Olympiad problems

2021 Stars of Mathematics, 2

Let $n{}$ be a positive integer. Show that there exists a polynomial $f{}$ of degree $n{}$ with integral coefficients such that \[f^2=(x^2-1)g^2+1,\] where $g{}$ is a polynomial with integral coefficients.

2009 China Girls Math Olympiad, 8

Tags: pell equation , diophantine equation , continued fraction , modular arithmetic , induction , floor function , number theory

For a positive integer $ n,$ $ a_{n}\equal{}n\sqrt{5}\minus{} \lfloor n\sqrt{5}\rfloor$. Compute the maximum value and the minimum value of $ a_{1},a_{2},\ldots ,a_{2009}.$

2014 IMAR Test, 2

Tags: pell equation , composite number , number theory

Let $\epsilon$ be a positive real number. A positive integer will be called $\epsilon$-squarish if it is the product of two integers $a$ and $b$ such that $1 < a < b < (1 +\epsilon )a$. Prove that there are infinitely many occurrences of six consecutive $\epsilon$ -squarish integers.

1989 Brazil National Olympiad, 2

Tags: number theory , pell equation , perfect square

Let $k$ a positive integer number such that $\frac{k(k+1)}{3}$ is a perfect square. Show that $\frac{k}{3}$ and $k+1$ are both perfect squares.

1994 Nordic, 4

Tags: pell equation , perfect square , integer , number theory

Determine all positive integers $n < 200$, such that $n^2 + (n+ 1)^2$ is the square of an integer.

2002 IMO Shortlist, 4

Tags: infinitely many solutions , vieta jumping , pell equation , equation , algebra , number theory

Is there a positive integer $m$ such that the equation \[ {1\over a}+{1\over b}+{1\over c}+{1\over abc}={m\over a+b+c} \] has infinitely many solutions in positive integers $a,b,c$?

2019 Middle European Mathematical Olympiad, 8

Tags: pell equation , number theory

Let $N$ be a positive integer such that the sum of the squares of all positive divisors of $N$ is equal to the product $N(N+3)$. Prove that there exist two indices $i$ and $j$ such that $N=F_iF_j$ where $(F_i)_{n=1}^{\infty}$ is the Fibonacci sequence defined as $F_1=F_2=1$ and $F_n=F_{n-1}+F_{n-2}$ for $n\geq 3$. [i]Proposed by Alain Rossier, Switzerland[/i]

2013 Iran MO (3rd Round), 4

Tags: polynomial , vieta , diophantine equation , pell equation , number theory , algebra

Prime $p=n^2 +1$ is given. Find the sets of solutions to the below equation: \[x^2 - (n^2 +1)y^2 = n^2.\] (25 points)

2018 Romania Team Selection Tests, 4

Tags: factorial , number theory , pell equation , sophie germain identity

Given an non-negative integer $k$, show that there are infinitely many positive integers $n$ such that the product of any $n$ consecutive integers is divisible by $(n+k)^2+1$.

2020 Korea - Final Round, P6

Tags: pell equation , number theory

Find all positive integers $n$ such that $6(n^4-1)$ is a square of an integer.

2019 Jozsef Wildt International Math Competition, W. 1

Tags: pell equation , split method , number theory

The Pell numbers $P_n$ satisfy $P_0 = 0$, $P_1 = 1$, and $P_n=2P_{n-1}+P_{n-2}$ for $n\geq 2$. Find $$\sum \limits_{n=1}^{\infty} \left (\tan^{-1}\frac{1}{P_{2n}}+\tan^{-1}\frac{1}{P_{2n+2}}\right )\tan^{-1}\frac{2}{P_{2n+1}}$$

2018 China Team Selection Test, 6

Tags: algebra , pell equation , number theory

Find all pairs of positive integers $(x, y)$ such that $(xy+1)(xy+x+2)$ be a perfect square .

2022 Cyprus TST, 2

Tags: perfect square , pell equation , number theory

Let $n, m$ be positive integers such that \[n(4n+1)=m(5m+1)\] (a) Show that the difference $n-m$ is a perfect square of a positive integer. (b) Find a pair of positive integers $(n, m)$ which satisfies the above relation. Additional part (not asked in the TST): Find all such pairs $(n,m)$.

2018 China Team Selection Test, 6

Tags: number theory , algebra , pell equation

Find all pairs of positive integers $(x, y)$ such that $(xy+1)(xy+x+2)$ be a perfect square .