Found problems: 471
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 $b$.
For $ x \in (0, 1)$ let $ y \in (0, 1)$ be the number whose $ n$-th digit after the decimal point is the $ 2^{n}$-th digit after the decimal point of $ x$. Show that if $ x$ is rational then so is $ y$.
[i]Proposed by J.P. Grossman, Canada[/i]
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 $p$
Let $p$ be a fixed prime. Determine all the integers $m$, as function of $p$, such that there exist $a_1, a_2, \ldots, a_p \in \mathbb{Z}$ satisfying
\[m \mid a_1^p + a_2^p + \cdots + a_p^p - (p+1).\]
We call a function $g$ [i]special [/i] if $g(x)=a^{f(x)}$ (for all $x$) where $a$ is a positive integer and $f$ is polynomial with integer coefficients such that $f(n)>0$ for all positive integers $n$.
A function is called an [i]exponential polynomial[/i] if it is obtained from the product or sum of special functions. For instance, $2^{x}3^{x^{2}+x-1}+5^{2x}$ is an exponential polynomial.
Prove that there does not exist a non-zero exponential polynomial $f(x)$ and a non-constant polynomial $P(x)$ with integer coefficients such that
$$P(n)|f(n)$$
for all positive integers $n$.
Does there exist a positive integer $ n$ such that $ n$ has exactly 2000 prime divisors and $ n$ divides $ 2^n \plus{} 1$?
Determine all functions $f: \mathbb{Z}\to\mathbb{Z}$ satisfying \[f\big(f(m)+n\big)+f(m)=f(n)+f(3m)+2014\] for all integers $m$ and $n$.
[i]Proposed by Netherlands[/i]
Suppose that the sum of the squares of two complex numbers $x$ and $y$ is 7 and the sum of the cubes is 10. What is the largest real value that $x + y$ can have?
Suppose that $p$ is an odd prime such that $2p+1$ is also prime. Show that the equation $x^{p}+2y^{p}+5z^{p}=0$ has no solutions in integers other than $(0,0,0)$.
Define the sequence of integers $\langle a_n\rangle$ as;
\[a_0=1, \quad a_1=-1, \quad \text{ and } \quad a_n=6a_{n-1}+5a_{n-2} \quad \forall n\geq 2.\]
Prove that $a_{2012}-2010$ is divisible by $2011.$
Which of the following expressions is never a prime number when $p$ is a prime number?
$\textbf{(A) } p^2+16 \qquad \textbf{(B) } p^2+24 \qquad \textbf{(C) } p^2+26 \qquad \textbf{(D) } p^2+46 \qquad \textbf{(E) } p^2+96$
Define the two sequences $a_0, a_1, a_2, \cdots$ and $b_0, b_1, b_2, \cdots$ by $a_0 = 3$ and $b_0 = 1$ with the recurrence relations $a_{n+1} = 3a_n + b_n$ and $b_{n+1} = 3b_n - a_n$ for all nonnegative integers $n.$ Let $r$ and $s$ be the remainders when $a_{32}$ and $b_{32}$ are divided by $31,$ respectively. Compute $100r + s.$
Determine all sequences of strictly positive integers $a_1, a_2, a_3, \ldots$ satisfying the following two conditions:
[list]
[*]There exists an integer $M > 0$ such that, for all indices $n \geqslant 1$, $0 < a_n \leqslant M$.
[*]For any prime number $p$ and for any index $n \geqslant 1$, the number
\[
a_n a_{n+1} \cdots a_{n+p-1} - a_{n+p}
\]
is a multiple of $p$.
[/list]
Determine all integers $ n > 1$ such that
\[ \frac {2^n \plus{} 1}{n^2}
\]
is an integer.
We define $\mathbb F_{101}[x]$ as the set of all polynomials in $x$ with coefficients in $\mathbb F_{101}$ (the integers modulo $101$ with usual addition and subtraction), so that two polynomials are equal if and only if the coefficients of $x^k$ are equal in $\mathbb F_{101}$ for each nonnegative integer $k$. For example, $(x+3)(100x+5)=100x^2+2x+15$ in $\mathbb F_{101}[x]$ because the corresponding coefficients are equal modulo $101$.
We say that $f(x)\in\mathbb F_{101}[x]$ is \emph{lucky} if it has degree at most $1000$ and there exist $g(x),h(x)\in\mathbb F_{101}[x]$ such that \[f(x)=g(x)(x^{1001}-1)+h(x)^{101}-h(x)\] in $\mathbb F_{101}[x]$. Find the number of lucky polynomials.
[i]Proposed by Michael Ren.[/i]
Prove that there are infinitely many positive integers $n$ such that $2^{2^n+1}+1$ is divisible by $n$ but $2^n+1$ is not.
[i](Russia) Valery Senderov[/i]
Find the largest positive integer $n <30$ such that $\frac{1}{2}(n^8 + 3n^4 -4)$ is not divisible by the square of any prime number.
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.
Let $ q \equal{} 2p\plus{}1$, $ p, q > 0$ primes. Prove that there exists a multiple of $ q$ whose digits sum in decimal base is positive and at most $ 3$.
Let $p$ be a prime number such that $\frac{28^p-1}{2p^2+2p+1}$ is an integer. Find all possible values of number of divisors of $2p^2+2p+1$.
Let $n{}$ be a positive integer. What is the smallest sum of digits that $5^n + 6^n + 2022^n$ can take?
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]
Prove that if $p$ and $q$ are two prime numbers, such that
$$p+p^2+p^3+...+p^q=q+q^2+q^3+...+q^p,$$
then $p=q$.
Let $p$ be an odd prime. Prove that \[\sum^{p-1}_{k=1} k^{2p-1} \equiv \frac{p(p+1)}{2}\pmod{p^2}.\]
Let $a$ and $b$ be fixed positive integers. We say that a prime $p$ is [i]fun[/i] if there exists a positive integer $n$ satisfying the following conditions:
[list]
[*]$p$ divides $a^{n!} + b$.
[*]$p$ divides $a^{(n + 1)!} + b$.
[*]$p < 2n^2 + 1$.
[/list]
Show that there are finitely many fun primes.