Found problems: 471
Find all positive integers $n$ such that $4^n+6^n+9^n$ is a square.
[i]David Yang, Alex Zhu.[/i]
Find the least positive integer $n$ such that $15$ divides the product
\[a_1a_2\dots a_{15}\left (a_1^n+a_2^n+\dots+a_{15}^n \right )\]
, for every positive integers $a_1, a_2, \dots, a_{15}$.
The number $21982145917308330487013369$ is the thirteenth power of a positive integer. Which positive integer?
For positive integral $k>1$, we let $p(k)$ be its smallest prime divisor. Given an integer $a_1>2$, we define an infinite sequence $a_n$ by $a_{n+1}=a_n^n-1$ for each $n\geq 1$. For which values of $a_1$ is the sequence $p(a_n)$ bounded?
Find all pairs of positive integers $(a,b)$ satisfying the following conditions:
[list]
[*] $a$ divides $b^4+1$,
[*] $b$ divides $a^4+1$,
[*] $\lfloor\sqrt{a}\rfloor=\lfloor \sqrt{b}\rfloor$.
[/list]
[i]Yang Liu[/i]
Find all integers $x$ and $y$ such that $x^3\pm y^3 =2001p$, where $p$ is prime.
Determine the remainder when $$2^6\cdot3^{10}\cdot5^{12}-75^4\left(26^2-1\right)^2+3^{10}-50^6+5^{12}$$ is divided by $1001$.
[i]2016 CCA Math Bonanza Lightning #4.1[/i]
Let $p$ be an odd prime number. Suppose $P$ and $Q$ are polynomials with integer coefficients such that $P(0)=Q(0)=1$, there is no nonconstant polynomial dividing both $P$ and $Q$, and
\[
1 + \cfrac{x}{1 + \cfrac{2x}{1 + \cfrac{\ddots}{1 +
(p-1)x}}}=\frac{P(x)}{Q(x)}.
\]
Show that all coefficients of $P$ except for the constant coefficient are divisible by $p$, and all coefficients of $Q$ are [i]not[/i] divisible by $p$.
[i]Andrew Gu[/i]
Determine all functions $f: \mathbb{Z}\to\mathbb{Z}$ satisfying \[f\big(f(m)+n\big)+f(m)=f(n)+f(3m)+2014\] for all integers $m$ and $n$.
[i]Proposed by Netherlands[/i]
Let $n$ be the smallest positive integer such that the number obtained by taking $n$’s rightmost digit (decimal expansion) and moving it to be the leftmost digit is $7$ times $n$. Determine the number of digits in $n$.
Let $a_1 < a_2 < \cdots <a_n$ be pairwise coprime positive integers with $a_1$ being prime and $a_1 \ge n + 2$. On the segment $I = [0, a_1 a_2 \cdots a_n ]$ of the real line, mark all integers that are divisible by at least one of the numbers $a_1 , \ldots , a_n$ . These points split $I$ into a number of smaller segments. Prove that the sum of the squares of the lengths of these segments is divisible by $a_1$.
[i]Proposed by Serbia[/i]
Let $a$ be a natural number, $a>3$. Prove there is an infinity of numbers n, for which $a+n|a^{n}+1$
If $n\ge 4,\ n\in\mathbb{N^*},\ n\mid (2^n-2)$. Prove that $\frac{2^n-2}{n}$ is not a prime number.
Find all triples $(a,b,c)$ of positive integers such that if $n$ is not divisible by any prime less than $2014$, then $n+c$ divides $a^n+b^n+n$.
[i]Proposed by Evan Chen[/i]
Prove that if $ p$ is a prime number, then $ 7p+3^{p}-4$ is not a perfect square.
Find all positive integers $a$ and $b$ for which there are three consecutive integers at which the polynomial \[ P(n) = \frac{n^5+a}{b} \] takes integer values.
Let $ p \geq 2$ be a prime number. Eduardo and Fernando play the following game making moves alternately: in each move, the current player chooses an index $i$ in the set $\{0,1,2,\ldots, p-1 \}$ that was not chosen before by either of the two players and then chooses an element $a_i$ from the set $\{0,1,2,3,4,5,6,7,8,9\}$. Eduardo has the first move. The game ends after all the indices have been chosen .Then the following number is computed:
$$M=a_0+a_110+a_210^2+\cdots+a_{p-1}10^{p-1}= \sum_{i=0}^{p-1}a_i.10^i$$.
The goal of Eduardo is to make $M$ divisible by $p$, and the goal of Fernando is to prevent this.
Prove that Eduardo has a winning strategy.
[i]Proposed by Amine Natik, Morocco[/i]
An integer $N$ is selected at random in the range $1\le N \le 2020.$ What is the probability that the remainder when $N^{16}$ is divided by $5$ is $1$?
$\textbf{(A)} \text{ }\frac{1}{5} \qquad \textbf{(B)} \text{ }\frac{2}{5} \qquad \textbf{(C)} \text{ }\frac{3}{5} \qquad \textbf{(D)} \text{ }\frac{4}{5} \qquad \textbf{(E)} \text{ 1}$
An infinite arithmetic progression whose terms are positive integers contains the square of an integer and the cube of an integer. Show that it contains the sixth power of an integer.
Define a function $f: \mathbb N \to \mathbb N$ by $f(1) = 1$, $f(n+1) = f(n) + 2^{f(n)}$ for every positive integer $n$. Prove that $f(1), f(2), \dots, f(3^{2013})$ leave distinct remainders when divided by $3^{2013}$.
For a given prime $ p > 2$ and positive integer $ k$ let \[ S_k \equal{} 1^k \plus{} 2^k \plus{} \ldots \plus{} (p \minus{} 1)^k\] Find those values of $ k$ for which $ p \, |\, S_k$.
Determine all functions $f: \mathbb{Z}\to\mathbb{Z}$ satisfying \[f\big(f(m)+n\big)+f(m)=f(n)+f(3m)+2014\] for all integers $m$ and $n$.
[i]Proposed by Netherlands[/i]
Let $p$ and $q$ be prime numbers and $\{a_{n}\}_{n=1}^{\infty}$ be a sequence of integers defined by:
\[a_{0}=0, a_{1}=1, a_{n+2}=pa_{n+1}-qa_{n}\quad\forall n\geq 0\]
Find $p$ and $q$ if there exists an integer $k$ such that $a_{3k}=-3$.
A natural number $n$ is said to have the property $P,$ if, for all $a, n^2$ divides $a^n - 1$ whenever $n$ divides $a^n - 1.$
a.) Show that every prime number $n$ has property $P.$
b.) Show that there are infinitely many composite numbers $n$ that possess property $P.$
Let u be a fixed positive integer. Prove that the equation $n! = u^{\alpha} - u^{\beta}$ has a finite number of solutions $(n, \alpha, \beta).$