Found problems: 341
Let $k$ be a fixed integer greater than 1, and let ${m=4k^2-5}$. Show that there exist positive integers $a$ and $b$ such that the sequence $(x_n)$ defined by \[x_0=a,\quad x_1=b,\quad x_{n+2}=x_{n+1}+x_n\quad\text{for}\quad n=0,1,2,\dots,\] has all of its terms relatively prime to $m$.
[i]Proposed by Jaroslaw Wroblewski, Poland[/i]
A triplet of polynomials $u,v,w \in \mathbb{R}[x,y,z]$ is called [i]smart[/i] if there exists polynomials $P,Q,R\in \mathbb{R}[x,y,z]$ such that the following polynomial identity holds :$$u^{2019}P +v^{2019 }Q+w^{2019} R=2019$$
a) Is the triplet of polynomials $$u=x+2y+3 , \;\;\;\; v=y+z+2, \;\;\;\;\;w=x+y+z$$ [i]smart[/i]?
b) Is the triplet of polynomials $$u=x+2y+3 , \;\;\;\; v=y+z+2, \;\;\;\;\;w=x+y-z$$ [i]smart[/i]?
[i]Proposed by Arturas Dubickas (Vilnius University).
[/i]
Call a rational number [i]short[/i] if it has finitely many digits in its decimal expansion. For a positive integer $m$, we say that a positive integer $t$ is $m-$[i]tastic[/i] if there exists a number $c\in \{1,2,3,\ldots ,2017\}$ such that $\dfrac{10^t-1}{c\cdot m}$ is short, and such that $\dfrac{10^k-1}{c\cdot m}$ is not short for any $1\le k<t$. Let $S(m)$ be the set of $m-$tastic numbers. Consider $S(m)$ for $m=1,2,\ldots{}.$ What is the maximum number of elements in $S(m)$?
What is the remainder when $2^{202} +202$ is divided by $2^{101}+2^{51}+1$?
$\textbf{(A) } 100 \qquad\textbf{(B) } 101 \qquad\textbf{(C) } 200 \qquad\textbf{(D) } 201 \qquad\textbf{(E) } 202$
Let $N$ be the number of functions $f$ from $\{1, 2, \dots, 101 \} \rightarrow \{1, 2, \dots, 101 \}$ such that $f^{101}(1) = 2.$ Find the remainder when $N$ is divided by $103.$
Find all natural numbers $a$ and $b$ such that \[a|b^2, \quad b|a^2 \mbox{ and } a+1|b^2+1.\]
Let $n$ be a positive integer and $a=2^n\cdot 7^{n+1}+11$ and $b=2^{n+1}\cdot 7^n+3$.
$a)$ Prove that fraction $\frac{a}{b}$ is irreducible
$b)$ Prove that number $a+b-7$ is not a perfect square for any positive integer $n$
Prove that $ 7^{2^{20}} + 7^{2^{19}} + 1 $ has at least $ 21 $ distinct prime divisors.
By dividing the integer $m$ by the integer $n, 22$ is the quotient and $5$ the remainder.
As the division of the remainder with $n$ continues, the new quotient is $0.4$ and the new remainder is $0.2$.
Find $m$ and $n$.
Let $n$ be a positive integer. We call a $n$-tuple $(a_1, . . . , a_n)$ of positive integers [i]nice [/i] if
$\bullet$ $gcd (a_1, . . . , a_n) = 1$, and
$\bullet$ $a_i|a_{i-1} + a_{i+1}$, for all $i = 1, . . . , n$ (we define $a_0 = a_n$ and $a_{n+1} = a1$ here).
Find the maximal possible value of the sum $a_1 +...+ a_n$ if $(a_1, . . . , a_n)$ is a nice $n$-tuple.
Let $p$ be an odd prime number and $\mathbb{Z}_{>0}$ be the set of positive integers. Suppose that a function $f:\mathbb{Z}_{>0}\times\mathbb{Z}_{>0}\to\{0,1\}$ satisfies the following properties:
[list]
[*] $f(1,1)=0$.
[*] $f(a,b)+f(b,a)=1$ for any pair of relatively prime positive integers $(a,b)$ not both equal to 1;
[*] $f(a+b,b)=f(a,b)$ for any pair of relatively prime positive integers $(a,b)$.
[/list]
Prove that
$$\sum_{n=1}^{p-1}f(n^2,p) \geqslant \sqrt{2p}-2.$$
Find all $l,m,n \in\mathbb{N}$ that satisfies the equation $5^l43^m+1=n^3$
Prove that for no integer $ n$ is $ n^7 \plus{} 7$ a perfect square.
The polynomials $P(z)$ and $Q(z)$ with complex coefficients have the same set of numbers for their zeros but possibly different multiplicities. The same is true for the polynomials
$$P(z)+1 \;\; \text{and} \;\; Q(z)+1.$$
Prove that $P(z)=Q(z).$
Let $n$ be the least positive integer greater than $1000$ for which $$\gcd(63, n+120) =21\quad \text{and} \quad \gcd(n+63, 120)=60.$$What is the sum of the digits of $n$?
$\textbf{(A) } 12 \qquad\textbf{(B) } 15 \qquad\textbf{(C) } 18 \qquad\textbf{(D) } 21\qquad\textbf{(E) } 24$
Internal angles of triangle are $(5x+3y)^{\circ}$, $(3x+20)^{\circ}$ and $(10y+30)^{\circ}$ where $x$ and $y$ are positive integers. Which values can $x+y$ get ?
Denote by $\mathbb{N}$ the set of all positive integers. Find all functions $f:\mathbb{N}\rightarrow \mathbb{N}$ such that for all positive integers $m$ and $n$, the integer $f(m)+f(n)-mn$ is nonzero and divides $mf(m)+nf(n)$.
[i]Proposed by Dorlir Ahmeti, Albania[/i]
For each integer $n \geq 2$, let $F(n)$ denote the greatest prime factor of $n$. A [i]strange pair[/i] is a pair of distinct primes $p$ and $q$ such that there is no integer $n \geq 2$ for which $F(n)F(n+1)=pq$.
Prove that there exist infinitely many strange pairs.
Find all positive integers $n$ with the following property: the $k$ positive divisors of $n$ have a permutation $(d_1,d_2,\ldots,d_k)$ such that for $i=1,2,\ldots,k$, the number $d_1+d_2+\cdots+d_i$ is a perfect square.
Find all positive integer solutions of the equation
$n^5+n^4=7^{m}-1$
Find all positive integers $n$ with the following property: the $k$ positive divisors of $n$ have a permutation $(d_1,d_2,\ldots,d_k)$ such that for $i=1,2,\ldots,k$, the number $d_1+d_2+\cdots+d_i$ is a perfect square.
A function $f$ from the positive integers to the positive integers is called [i]Canadian[/i] if it satisfies $$\gcd\left(f(f(x)), f(x+y)\right)=\gcd(x, y)$$ for all pairs of positive integers $x$ and $y$.
Find all positive integers $m$ such that $f(m)=m$ for all Canadian functions $f$.
Let $a$ and $b$ be integers such that $a - b = a^2c - b^2d$ for some consecutive integers $c$ and $d$. Prove that $|a - b|$ is a perfect square.
The expression $\frac{2021}{2020} - \frac{2020}{2021}$ is equal to the fraction $\frac{p}{q}$, where $p$ and $q$ are positive integers whose greatest common divisor is $1$. What is $p$?
$\textbf{(A) }1\qquad\textbf{(B) }9\qquad\textbf{(C) }2020\qquad\textbf{(D) }2021\qquad\textbf{(E) }4041$
Find all positive integers $(a,b,c)$ such that
$$ab-c,\quad bc-a,\quad ca-b$$ are all powers of $2$.
[i]Proposed by Serbia[/i]