Found problems: 364
Find all triples $(p, x, y)$ consisting of a prime number $p$ and two positive integers $x$ and $y$ such that $x^{p -1} + y$ and $x + y^ {p -1}$ are both powers of $p$.
[i]Proposed by Belgium[/i]
Let $p > 3$ be a prime number and $ x$ an integer, denote by $r ( x )\in \{ 0 , 1 , ... , p - 1 \}$ to the rest of $x$ modulo $p$ . Let $x_1, x_2, ... , x_k$ ( $2 < k < p$) different integers modulo $p$ and not divisible by $p$.
We say that a number $a \in \{ 1 , 2 ,..., p -1 \}$ is [i]good [/i] if $r ( a x_1) < r ( a x_2) <...< r ( a x_k)$.
Show that there are at most $\frac{2 p}{k + 1}-{ 1}$ [i]good [/i] numbers.
Let $p$ be a prime number. Determine all triples $(a,b,c)$ of positive integers such that $a + b + c < 2p\sqrt{p}$ and
$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{p}$
Let $n$ and $k$ be positive integers for which we have $4$ statements:
$i)$ $n+1$ is divisible with $k$
$ii)$ $n=2k+5$
$iii)$ $n+k$ is divisible with $3$
$iv)$ $n+7k$ is prime
Determine all possible values for $n$ and $k$, if out of the $4$ statements, three of them are true and one is false
For which $k$ the number $N = 101 ... 0101$ with $k$ ones is a prime?
Let $a, b, c, n$ be four integers, where n$\ge 2$, and let $p$ be a prime dividing both $a^2+ab+b^2$ and $a^n+b^n+c^n$, but not $a+b+c$. for instance, $a \equiv b \equiv -1 (mod \,\, 3), c \equiv 1 (mod \,\, 3), n$ a positive even integer, and $p = 3$ or $a = 4, b = 7, c = -13, n = 5$, and $p = 31$ satisfy these conditions. Show that $n$ and $p - 1$ are not coprime.
If $p$ and $q$ are natural numbers so that \[ \frac{p}{q}=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+ \ldots -\frac{1}{1318}+\frac{1}{1319}, \] prove that $p$ is divisible with $1979$.
A set S consisting of $2019$ (different) positive integers has the following property:
[i]the product of every 100 elements of $S$ is a divisor of the product of the remaining $1919$ elements[/i].
What is the maximum number of prime numbers that $S$ can contain?
Suppose $n$ is a natural number such that $4^n + 2^n + 1$ is a prime. Prove that $n = 3^k$ for some nonnegative integer $k$.
Find all prime numbers $ p $ for which $ 1 + p\cdot 2^{p} $ is a perfect square.
Find all positive prime numbers $p,q,r,s$ so that $p^2+2019=26(q^2+r^2+s^2)$.
Let $p > 3$ be a prime number, and let $F_p$ denote the (fi nite) set of residue classes modulo $p$.
Let $S_d$ denote the set of $2$-variable polynomials $P(x, y)$ with coefficients in $F_p$, total degree $\le d$, and satisfying $P(x, y) = P(y,- x -y)$. Show that $$|S_d| = p^{\lceil (d+1)(d+2)/6 \rceil}$$.
[i]The total degree of a $2$-variable polynomial $P(x, y)$ is the largest value of $i + j$ among monomials $x^iy^j$
[/i] appearing in $P$.
Let $p>3$ be a prime number and $n=\frac{2^{2p}-1}3$. Show that $n$ divides $2^n-2$.
Find all pairs of prime numbers $(p, q)$ for which $7pq^2 + p = q^3 + 43p^3 + 1$
A book is published in three volumes, the pages being numbered from $1$ onwards. The page numbers are continued from the first volume to the second volume to the third. The number of pages in the second volume is $50$ more than that in the first volume, and the number pages in the third volume is one and a half times that in the second. The sum of the page numbers on the first pages of the three volumes is $1709$. If $n$ is the last page number, what is the largest prime factor of $n$?
Let $p$ be a fixed prime. Find all pairs $(x,y)$ of positive numbers satisfying $p(x-y) = xy$.
Given $N = 2^ap_1p_2...p_m$, $m \ge 1$, $a \in N$ with $p_1, p_2,..., p_m$ are different primes. It is known that $\sigma (N) = 3N $ where $\sigma (N)$ is the sum of all positive integers which are factors of $N$. Show that there exists a prime number $p$ such that $2^p- 1$ is also a prime, and $2^p - 1|N$.
Vasya has a calculator that works with pairs of numbers. The calculator knows hoe to make a pair $(x+y,x)$ or a pair $(2x+y+1,x+y+1)$ from a pair $(x,y).$ At the beginning, the pair $(1,1)$ is presented on the calculator. Prove that for any natural $n$ there is exactly one pair $(n,k)$ that can be obtained using a calculator.
Suppose $\overline{a_1a_2...a_{2009}}$ is a $2009$-digit integer such that for each $i = 1,2,...,2007$, the $2$-digit integer $\overline{a_ia_{i+1}}$ contains $3$ distinct prime factors. Find $a_{2008}$
(Note: $\overline{xyz...}$ denotes an integer whose digits are $x, y,z,...$.)
Let $p > 10^9$ be a prime number such that $4p + 1$ is also prime.
Prove that the decimal expansion of $\frac{1}{4p+1}$ contains all the digits $0,1, \ldots, 9$.
A [i]semiprime [/i] is a positive integer that is a product of two prime numbers. For example, $9$ and $10$ are semiprimes. How many semiprimes less than $100$ are there?
A prime number has the property that however its decimal digits are permuted, the obtained number is also prime. Prove that this number has at most three different digits. Also prove a stronger statement.
We observe that number $10001=73\cdot137$ is not prime. Show that every member of infinite sequence $10001, 100010001, 1000100010001,...$ is not prime
Let $A$ be a set of $2025$ non-negative integers and $f: \mathbb{Z}_{>0} \to \mathbb{Z}_{>0}$ be a function with the following two properties:
1) For every two distinct positive integers $x,y$ there exists $a\in A$, such that $x-y$ divides $f(x+a) - f(y+a)$.
2) For every positive integer $N$ there exists a positive integer $t$ such that $f(x) \neq f(y)$ whenever $x,y \in [t, t+N]$ are distinct.
Prove that there are infinitely many primes $p$ such that $p$ divides $f(x)$ for some positive integer $x$.
Let $a,b$ be integers and $p$ be a prime number such that:
(i) $p$ is the greatest common divisor of $a$ and $b$;
(ii) $p^2$ divides $a$.
Prove that the polynomial $x^{n+2}+ax^{n+1}+bx^{n}+a+b$ cannot be decomposed into the product of two polynomials with integer coefficients and degree greater than $1$.