Found problems: 471
Determine all functions $f$ defined on the set of all positive integers and taking non-negative integer values, satisfying the three conditions:
[list]
[*] $(i)$ $f(n) \neq 0$ for at least one $n$;
[*] $(ii)$ $f(x y)=f(x)+f(y)$ for every positive integers $x$ and $y$;
[*] $(iii)$ there are infinitely many positive integers $n$ such that $f(k)=f(n-k)$ for all $k<n$.
[/list]
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]
Determine all functions $f$ defined on the set of all positive integers and taking non-negative integer values, satisfying the three conditions:
[list]
[*] $(i)$ $f(n) \neq 0$ for at least one $n$;
[*] $(ii)$ $f(x y)=f(x)+f(y)$ for every positive integers $x$ and $y$;
[*] $(iii)$ there are infinitely many positive integers $n$ such that $f(k)=f(n-k)$ for all $k<n$.
[/list]
Let $p \geq 5$ be a prime number. Prove that there exists an integer $a$ with $1 \leq a \leq p-2$ such that neither $a^{p-1}-1$ nor $(a+1)^{p-1}-1$ is divisible by $p^2$.
Find all prime numbers p such that $2^p+p^2 $ is also a prime number.
Let $\mathcal{P}$ be the set of all primes, and let $M$ be a subset of $\mathcal{P}$, having at least three elements, and such that for any proper subset $A$ of $M$ all of the prime factors of the number $ -1+\prod_{p\in A}p$ are found in $M$. Prove that $M= \mathcal{P}$.
[i]Valentin Vornicu[/i]
Given a prime number $p{}$ and integers $x{}$ and $y$, find the remainder of the sum $x^0y^{p-1}+x^1y^{p-2}+\ldots+x^{p-2}y^1+x^{p-1}y^0$ upon division by $p{}$.
Let $a, b$ and $n$ be positive integers with $a>b$ such that all of the following hold:
i. $a^{2021}$ divides $n$,
ii. $b^{2021}$ divides $n$,
iii. 2022 divides $a-b$.
Prove that there is a subset $T$ of the set of positive divisors of the number $n$ such that the sum of the elements of $T$ is divisible by 2022 but not divisible by $2022^2$.
[i]Proposed by Silouanos Brazitikos, Greece[/i]
Let $A$ be a subset of $\{2,3, \ldots, 28 \}$ such that if $a \in A$, then the residue obtained when we divide $a^2$ by $29$ also belongs to $A$.
Find the minimum possible value of $|A|$.
Find all positive integers $n$ such that $\varphi(n)$ divides $n^2 + 3$.
Prove that any integer has a multiple consisting of all ten digits $\{0,1,2,3,4,5,6,7,8,9\}$.
\\
[i]Note: Any digit can be repeated any number of times[/i]
Let $a > 1$ be a positive integer. Prove that for some nonnegative integer $n$, the number $2^{2^n}+a$ is not prime.
[i]Proposed by Jack Gurev[/i]
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]
Given that there are $24$ primes between $3$ and $100$, inclusive, what is the number of ordered pairs $(p, a)$ with $p$ prime, $3 \le p < 100$, and $1 \le a < p$ such that the sum
\[a+a^2+a^3+\cdots+a^{(p-2)!} \]is not divisible by $p$?
Let $p,q$ be prime numbers ($q$ is odd). Prove that there exists an integer $x$ such that:
$$q |(x+1)^p-x^p$$
If and only if $$q \equiv 1 \pmod p$$
Prove that $m^7- m$ is divisible by $42$ for any positive integer $m$.
We say that a square-free positive integer $n$ is [i]almost prime[/i] if
\[n \mid x^{d_1}+x^{d_2}+\dots+x^{d_k}-kx\]
for all integers $x$, where $1=d_1<d_2<\dots<d_k=n$ are all the positive divisors of $n$. Suppose that $r$ is a Fermat prime (i.e. it is a prime of the form $2^{2^m}+1$ for an integer $m \ge 0$), $p$ is a prime divisor of an almost prime integer $n$, and $p \equiv 1 \pmod{r}$. Show that, with the above notation, $d_i \equiv 1 \pmod{r}$ for all $1 \le i \le k$.
(An integer $n$ is called [i]square-free[/i] if it is not divisible by $d^2$ for any integer $d>1$.)
Find all pairs $(p,q)$ of prime numbers which $p>q$ and
$$\frac{(p+q)^{p+q}(p-q)^{p-q}-1}{(p+q)^{p-q}(p-q)^{p+q}-1}$$
is an integer.
Find all ordered triples of primes $(p, q, r)$ such that \[ p \mid q^r + 1, \quad q \mid r^p + 1, \quad r \mid p^q + 1. \] [i]Reid Barton[/i]
We say that a set $S$ of integers is [i]rootiful[/i] if, for any positive integer $n$ and any $a_0, a_1, \cdots, a_n \in S$, all integer roots of the polynomial $a_0+a_1x+\cdots+a_nx^n$ are also in $S$. Find all rootiful sets of integers that contain all numbers of the form $2^a - 2^b$ for positive integers $a$ and $b$.
Determine all triples $(m , n , p)$ satisfying :
\[n^{2p}=m^2+n^2+p+1\]
where $m$ and $n$ are integers and $p$ is a prime number.
$\boxed{N2}$ Let $p$ be a prime numbers and $x_1,x_2,...,x_n$ be integers.Show that if
\[x_1^n+x_2^n+...+x_p^n\equiv 0 \pmod{p}\]
for all positive integers n then $x_1\equiv x_2 \equiv...\equiv x_p \pmod{p}.$
Given a prime number $p{}$ and integers $x{}$ and $y$, find the remainder of the sum $x^0y^{p-1}+x^1y^{p-2}+\ldots+x^{p-2}y^1+x^{p-1}y^0$ upon division by $p{}$.
For each prime $p$, construct a graph $G_p$ on $\{1,2,\ldots p\}$, where $m\neq n$ are adjacent if and only if $p$ divides $(m^{2} + 1-n)(n^{2} + 1-m)$. Prove that $G_p$ is disconnected for infinitely many $p$
Determine all positive integers $x$, $y$ and $z$ such that $x^5 + 4^y = 2013^z$.
([i]Serbia[/i])