Found problems: 1048
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 $p>$ be a prime number and $S$ be a set of $p+1$ integers. Prove that there exist pairwise distinct numbers $a_1,a_2,...,a_{p-1}\in S$ that
$$ a_1+2a_2+3a_3+...+(p-1)a_{p-1}$$ is divisible by $p$.
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 $P$.
Consider the set $E$ of all natural numbers $n$ such that whenn divided by $11, 12, 13$, respectively, the remainders, int that order, are distinct prime numbers in an arithmetic progression. If $N$ is the largest number in $E$, find the sum of digits of $N$.
Let $p>3$ be a prime number and $n=\frac{2^{2p}-1}3$. Show that $n$ divides $2^n-2$.
The sum of two prime numbers is $85$. What is the product of these two prime numbers?
$\textbf{(A) }85\qquad\textbf{(B) }91\qquad\textbf{(C) }115\qquad\textbf{(D) }133\qquad \textbf{(E) }166$
Find all pairs of prime numbers $(p, q)$ for which $7pq^2 + p = q^3 + 43p^3 + 1$
Determine all integers $m \geq 2$ such that every $n$ with $\frac{m}{3} \leq n \leq \frac{m}{2}$ divides the binomial coefficient $\binom{n}{m-2n}$.
Let be a prime number $ p, $ the quotient ring $ R=\mathbb{Z}[X,Y]/(pX,pY), $ and a prime ideal $ I\supset pA $ that is not maximal. Show that the ring $ \left\{ r/i|r\in R, i\in I \right\} $ is factorial.
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 $p(x-y) = xy$.
A positive integer \(n\) is called perfect if the sum of its positive divisors \(\sigma(n)\) is twice \(n\), that is, \(\sigma(n) = 2n\). For example, \(6\) is a perfect number since the sum of its positive divisors is \(1 + 2 + 3 + 6 = 12\), which is twice \(6\). Prove that if \(n\) is a positive perfect integer, then:
\[
\sum_{p|n} \frac{1}{p + 1} < \ln 2 < \sum_{p|n} \frac{1}{p - 1}
\]
where the sums are taken over all prime divisors \(p\) of \(n\).
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 $2^p - 1|N$.
Let $n \geq 4$ be an integer. Consider a regular $2n-$gon for which to every vertex, an integer is assigned, which we call the value of said vertex. If four distinct vertices of this $2n-$gon form a rectangle, we say that the sum of the values of these vertices is a rectangular sum.
Determine for which (not necessarily positive) integers $m$ the integers $m + 1, m + 2, . . . , m + 2n$ can be assigned to the vertices (in some order) in such a way that every rectangular sum is a prime number. (Prime numbers are positive by definition.)
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 $p \neq 5$ be a prime number. Prove that $p^5-1$ has a prime divisor of the form $5x+1$.
Determine the maximal size of the set $S$ such that:
i) all elements of $S$ are natural numbers not exceeding $100$;
ii) for any two elements $a,b$ in $S$, there exists $c$ in $S$ such that $(a,c)=(b,c)=1$;
iii) for any two elements $a,b$ in $S$, there exists $d$ in $S$ such that $(a,d)>1,(b,d)>1$.
[i]Yao Jiangang[/i]
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 $0,1, \ldots, 9$.
Positive integer numbers $k$ and $n$ satisfy the inequality $k > n!$. Prove that there exist pairwisely different prime numbers $p_1, p_2, \ldots, p_n$ which are divisors of the numbers $k+1, k+2, \ldots, k+n$ respectively (i.e. $p_i|k+i$).
Find all pairs of $p,q$ prime numbers that satisfy the equation
$$p(p^4+p^2+10q)=q(q^2+3)$$
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?
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 $p_k\ge 3k/4$. Let $n$ be a composite integer. Prove:
(a) if $n=2p_k$, then $n$ does not divide $(n-k)!$;
(b) if $n>2p_k$, then $n$ divides $(n-k)!$.
Let $ (a_{n})_{n\ge 1}$ be a sequence of positive integers satisfying $ (a_{m},a_{n}) = a_{(m,n)}$ (for all $ m,n\in N^ +$). Prove that for any $ n\in N^ + ,\prod_{d|n}{a_{d}^{\mu (\frac {n}{d})}}$ is an integer. where $ d|n$ denotes $ d$ take all positive divisors of $ n.$ Function $ \mu (n)$ is defined as follows: if $ n$ can be divided by square of certain prime number, then $ \mu (1) = 1;\mu (n) = 0$; if $ n$ can be expressed as product of $ k$ different prime numbers, then $ \mu (n) = ( - 1)^k.$