Found problems: 700
Find all triples $(a,b,c)$ of positive integers with the following property: for every prime $p$, if $n$ is a quadratic residue $\mod p$, then $an^2+bn+c$ is a quadratic residue $\mod p$.
Find all prime numbers $p$ such that there exists a unique $a \in \mathbb{Z}_p$ for which $a^3 - 3a + 1 = 0.$
How many primes $p$ are there such that $2p^4-7p^2+1$ is equal to square of an integer?
$
\textbf{(A)}\ 0
\qquad\textbf{(B)}\ 1
\qquad\textbf{(C)}\ 4
\qquad\textbf{(D)}\ \text{Infinitely many}
\qquad\textbf{(E)}\ \text{None of the preceding}
$
Find the 2002nd positive integer that is not the difference of two square integers
Prove that among $20$ consecutive positive integers there is an integer $d$ such that for every positive integer $n$ the following inequality holds
$$n \sqrt{d} \left\{n \sqrt {d} \right \} > \dfrac{5}{2}$$
where by $\left \{x \right \}$ denotes the fractional part of the real number $x$. The fractional part of the real number $x$ is defined as the difference between the largest integer that is less than or equal to $x$ to the actual number $x$.
[i](Serbia)[/i]
An amusement park has a collection of scale models, with ratio $1 : 20$, of buildings and other sights from around the country. The height of the United States Capitol is 289 feet. What is the height in feet of its replica to the nearest whole number?
$\textbf{(A) }14\qquad\textbf{(B) }15\qquad\textbf{(C) }16\qquad\textbf{(D) }18\qquad\textbf{(E) }20$
Does there exist a strictly increasing infinite sequence of perfect squares $a_1, a_2, a_3, ...$ such that for all $k\in \mathbb{Z}^+$ we have that $13^k | a_k+1$?
[i]Proposed by Jesse Zhang[/i]
Let $p=1601$. Prove that if
\[\dfrac {1} {0^2+1}+\dfrac{1}{1^2+1}+\cdots+\dfrac{1}{(p-1)^2+1}=\dfrac{m} {n},\]
where we only sum over terms with denominators not divisible by $p$ (and the fraction $\dfrac {m} {n}$ is in reduced terms) then $p \mid 2m+n$.
[i]Proposed by A. Golovanov[/i]
Prove that for every natural number $k$, at least one of the integers
\[ 2k-1, \quad 5k-1 \quad \text{and} \quad 13k-1\]
is not a perfect square.
Let $p = 2017$ be a prime and $\mathbb{F}_p$ be the integers modulo $p$. A function $f: \mathbb{Z}\rightarrow\mathbb{F}_p$ is called [i]good[/i] if there is $\alpha\in\mathbb{F}_p$ with $\alpha\not\equiv 0\pmod{p}$ such that
\[f(x)f(y) = f(x + y) + \alpha^y f(x - y)\pmod{p}\]
for all $x, y\in\mathbb{Z}$. How many good functions are there that are periodic with minimal period $2016$?
[i]Ashwin Sah[/i]
Find all pairs of $(a,b)$ positive integers satisfying the equation:
$$\frac {a^3+b^3}{ab+4}=2020$$
Given are integers $a, b, c$ and an odd prime $p.$ Prove that $p$ divides $x^2 + y^2 + ax + by + c$ for some integers $x$ and $y.$
[i](A. Golovanov )[/i]
An integer $n > 1$ is called $\emph{good}$ if there exists a permutation $a_1, a_2, a_3, \dots, a_n$ of the numbers $1, 2, 3, \dots, n$, such that:
$(i)$ $a_i$ and $a_{i+1}$ have different parities for every $1 \leq i \leq n-1$;
$(ii)$ the sum $a_1 + a_2 + \cdots + a_k$ is a quadratic residue modulo $n$ for every $1 \leq k \leq n$.
Prove that there exist infinitely many good numbers, as well as infinitely many positive integers which are not good.
Let $S$ be a finite set of positive integers. Assume that there are precisely 2023 ordered pairs $(x,y)$ in $S\times S$ so that the product $xy$ is a perfect square. Prove that one can find at least four distinct elements in $S$ so that none of their pairwise products is a perfect square.
[i]Note:[/i] As an example, if $S=\{1,2,4\}$, there are exactly five such ordered pairs: $(1,1)$, $(1,4)$, $(2,2)$, $(4,1)$, and $(4,4)$.
[i]Proposed by Sutanay Bhattacharya[/i]
Find the smallest positive prime that divides $n^2 + 5n + 23$ for some integer $n$.
Are there natural number(s) $n$, such that $3^n+1$ has a divisor in the form $24k+20$
Find all positive integers $a$ such that for any positive integer $n\ge 5$ we have $2^n-n^2\mid a^n-n^a$.
Find all positive integers $ n$ for which the numbers in the set $ S \equal{} \{1,2, \ldots,n \}$ can be colored red and blue, with the following condition being satisfied: The set $ S \times S \times S$ contains exactly $ 2007$ ordered triples $ \left(x, y, z\right)$ such that:
[b](i)[/b] the numbers $ x$, $ y$, $ z$ are of the same color,
and
[b](ii)[/b] the number $ x \plus{} y \plus{} z$ is divisible by $ n$.
[i]Author: Gerhard Wöginger, Netherlands[/i]
Let $n{}$ be a positive integer. What is the smallest sum of digits that $5^n + 6^n + 2022^n$ can take?
Let $p$ be a given odd prime. Find the largest integer $k'$ such that it is possible to partition $\{1,2,\cdots,p-1\}$ into two sets $X,Y$ such that for any $k$ with $0 \le k \le k'$,
$$\sum_{a \in X}a^k \equiv \sum_{b \in Y}b^k \pmod p$$
[i]houkai[/i]
For integers $x$, $y$, and $z$, we have $(x-y)(y-z)(z-x)=x+y+z$. Prove that $27|x+y+z$.
Prove that for any n natural, the number \[ \sum \limits_{k=0}^{n} \binom{2n+1}{2k+1} 2^{3k} \]
cannot be divided by $5$.
If $a$ , $b$ , $c$ , $d$ are integers and $A=2(a-2b+c)^4+2(b-2c+a)^4+2(c-2a+b)^4$ , $B=d(d+1)(d+2)(d+3)+1$ , then prove that $\left (\sqrt{A}+1 \right )^2 +B$ cannot be a perfect square.
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]
Given are distinct natural numbers $a$, $b$, and $c$. Prove that
\[ \gcd(ab+1, ac+1, bc+1)\le \frac{a+b+c}{3}\]