Found problems: 526
Prove that there exist infinitely many positive integers $n$ such that the largest prime divisor of $n^4 + n^2 + 1$ is equal to the largest prime divisor of $(n+1)^4 + (n+1)^2 +1$.
Let $n > 1$ be a given integer. Prove that infinitely many terms of the sequence $(a_k )_{k\ge 1}$, defined by \[a_k=\left\lfloor\frac{n^k}{k}\right\rfloor,\] are odd. (For a real number $x$, $\lfloor x\rfloor$ denotes the largest integer not exceeding $x$.)
[i]Proposed by Hong Kong[/i]
How many positive integers at most $420$ leave different remainders when divided by each of $5$, $6$, and $7$?
[i]Proposed by Milan Haiman.[/i]
A sequence of polynomials is defined by the recursion $P_1(x) = x+1$ and$$P_{n}(x) = \frac{(P_{n-1}(x)+1)^5 - (P_{n-1}(-x)+1)^5}{2}$$for all $n \geq 2$. Find the remainder when $P_{2022}(1)$ is divided by $1000$.
[i]Proposed by [b]treemath[/b][/i]
A set of positive integers is called [i]fragrant[/i] if it contains at least two elements and each of its elements has a prime factor in common with at least one of the other elements. Let $P(n)=n^2+n+1$. What is the least possible positive integer value of $b$ such that there exists a non-negative integer $a$ for which the set $$\{P(a+1),P(a+2),\ldots,P(a+b)\}$$ is fragrant?
Determine the smallest positive prime $p$ which satisfies the congruence \[p+p^{-1}\equiv 25\pmod{143}.\] Here, $p^{-1}$ as usual denotes multiplicative inverse.
An integer $n \geq 3$ is [i]fabulous[/i] when there exists an integer $a$ with $2 \leq a \leq n - 1$ for which $a^n - a$ is divisible by $n$. Find all the [i]fabulous[/i] integers.
Let $a_1$ be any positive integer. For all $i$, write $5^{2020}$ times $a_i$ in base $10$, replace each digit with its remainder when divided by $2$, read off the result in binary, and call that $a_{i+1}$. Prove that $a_N = a_{N+2^{2020}}$ for all sufficiently large $N$.
Show that, for every positive integer $n$, there exist $n$ consecutive positive integers such that none is divisible by the sum of its digits.
(Alternative Formulation: Call a number good if it's not divisible by the sum of its digits. Show that for every positive integer $n$ there are $n$ consecutive good numbers.)
The set of positive integers is partitioned into $n$ disjoint infinite arithmetic progressions $S_1, S_2, \ldots, S_n$ with common differences $d_1, d_2, \ldots, d_n$, respectively. Prove that there exists exactly one index $1\leq i \leq n$ such that\[ \frac{1}{d_i}\prod_{j=1}^n d_j \in S_i.\]
Let $\mathbb{Z}_{>0}$ denote the set of positive integers. For any positive integer $k$, a function $f: \mathbb{Z}_{>0} \to \mathbb{Z}_{>0}$ is called [i]$k$-good[/i] if $\gcd(f(m) + n, f(n) + m) \le k$ for all $m \neq n$. Find all $k$ such that there exists a $k$-good function.
[i]Proposed by James Rickards, Canada[/i]
Prove that there exists two strictly increasing sequences $(a_{n})$ and $(b_{n})$ such that $a_{n}(a_{n}+1)$ divides $b^{2}_{n}+1$ for every natural n.
Is there an integer coefficients polynomial $P(x)$ satisfying \[ \begin{cases} P(1+\sqrt[3]{2})=1+\sqrt[3]{2}\\ P(1+\sqrt{5})=2+3\sqrt{5}\end{cases} \]
The function $f$ from the set $\mathbb{N}$ of positive integers into itself is defined by the equality \[f(n)=\sum_{k=1}^{n} \gcd(k,n),\qquad n\in \mathbb{N}.\]
a) Prove that $f(mn)=f(m)f(n)$ for every two relatively prime ${m,n\in\mathbb{N}}$.
b) Prove that for each $a\in\mathbb{N}$ the equation $f(x)=ax$ has a solution.
c) Find all ${a\in\mathbb{N}}$ such that the equation $f(x)=ax$ has a unique solution.
Given an odd integer $n>3$, let $k$ and $t$ be the smallest positive integers such that both $kn+1$ and $tn$ are squares. Prove that $n$ is prime if and only if both $k$ and $t$ are greater than $\frac{n}{4}$
Show that if an infinite arithmetic progression of positive integers contains a square and a cube, it must contain a sixth power.
Compute $3^{3^{\ldots^3}} \mod{333},$ where there are $3^{3^3}$ $3$'s in the exponent.
Let $p \in \mathbb{N} \setminus \{0, 1\}$ be a fixed positive integer. Prove that for every $K > 0$, there exist infinitely many $n$ and $N$ such that there are atleast $\dfrac{KN}{\log(N)}$ primes among the following $N$ numbers given by
\[n + 1, n + 2^p, n + 3^p, \cdots, n + N^p.\]
[i]Proposed by Bimit Mandal[/i]
Consider the triangular array of numbers with $0,1,2,3,...$ along the sides and interior numbers obtained by adding the two adjacent numbers in the previous row. Rows $1$ through $6$ are shown.
\begin{tabular}{ccccccccccc} & & & & & 0 & & & & & \\ & & & & 1 & & 1 & & & & \\ & & & 2 & & 2 & & 2 & & & \\ & & 3 & & 4 & & 4 & & 3 & & \\ & 4 & & 7 & & 8 & & 7 & & 4 & \\ 5 & & 11 & & 15 & & 15 & & 11 & & 5 \end{tabular}
Let $f(n)$ denote the sum of the numbers in row $n$. What is the remainder when $f(100)$ is divided by $100$?
$\textbf{(A)}\ 12\qquad
\textbf{(B)}\ 30 \qquad
\textbf{(C)}\ 50 \qquad
\textbf{(D)}\ 62 \qquad
\textbf{(E)}\ 74$
Let $a_1,a_2,a_3,\cdots $ be an infinite sequence of distinct integers. Prove that there are infinitely many primes $p$ that distinct positive integers $i,j,k$ can be found such that $p\mid a_ia_ja_k-1$.
[i]Proposed by Mohsen Jamali[/i]
Let $N = 123456789101112\dots4344$ be the $79$-digit number obtained that is formed by writing the integers from $1$ to $44$ in order, one after the other. What is the remainder when $N$ is divided by $45$?
$\textbf{(A)}\ 1 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 9 \qquad\textbf{(D)}\ 18 \qquad\textbf{(E)}\ 44$
Does there exist a sequence of natural numbers in which every natural number occurs exactly once, such that for each $k = 1, 2, 3, \dots$ the sum of the first $k$ terms of the sequence is divisible by $k$?
[i]A. Shapovalov[/i]
Given a prime $p$, consider integers $0<a<b<c<d<p$ such that $a^4\equiv b^4\equiv c^4\equiv d^4\pmod{p}$. Show that \[a+b+c+d\mid a^{2013}+b^{2013}+c^{2013}+d^{2013}\]
Let $P(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\ldots+a_{0}$, where $a_{0},\ldots,a_{n}$ are integers, $a_{n}>0$, $n\geq 2$. Prove that there exists a positive integer $m$ such that $P(m!)$ is a composite number.
Arrange all square-free positive integers in ascending order $a_1,a_2,a_3,\ldots,a_n,\ldots$. Prove that there are infinitely many positive integers $n$, such that $a_{n+1}-a_n=2020$.