Found problems: 1048
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.
a) Suppose that $n$ is an odd integer. Prove that $k(n-k)$ is divisible by $2$ for all positive integers $k$.
b) Find an integer $k$ such that $k(100-k)$ is not divisible by $11$.
c) Suppose that $p$ is an odd prime, and $n$ is an integer.
Prove that there is an integer $k$ such that $k(n-k)$ is not divisible by $p$.
d) Suppose that $p,q$ are two different odd primes, and $n$ is an integer.
Prove that there is an integer $k$ such that $k(n-k)$ is not divisible by any of $p,q$.
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 $2pq$.
The radius $r$ of a circle with center at the origin is an odd integer.
There is a point ($p^m, q^n$) on the circle, with $p,q$ prime numbers and $m,n$ positive integers.
Determine $r$.
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
Prove that when dividing a prime number with $30$, remainder is always not a composite number
Find all prime numbers $p$ for which $\frac{2^{p-1} -1}{p}$ is a perfect square.
Let $n$ be a positive integer.
Prove that every prime $p > 2$ that divides $(2-\sqrt{3})^n + (2+\sqrt{3})^n$ satisfy $p=1 (mod3)$
Let $n$ be a natural number, $n > 2$. Prove that if $\frac{b^n-1}{b-1}$ is a prime power for some positive integer $b$ then $n$ is prime.
Construct a tetromino by attaching two $2 \times 1$ dominoes along their longer sides such that the midpoint of the longer side of one domino is a corner of the other domino. This construction yields two kinds of tetrominoes with opposite orientations. Let us call them $S$- and $Z$-tetrominoes, respectively.
Assume that a lattice polygon $P$ can be tiled with $S$-tetrominoes. Prove that no matter how we tile $P$ using only $S$- and $Z$-tetrominoes, we always use an even number of $Z$-tetrominoes.
[i]Proposed by Tamas Fleiner and Peter Pal Pach, Hungary[/i]
Find a constant $ c > 1$ with the property that, for arbitrary positive integers $ n$ and $ k$ such that $ n>c^k$, the number of distinct prime factors of $ \binom{n}{k}$ is at least $ k$.
[i]P. Erdos[/i]
Determine the smallest positive integer $m$ with the property that $m^3-3m^2+2m$ is divisible by both $79$ and $83$.
Prove that if a triangle has integral side lengths and its circumradius is a prime number then the triangle is right-angled.
Determine all non-constant monic polynomials $P(x)$ with integer coefficients such that no prime $p>10^{100}$ divides any number of the form $P(2^n)$
There are $n$ children in a room. Each child has at least one piece of candy. In Round $1$, Round $2$, etc., additional pieces of candy are distributed among the children according to the following rule:
In Round $k$, each child whose number of pieces of candy is relatively prime to $k$ receives an additional piece.
Show that after a sufficient number of rounds the children in the room have at most two different numbers of pieces of candy.
[i](Proposed by Theresia Eisenkölbl)[/i]