Found problems: 121
Determine all pairs of integers $(a, b)$ such that \[\frac{a^{2}}{2ab^{2}-b^{3}+1}\] is a positive integer.
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 $a^n +b^n +c^n$.
Let $p(n)$ be the greatest odd divisor of $n$. Prove that \[\frac{1}{2^{n}}\sum_{k=1}^{2^{n}}\frac{p(k)}{k}> \frac{2}{3}.\]
Find infinitely many triples $(a, b, c)$ of positive integers such that $a$, $b$, $c$ are in arithmetic progression and such that $ab+1$, $bc+1$, and $ca+1$ are perfect squares.
Suppose that $a$ and $b$ are natural numbers such that \[p=\frac{b}{4}\sqrt{\frac{2a-b}{2a+b}}\] is a prime number. What is the maximum possible value of $p$?
Prove that for every $n \in \mathbb{N}$ the following proposition holds: $7|3^n +n^3$ if and only if $7|3^{n} n^3 +1$.
If $n$ is a natural number, prove that the number $(n+1)(n+2)\cdots(n+10)$ is not a perfect square.
Find all integers $\,a,b,c\,$ with $\,1<a<b<c\,$ such that \[(a-1)(b-1)(c-1)\hspace{0.2in}\text{is a divisor of}\hspace{0.2in}abc-1.\]
Let $a,b,c$ and $d$ be odd integers such that $0<a<b<c<d$ and $ad=bc$. Prove that if $a+d=2^{k}$ and $b+c=2^{m}$ for some integers $k$ and $m$, then $a=1$.
Show that every integer $k>1$ has a multiple less than $k^4$ whose decimal expansion has at most four distinct digits.
Prove that the expression \[\frac{\gcd(m, n)}{n}{n \choose m}\] is an integer for all pairs of positive integers $(m, n)$ with $n \ge m \ge 1$.
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.
Show that for every natural number $n$ the product \[\left( 4-\frac{2}{1}\right) \left( 4-\frac{2}{2}\right) \left( 4-\frac{2}{3}\right) \cdots \left( 4-\frac{2}{n}\right)\] is an integer.
Let $n$ be a positive integer with $n \ge 3$. Show that \[n^{n^{n^{n}}}-n^{n^{n}}\] is divisible by $1989$.
For each integer $n>1$, let $p(n)$ denote the largest prime factor of $n$. Determine all triples $(x, y, z)$ of distinct positive integers satisfying [list] [*] $x, y, z$ are in arithmetic progression, [*] $p(xyz) \le 3$. [/list]
Suppose that $p$ is a prime number and is greater than $3$. Prove that $7^{p}-6^{p}-1$ is divisible by $43$.
There is a large pile of cards. On each card one of the numbers $1$, $2$, $\cdots$, $n$ is written. It is known that the sum of all numbers of all the cards is equal to $k \cdot n!$ for some integer $k$. Prove that it is possible to arrange cards into $k$ stacks so that the sum of numbers written on the cards in each stack is equal to $n!$.
(Wolstenholme's Theorem) Prove that if \[1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{p-1}\] is expressed as a fraction, where $p \ge 5$ is a prime, then $p^{2}$ divides the numerator.
Show that there exist infinitely many positive integers $n$ such that $n^{2}+1$ divides $n!$.
Let $f(x)=x^3 +17$. Prove that for each natural number $n \ge 2$, there is a natural number $x$ for which $f(x)$ is divisible by $3^n$ but not $3^{n+1}$.
Determine all three-digit numbers $N$ having the property that $N$ is divisible by $11,$ and $\frac{N}{11}$ is equal to the sum of the squares of the digits of $N.$