n\geq 0$.

PEN A Problems, 80
Tags: inequalities , algebra , polynomial , divisibility theory
Find all pairs of positive integers $m, n \ge 3$ for which there exist infinitely many positive integers $a$ such that \[\frac{a^{m}+a-1}{a^{n}+a^{2}-1}\] is itself an integer.

PEN A Problems, 78
Tags: divisibility theory
Determine all ordered pairs $(m, n)$ of positive integers such that \[\frac{n^{3}+1}{mn-1}\] is an integer.

PEN A Problems, 18
Tags: quadratic , algebra , difference of squares , special factorizations , divisibility theory
Let $m$ and $n$ be natural numbers and let $mn+1$ be divisible by $24$. Show that $m+n$ is divisible by $24$.

PEN A Problems, 60
Tags: divisibility theory
Prove that there exist an infinite number of ordered pairs $(a,b)$ of integers such that for every positive integer $t$, the number $at+b$ is a triangular number if and only if $t$ is a triangular number.

PEN A Problems, 106
Tags: floor function , divisibility theory
Determine the least possible value of the natural number $n$ such that $n!$ ends in exactly $1987$ zeros.

PEN A Problems, 38
Tags: floor function , divisibility theory
Let $p$ be a prime with $p>5$, and let $S=\{p-n^2 \vert n \in \mathbb{N}, {n}^{2}<p \}$. Prove that $S$ contains two elements $a$ and $b$ such that $a \vert b$ and $1<a<b$.

PEN A Problems, 118
Tags: function , divisibility theory
Determine the highest power of $1980$ which divides \[\frac{(1980n)!}{(n!)^{1980}}.\]

PEN A Problems, 117
Tags: divisibility theory
Find the smallest positive integer $n$ such that \[2^{1989}\; \vert \; m^{n}-1\] for all odd positive integers $m>1$.

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 $d=\gcd(b,p-1)$.

PEN A Problems, 100
Tags: calculus , integration , modular arithmetic , divisibility theory
Find all positive integers $n$ such that $n$ has exactly $6$ positive divisors $1<d_{1}<d_{2}<d_{3}<d_{4}<n$ and $1+n=5(d_{1}+d_{2}+d_{3}+d_{4})$.

2018 Moldova Team Selection Test, 12
Tags: floor function , modular arithmetic , divisibility theory
Let $p>3$ is a prime number and $k=\lfloor\frac{2p}{3}\rfloor$. Prove that \[{p \choose 1}+{p \choose 2}+\cdots+{p \choose k}\] is divisible by $p^{2}$.

PEN A Problems, 93
Tags: divisibility theory
Find the largest positive integer $n$ such that $n$ is divisible by all the positive integers less than $\sqrt[3]{n}$.

PEN A Problems, 109
Tags: divisibility theory
Find all positive integers $a$ and $b$ such that \[\frac{a^{2}+b}{b^{2}-a}\text{ and }\frac{b^{2}+a}{a^{2}-b}\] are both integers.

PEN A Problems, 33
Tags: divisibility theory
Let $a,b,x\in \mathbb{N}$ with $b>1$ and such that $b^{n}-1$ divides $a$. Show that in base $b$, the number $a$ has at least $n$ non-zero digits.

PEN A Problems, 92
Tags: divisibility theory
Let $a$ and $b$ be positive integers. When $a^{2}+b^{2}$ is divided by $a+b,$ the quotient is $q$ and the remainder is $r.$ Find all pairs $(a,b)$ such that $q^{2}+r=1977$.

PEN A Problems, 30
Tags: divisibility theory
Show that if $n \ge 6$ is composite, then $n$ divides $(n-1)!$.

PEN A Problems, 4
Tags: search , polynomial , vieta , symmetry , quadratic , divisibility theory , algebra
If $a, b, c$ are positive integers such that \[0 < a^{2}+b^{2}-abc \le c,\] show that $a^{2}+b^{2}-abc$ is a perfect square.

PEN A Problems, 29
Tags: divisibility theory
For which positive integers $k$, is it true that there are infinitely many pairs of positive integers $(m, n)$ such that \[\frac{(m+n-k)!}{m! \; n!}\] is an integer?

PEN A Problems, 39
Tags: divisibility theory
Let $n$ be a positive integer. Prove that the following two statements are equivalent. [list][*] $n$ is not divisible by $4$ [*] There exist $a, b \in \mathbb{Z}$ such that $a^{2}+b^{2}+1$ is divisible by $n$. [/list]

PEN A Problems, 56
Tags: modular arithmetic , induction , divisibility theory
Let $a, b$, and $c$ be integers such that $a+b+c$ divides $a^2 +b^2 +c^2$. Prove that there are infinitely many positive integers $n$ such that $a+b+c$ divides $a^n +b^n +c^n$.

PEN A Problems, 87
Tags: modular arithmetic , divisibility theory
Find all positive integers $n$ such that $3^{n}-1$ is divisible by $2^n$.

PEN A Problems, 81
Tags: divisibility theory
Determine all triples of positive integers $(a, m, n)$ such that $a^m +1$ divides $(a+1)^n$.

PEN A Problems, 16
Tags: induction , divisibility theory
Determine if there exists a positive integer $n$ such that $n$ has exactly $2000$ prime divisors and $2^{n}+1$ is divisible by $n$.