Found problems: 143
Consider all ordered pairs $(a, b)$ of positive integers such that $\frac{a^2 + b^2 + 2}{ab}$ is an integer and $a\le b$. We label all such pairs in increasing order by their distance from the origin. (It is guaranteed that no ties exist.) Thus $P_1 = (1, 1), P_2 = (1, 3)$, and so on. If $P_{2020} = (x, y),$ then compute the remainder when $x + y$ is divided by $2017$.
[i]Proposed by Ashwin Sah[/i]
Positive integer $k(\ge 8)$ is given. Prove that if there exists a pair of positive integers $(x,y)$ that satisfies the conditions below, then there exists infinitely many pairs $(x,y)$.
(1) $ $ $x\mid y^2-3, y\mid x^2-2$
(2) $ $ $gcd\left(3x+\frac{2(y^2-3)}{x},2y+\frac{3(x^2-2)}{y}\right)=k$ $ $
Prove that there are infinitely many triples $(a, b, p)$ of positive integers with $p$ prime, $a < p$, and $b < p$, such that $(a + b)^p - a^p - b^p$ is a multiple of $p^3$.
[i]Noam Elkies[/i]
Find the smallest positive integer $n$ or show no such $n$ exists, with the following property: there are infinitely many distinct $n$-tuples of positive rational numbers $(a_1, a_2, \ldots, a_n)$ such that both
$$a_1+a_2+\dots +a_n \quad \text{and} \quad \frac{1}{a_1} + \frac{1}{a_2} + \dots + \frac{1}{a_n}$$
are integers.
Find the smallest positive integer $n$ or show no such $n$ exists, with the following property: there are infinitely many distinct $n$-tuples of positive rational numbers $(a_1, a_2, \ldots, a_n)$ such that both
$$a_1+a_2+\dots +a_n \quad \text{and} \quad \frac{1}{a_1} + \frac{1}{a_2} + \dots + \frac{1}{a_n}$$
are integers.
Determine for which $ n$ positive integer the equation: $ a \plus{} b \plus{} c \plus{} d \equal{} n \sqrt {abcd}$ has positive integer solutions.
Find all pairs $(m,n)$ of natural numbers with $m<n$ such that $m^2+1$ is a multiple of $n$ and $n^2+1$ is a multiple of $m$.
Let $k$ be an integer and define a sequence $a_0 , a_1 ,a_2 ,\ldots$ by
$$ a_0 =0 , \;\; a_1 =k \;\;\text{and} \;\; a_{n+2} =k^{2}a_{n+1}-a_n \; \text{for} \; n\geq 0.$$
Prove that $a_{n+1} a_n +1$ divides $a_{n+1}^{2} +a_{n}^{2}$ for all $n$.
Find all ordered pairs $ (m,n)$ where $ m$ and $ n$ are positive integers such that $ \frac {n^3 \plus{} 1}{mn \minus{} 1}$ is an integer.
Let $a$ and $b$ be two positive integers. Prove that the integer
\[a^2+\left\lceil\frac{4a^2}b\right\rceil\]
is not a square. (Here $\lceil z\rceil$ denotes the least integer greater than or equal to $z$.)
[i]Russia[/i]
Find all positive integers $k$ such that there exists positive integers $a, b$ such that
\[a^2+4=(k^2-4)b^2.\]
Find all positive integers $n$ such that there exists a sequence of positive integers $a_1$, $a_2$,$\ldots$, $a_n$ satisfying: \[a_{k+1}=\frac{a_k^2+1}{a_{k-1}+1}-1\] for every $k$ with $2\leq k\leq n-1$.
[i]Proposed by North Korea[/i]
A polynomial $P(x, y, z)$ in three variables with real coefficients satisfies the identities
$$P(x, y, z)=P(x, y, xy-z)=P(x, zx-y, z)=P(yz-x, y, z).$$
Prove that there exists a polynomial $F(t)$ in one variable such that
$$P(x,y,z)=F(x^2+y^2+z^2-xyz).$$
Find all ordered pairs $(a,b)$ of positive integers for which the numbers $\dfrac{a^3b-1}{a+1}$ and $\dfrac{b^3a+1}{b-1}$ are both positive integers.
Find all triples $(p,q,n)$ that satisfy
\[q^{n+2} \equiv 3^{n+2} (\mod p^n) ,\quad p^{n+2} \equiv 3^{n+2} (\mod q^n)\]
where $p,q$ are odd primes and $n$ is an positive integer.
Let $a$ and $b$ be two positive integers. Prove that the integer
\[a^2+\left\lceil\frac{4a^2}b\right\rceil\]
is not a square. (Here $\lceil z\rceil$ denotes the least integer greater than or equal to $z$.)
[i]Russia[/i]
Find all ordered pairs $(a,b)$ of positive integers for which the numbers $\dfrac{a^3b-1}{a+1}$ and $\dfrac{b^3a+1}{b-1}$ are both positive integers.
Any positive integer $N$ which can be expressed as the sum of three squares can obviously be written as \[N=\frac{a^2+b^2+c^2+d^2}{1+abcd}\]where $a,b,c,d$ are nonnegative integers. Is the mutual assertion true?