Found problems: 721
An increasing arithmetic progression consists of one hundred positive integers. Is it possible that every two of them are relatively prime?
Let $p$ and $q$ be distinct odd primes. Show that
$$\bigg\lceil \dfrac{p^q+q^p-pq+1}{pq} \bigg\rceil$$ is even.
Let $p$ be a prime number and $f$ be a bijection from $\left\{0,1,\ldots,p-1\right\}$ to itself. Suppose that for integers $a,b \in \left\{0,1,\ldots,p-1\right\}$, $|f(a) - f(b)|\leqslant 2024$ if $p \mid a^2 - b$. Prove that there exists infinite many $p$ such that there exists such an $f$ and there also exists infinite many $p$ such that there doesn't exist such an $f$.
For a positive integer $k$, let $p(k)$ be the smallest prime that does not divide $k$. Given a positive integer $a$, define the infinite sequence $a_0, a_1, \ldots$ by $a_0 = a$ and, for $n > 0$, $a_n$ is the smallest positive integer with the following properties:
• $a_n$ has not yet appeared in the sequence, that is, $a_n \neq a_i$ for $0 \leq i < n$;
• $(a_{n-1})^{a_n} - 1$ is a multiple of $p(a_{n-1})$.
Prove that every positive integer appears as a term in the sequence, that is, for every positive integer $m$ there is $n$ such that $a_n = m$.
Suppose the prime numbers $p$ and $q$ satisfy $q^2+3p=197p^2+q$.Write $\frac{p}{q}$ as $l+\frac{m}{n}$, where $l,m,n$ are positive integers , $m<n$ and $GCD(m,n)=1$. Find the maximum value of $l+m+n$.
The diagonals of some convex quadrilateral are mutually perpendicular and divide the quadrangle into $4$ triangles, the areas of which are expressed by prime numbers. Prove that a circle can be inscribed in this quadrilateral.
Let $p$ and $q$ be prime numbers. Show that $p^2+q^2+2020$ is composite.
Let $ b, m, n$ be positive integers such that $ b > 1$ and $ m \neq n.$ Prove that if $ b^m \minus{} 1$ and $ b^n \minus{} 1$ have the same prime divisors, then $ b \plus{} 1$ is a power of 2.
A prime $p$ and a positive integer $n$ are given. The product $$(1^3+1)(2^3+1)...((n-1)^3+1)(n^3+1)$$ is divisible by $p^3$. Prove that $p \leq n+1$.
[i]Proposed by Z. Luria[/i]
Find all prime numbers $p$ and $q$ such that $3p^2q+2pq^2=483$
Find all prime numbers $p$ such that $3^p + 5^p -1$ is a prime number.
Find all prime numbers $p$ such that $p^3$ divides the determinant
\[\begin{vmatrix} 2^2 & 1 & 1 & \dots & 1\\1 & 3^2 & 1 & \dots & 1\\ 1 & 1 & 4^2 & & 1\\ \vdots & \vdots & & \ddots & \\1 & 1 & 1 & & (p+7)^2 \end{vmatrix}.\]
Do there exist prime numbers $p$ and $q$ such that $p^2(p^3-1)=q(q+1)$ ?
Is there an infinite sequence of prime numbers $p_1$, $p_2$, $\ldots$, $p_n$, $p_{n+1}$, $\ldots$ such that $|p_{n+1}-2p_n|=1$ for each $n \in \mathbb{N}$?
Find the primes ${p, q, r}$, given that one of the numbers ${pqr}$ and ${p + q + r}$ is ${101}$ times the other.
Let $p$ be an odd prime number and suppose that $2^h \not \equiv 1 \text{ (mod } p\text{)}$ for all integer $1 \leq h \leq p-2$. Let $a$ be an even number such that $\frac{p}{2} < a < p$. Define the sequence $a_0, a_1, a_2, \ldots$ as $$a_0 = a, \qquad a_{n+1} = p -b_n, \qquad n = 0,1,2, \ldots,$$ where $b_n$ is the greatest odd divisor of $a_n$. Show that the sequence is periodic and determine its period.
Find all pairs of prime numbers $p,\,q(p\le q)$ satisfying the following condition:
There exists a natural number $n$ such that $2^{n}+3^{n}+\cdots+(2pq-1)^{n}$ is a multiple of $2pq$.
Find all the pairs of positive integers $(x,y)$ such that $x\leq y$ and \[\frac{(x+y)(xy-1)}{xy+1}=p,\]where $p$ is a prime number.
Let $p{}$ be a prime number greater than 3. Prove that there exists a natural number $y{}$ less than $p/2$ and such that the number $py + 1$ cannot be represented as a product of two integers, each of which is greater than $y{}$.
[i]Proposed by M. Antipov[/i]
Let $(a_{n})_{n\geq 1}$ be a sequence defined by $a_{n}=2^{n}+49$. Find all values of $n$ such that $a_{n}=pg, a_{n+1}=rs$, where $p,q,r,s$ are prime numbers with $p<q, r<s$ and $q-p=s-r$.
Let $p_1$, $p_2$ be two odd prime numbers and $\alpha $, $n$ be positive integers with $\alpha >1$, $n>1$. Prove that if the equation $\left (\frac{p_2 -1}{2} \right )^{p_1} + \left (\frac{p_2 +1}{2} \right )^{p_1} = \alpha^n$ does not have integer solutions for both $p_1 =p_2$ and $p_1 \neq p_2$.
Higher Secondary P7
If there exists a prime number $p$ such that $p+2q$ is prime for all positive integer $q$ smaller than $p$, then $p$ is called an "awesome prime". Find the largest "awesome prime" and prove that it is indeed the largest such prime.
$p>30$ is a prime number. Prove that one of the following numbers is in form of $x^2+y^2$.
$$ p+1 , 2p+1 , 3p+1 , .... , (p-3)p+1$$
Given a $6\times 6$ square consisting of unit squares, denote its rows and columns from $1$ to $6$. Figure [i]p-horse[/i] can move from square $(x; y)$ to $(x’; y’)$ if and only if both $x + x’$ and $y + y’$ are primes. At the start the [i]p-horse[/i] is located in one of the unit squares.
$a)$ Can the [i]p-horse[/i] visit every unit square exactly once?
$b$) Can the [i]p-horse[/i] visit every unit square exactly once and with the last move return to the initial starting position?
Let $ a$ be a fix natural number . Prove that the set of prime divisors of $ 2^{2^{n}} \plus{} a$ for $ n \equal{} 1,2,\cdots$ is infinite