Found problems: 1048
Let $P(x)$ be a polynomial with real coefficients. Prove that there exist positive integers $n$ and $k$ such that $k$ has $n$ digits and more than $P(n)$ positive divisors.
$a)$ Let $p$ and $q$ be distinct prime numbers such that $p+q^2$ divides $p^2+q$. Prove that $p+q^2$ divides $pq-1$.
$b)$ Find all prime numbers $p$ such that $p+121$ divides $p^2+11$.
How many integers $n$ are there such that $n^4 + 2n^3 + 2n^2 + 2n + 1$ is a prime number?
Let $a$, $b$, $c$, $d$, $e$, $f$ and $g$ be seven distinct positive integers not bigger than $7$. Find all primes which can be expressed as $abcd+efg$
Let $p$ be a prime number and let $k$ be a positive integer. Suppose that the numbers $a_i=i^k+i$ for $i=0,1, \ldots,p-1$ form a complete residue system modulo $p$. What is the set of possible remainders of $a_2$ upon division by $p$?
Let \(p\) a prime number, and \(N\) the number of matrices \(p \times p\)
\[\begin{array}{cccc}
a_{11} & a_{12} & \ldots & a_{1p}\\
a_{21} & a_{22} & \ldots & a_{2p}\\
\vdots & \vdots & \ddots & \vdots \\
a_{p1} & a_{p2} & \ldots & a_{pp}
\end{array}\]
such that \(a_{ij} \in \{0,1,2,\ldots,p\} \) and if \(i \leq i^\prime\) and \(j \leq j^\prime\), then \(a_{ij} \leq a_{i^\prime j^\prime}\).
Find \(N \pmod{p}\).
Find all positive integers $a$, $b$, $n$ and prime numbers $p$ that satisfy
\[ a^{2013} + b^{2013} = p^n\text{.}\]
[i]Proposed by Matija Bucić.[/i]
How many of the first ten numbers of the sequence $121$, $11211$, $1112111$, ... are prime numbers?
$\textbf{(A) } 0 \qquad \textbf{(B) }1 \qquad \textbf{(C) }2 \qquad \textbf{(D) }3 \qquad \textbf{(E) }4$
Tita the Frog sits on the number line. She is initially on the integer number $k>1$. If she is sitting on the number $n$, she hops to the number $f(n)+g(n)$, where $f(n)$ and $g(n)$ are, respectively, the biggest and smallest positive prime numbers that divide $n$. Find all values of $k$ such that Tita can hop to infinitely many distinct integers.
On Qingqing Grassland, there are 7 sheep numberd $1,2,3,4,5,6,7$ and 2017 wolves numberd $1,2,\cdots,2017$. We have such strange rules:
(1) Define $P(n)$: the number of prime numbers that are smaller than $n$. Only when $P(i)\equiv j\pmod7$, wolf $i$ may eat sheep $j$ (he can also choose not to eat the sheep).
(2) If wolf $i$ eat sheep $j$, he will immediately turn into sheep $j$.
(3) If a wolf can make sure not to be eaten, he really wants to experience life as a sheep.
Assume that all wolves are very smart, then how many wolves will remain in the end?
The numbers from $1$ to $2015$ are written on sheets so that if if $n-m$ is a prime, then $n$ and $m$ are on different sheets. What is the minimum number of sheets required?
The positive difference between a pair of primes is equal to $2$, and the positive difference between the cubes of the two primes is $31106$. What is the sum of the digits of the least prime that is greater than those two primes?
$\textbf{(A) } 8 \qquad \textbf{(B) } 10 \qquad \textbf{(C) } 11 \qquad \textbf{(D) } 13 \qquad \textbf{(E) } 16$
Determine all prime numbers $p$ and all positive integers $x$ and $y$ satisfying $$x^3+y^3=p(xy+p).$$
Let $a$ be any integer. Prove that there are infinitely many primes $p$ such that \[ p\,|\,n^2+3\qquad\text{and}\qquad p\,|\,m^3-a \] for some integers $n$ and $m$.
Proove that if for a positive integer $n$ , both $3n + 1$ and $10n + 1$ are perfect squares , then $29n + 11$ is not a prime number.
We'll call a positive integer "almost prime", if it is not divisible by any prime from the interval $[3,19]$. We'll call a number "very non-prime", if it has at least 2 primes from interval $[3,19]$ dividing it. What is the greatest amount of almost prime numbers can be selected, such that the sum of any two of them is a very non-prime number?
[I]Proposed by S. Berlov, S. Ivanov[/i]
Determine all positive integer numbers $n$ satisfying the following condition:
the sum of the squares of any $n$ prime numbers greater than $3$ is divisible by $n$.
Consider an odd prime $p$ and a positive integer $N < 50p$. Let $a_1, a_2, \ldots , a_N$ be a list of positive integers less than $p$ such that any specific value occurs at most $\frac{51}{100}N$ times and $a_1 + a_2 + \cdots· + a_N$ is not divisible by $p$. Prove that there exists a permutation $b_1, b_2, \ldots , b_N$ of the $a_i$ such that, for all $k = 1, 2, \ldots , N$, the sum $b_1 + b_2 + \cdots + b_k$ is not divisible by $p$.
[i]Will Steinberg, United Kingdom[/i]
Find all pairs of natural numbers $(n,k)$ for which
$(n+1)^{k}-1 = n!$.
Prove that $n^4 + 4^{n}$ is composite for all values of $n$ greater than $1$.
For how many pairs of primes $(p, q)$, is $p^2 + 2pq^2 + 1$ also a prime?
A pair of positive integers $(x, y)$ is [i] good [/i] if they satisfy $\text{rad}(x) = \text{rad}(y)$ and they do not divide each-other. Given coprime positive integers $a$ and $b$, show that there exist infinitely many $n$ for which there exists a positive integer $m$ such that $(a^n + bm, b^n + am)$ is [i] good[/i].
(Here, $\text{rad}(x)$ denotes the product of $x$'s prime divisors, as usual.)
Let $n\ge2$ be an integer. Prove that if $k^2+k+n$ is prime for all integers $k$ such that $0\le k\le\sqrt{n\over3}$, then $k^2+k+n$ is prime for all integers $k$ such that $0\le k\le n-2$.[i](IMO Problem 6)[/i]
[b][i]Original Formulation[/i][/b]
Let $f(x) = x^2 + x + p$, $p \in \mathbb N.$ Prove that if the numbers $f(0), f(1), \cdots , f( \sqrt{p\over 3} )$ are primes, then all the numbers $f(0), f(1), \cdots , f(p - 2)$ are primes.
[i]Proposed by Soviet Union. [/i]
Let $p_1,p_2,\ldots,p_n$ be distinct primes greater than $3$. Show that $2^{p_1p_2\cdots p_n}+1$ has at least $4^n$ divisors.
$p$ is a prime and $m$ is a non-negative integer $< p-1$.
Show that $ \sum_{j=1}^p j^m$ is divisible by $p$.