Found problems: 721
Let $p$ be a prime number. Troy and Abed are playing a game. Troy writes a positive integer $X$ on the board, and gives a sequence $(a_n)_{n\in\mathbb{N}}$ of positive integers to Abed. Abed now makes a sequence of moves. The $n$-th move is the following:
$$\text{ Replace } Y \text{ currently written on the board with either } Y + a_n \text{ or } Y \cdot a_n.$$
Abed wins if at some point the number on the board is a multiple of $p$. Determine whether Abed can win, regardless of Troy’s choices, if
$a) p = 10^9 + 7$;
$b) p = 10^9 + 9$.
[i]Remark[/i]: Both $10^9 + 7$ and $10^9 + 9$ are prime.
[i]Proposed by Ivan Novak[/i]
Exists a positive integer $n$ such that the number $\underbrace{1...1}_{n \,ones} 2 \underbrace{1...1}_{n \, ones}$ is a prime number?
Let $p,q$ be prime numbers such that $n^{3pq}-n$ is a multiple of $3pq$ for [b]all[/b] positive integers $n$. Find the least possible value of $p+q$.
Denote by $d(n)$ the number of divisors of the positive integer $n$. A positive integer $n$ is called highly divisible if $d(n) > d(m)$ for all positive integers $m < n$.
Two highly divisible integers $m$ and $n$ with $m < n$ are called consecutive if there exists no highly divisible integer $s$ satisfying $m < s < n$.
(a) Show that there are only finitely many pairs of consecutive highly divisible
integers of the form $(a, b)$ with $a\mid b$.
(b) Show that for every prime number $p$ there exist infinitely many positive highly divisible integers $r$ such that $pr$ is also highly divisible.
Find all triples $(p, q, r)$ of prime numbers for which $4q - 1$ is a prime number and $$\frac{p + q}{p + r} = r - p$$
holds.
[i](Walther Janous)[/i]
Which of the followings is false for the sequence $9,99,999,\dots$?
$\textbf{(A)}$ The primes which do not divide any term of the sequence are finite.
$\textbf{(B)}$ Infinitely many primes divide infinitely many terms of the sequence.
$\textbf{(C)}$ For every positive integer $n$, there is a term which is divisible by at least $n$ distinct prime numbers.
$\textbf{(D)}$ There is an inteter $n$ such that every prime number greater than $n$ divides infinitely many terms of the sequence.
$\textbf{(E)}$ None of above
Let $m$ be a positive integer for which there exists a positive integer $n$ such that the multiplication $mn$ is a perfect square and $m- n$ is prime. Find all $m$ for $1000\leq m \leq 2021.$
Show that [list=a][*] infinitely many perfect squares are a sum of a perfect square and a prime number, [*] infinitely many perfect squares are not a sum of a perfect square and a prime number. [/list]
Let $n$ be a given number greater than 2. We consider the set $V_n$ of all the integers of the form $1 + kn$ with $k = 1, 2, \ldots$ A number $m$ from $V_n$ is called indecomposable in $V_n$ if there are not two numbers $p$ and $q$ from $V_n$ so that $m = pq.$ Prove that there exist a number $r \in V_n$ that can be expressed as the product of elements indecomposable in $V_n$ in more than one way. (Expressions which differ only in order of the elements of $V_n$ will be considered the same.)
For a given positive integer $n$ and prime number $p$, find the minimum value of positive integer $m$ that satisfies the following property: for any polynomial $$f(x)=(x+a_1)(x+a_2)\ldots(x+a_n)$$ ($a_1,a_2,\ldots,a_n$ are positive integers), and for any non-negative integer $k$, there exists a non-negative integer $k'$ such that $$v_p(f(k))<v_p(f(k'))\leq v_p(f(k))+m.$$ Note: for non-zero integer $N$,$v_p(N)$ is the largest non-zero integer $t$ that satisfies $p^t\mid N$.
For a positive integer $n$ let $P(n)$ denote the set of primes $p$ for which there exist
positive integers $a, b$ such that $n=a^p+b^p$ . Is it true that for any finite set $H$ consisting of primes, there is an n such that $P(n) = H$?
Let $p_i$ for $i=1,2,..., k$ be a sequence of smallest consecutive prime numbers ($p_1=2$, $p_2=3$, $p_3=3$ etc. ). Let $N=p_1\cdot p_2 \cdot ... \cdot p_k$. Prove that in a set $\{ 1,2,...,N \}$ there exist exactly $\frac{N}{2}$ numbers which are divisible by odd number of primes $p_i$.
[hide=example]For $k=2$ $p_1=2$, $p_2=3$, $N=6$. So in set $\{ 1,2,3,4,5,6 \}$ we can find $3$ number satisfying thesis: $2$, $3$ and $4$. ($1$ and $5$ are not divisible by $2$ or $3$, and $6$ is divisible by both of them so by even number of primes )[/hide]
Find all natural numbers $ n$ for which every natural number whose decimal representation has $ n \minus{} 1$ digits $ 1$ and one digit $ 7$ is prime.
Call a positive integer [b]good[/b] if either $N=1$ or $N$ can be written as product of [i]even[/i] number of prime numbers, not necessarily distinct.
Let $P(x)=(x-a)(x-b),$ where $a,b$ are positive integers.
(a) Show that there exist distinct positive integers $a,b$ such that $P(1),P(2),\cdots ,P(2010)$ are all good numbers.
(b) Suppose $a,b$ are such that $P(n)$ is a good number for all positive integers $n$. Prove that $a=b$.
Find all prime numbers $p$ which satisfy the following condition: For any prime $q < p$, if $p = kq + r, 0 \leq r < q$, there does not exist an integer $q > 1$ such that $a^{2} \mid r$.
A positive integer is called [i]cool[/i] if it can be expressed in the form $a!\cdot b!+315$ where $a,b$ are positive integers. For example, $1!\cdot 1!+315=316$ is a cool number. Find the sum of all cool numbers that are also prime numbers.
[i]Proposed by Evan Fang
Given a prime number $p$ such that $2p$ is equal to the sum of the squares of some four consecutive positive integers. Prove that $p-7$ is divisible by 36.
Suppose that $p_1<p_2<\dots <p_{15}$ are prime numbers in arithmetic progression, with common difference $d$. Prove that $d$ is divisible by $2,3,5,7,11$ and $13$.
250 numbers are chosen among positive integers not exceeding 501. Prove that for every integer $ t$ there are four chosen numbers $ a_1$, $ a_2$, $ a_3$, $ a_4$, such that $ a_1 \plus{} a_2 \plus{} a_3 \plus{} a_4 \minus{} t$ is divisible by 23.
[i]Author: K. Kokhas[/i]
How many subsets of $\{2,3,4,5,6,7,8,9\}$ contain at least one prime number?
$\textbf{(A)} \text{ 128} \qquad \textbf{(B)} \text{ 192} \qquad \textbf{(C)} \text{ 224} \qquad \textbf{(D)} \text{ 240} \qquad \textbf{(E)} \text{ 256}$
Alice is once again very bored in class. On a whim, she chooses three primes $p$, $q$, $r$ independently and uniformly at random from the set of primes at most 30. She then calculates the roots of $px^2+qx+r$. What is the probability that at least one of her roots is an integer?
Find the number of polynomials $f(x)=ax^3+bx$ satisfying both following conditions:
(i) $a,b\in\{1,2,\ldots,2013\}$;
(ii) the difference between any two of $f(1),f(2),\ldots,f(2013)$ is not a multiple of $2013$.
Distinct prime numbers $p,q,r$ satisfy the equation $$2pqr+50pq=7pqr+55pr=8pqr+12qr=A$$ for some positive integer $A.$ What is $A$?
Let $p$ be a fixed prime number. Find all integers $n \ge 1$ with the following property: One can partition the positive divisors of $n$ in pairs $(d,d')$ satisfying $d<d'$ and $p \mid \left\lfloor \frac{d'}{d}\right\rfloor$.
Consider the numbers from $1$ to $32$. A game is made by placing all the numbers in pairs and replacing each pair with the largest prime divisor of the sum of the numbers of that couple. For example, if we match the $32$ numbers as: $(1, 2), (3,4),(5, 6), (7, 8),..., (27, 28),(29, 30), (31,32)$, we get the following list of $16$ numbers: $3,7,11,5,...,11,59,7$. where there are repetitions. The game continues in a similar way until in the end only one number remains. Determine the highest possible value from the number that remains at the end.