Olympiad problems

2017 South East Mathematical Olympiad, 7

Find the maximum value of $n$, such that there exist $n$ pairwise distinct positive numbers $x_1,x_2,\cdots,x_n$, satisfy $$x_1^2+x_2^2+\cdots+x_n^2=2017$$

1995 Rioplatense Mathematical Olympiad, Level 3, 1

Tags: number theory , perfect square

Let $n$ and $p$ be two integers with $p$ positive prime, such that $pn + 1$ is a perfect square. Show that $n + 1$ is the sum of $p$ perfect squares, not necessarily distinct.

2001 All-Russian Olympiad Regional Round, 9.8

Tags: number theory , perfect square , digit

Sasha wrote a non-zero number on the board and added it to it on the right, one non-zero digit at a time, until he writes out a million digits. Prove that an exact square has been written on the board no more than $100$ times.

1998 Belarus Team Selection Test, 2

Tags: number theory , diophantine , perfect square , diophantine equation

a) Given that integers $a$ and $b$ satisfy the equality $$a^2 - (b^2 - 4b + 1) a - (b^4 - 2b^3) = 0 \,\,\, (*)$$, prove that $b^2 + a$ is a square of an integer. b) Do there exist an infinitely many of pairs $(a,b)$ satisfying (*)?

2008 Postal Coaching, 2

Tags: polynomial , integer polynomial , perfect square , algebra

Find all polynomials $P$ with integer coefficients such that wherever $a, b \in N$ and $a+b$ is a square we have $P(a) + P(b)$ is also a square.

2022 Saudi Arabia JBMO TST, 4

Tags: number theory , perfect square

Determine the smallest positive integer $a$ for which there exist a prime number $p$ and a positive integer $b \ge 2$ such that $$\frac{a^p -a}{p}=b^2.$$

2020-21 IOQM India, 8

Tags: perfect square , digit , number theory

A $5$-digit number (in base $10$) has digits $k, k + 1, k + 2, 3k, k + 3$ in that order, from left to right. If this number is $m^2$ for some natural number $m$, find the sum of the digits of $m$.

1989 Spain Mathematical Olympiad, 4

Tags: number theory , power of number , perfect square

Show that the number $1989$ as well as each of its powers $1989^n$ ($n \in N$), can be expressed as a sum of two positive squares in at least two ways.

2017 Hanoi Open Mathematics Competitions, 3

Tags: perfect square , number theory

Suppose $n^2 + 4n + 25$ is a perfect square. How many such non-negative integers $n$'s are there? (A): $1$ (B): $2$ (C): $4$ (D): $6$ (E): None of the above.

2019 Nigerian Senior MO Round 4, 4

Tags: algebra , recurrence relation , perfect square , odd , sequence , number theory

We consider the real sequence ($x_n$) defined by $x_0=0, x_1=1$ and $x_{n+2}=3x_{n+1}-2 x_{n}$ for $n=0,1,2,...$ We define the sequence ($y_n$) by $y_n=x^2_n+2^{n+2}$ for every nonnegative integer $n$. Prove that for every $n>0, y_n$ is the square of an odd integer.

1998 Junior Balkan MO, 1

Tags: induction , perfect square

Prove that the number $\underbrace{111\ldots 11}_{1997}\underbrace{22\ldots 22}_{1998}5$ (which has 1997 of 1-s and 1998 of 2-s) is a perfect square.

2012 Danube Mathematical Competition, 1

Tags: number theory , perfect square

a) Exist $a, b, c, \in N$, such that the numbers $ab+1,bc+1$ and $ca+1$ are simultaneously even perfect squares ? b) Show that there is an infinity of natural numbers (distinct two by two) $a, b, c$ and $d$, so that the numbers $ab+1,bc+1, cd+1$ and $da+1$ are simultaneously perfect squares.

2019 Olympic Revenge, 2

Tags: number theory , perfect square , prime number , my 100th post

Prove that there exist infinitely many positive integers $n$ such that the greatest prime divisor of $n^2+1$ is less than $n \cdot \pi^{-2019}.$

2006 Estonia National Olympiad, 2

Tags: perfect square , divisible , number theory

Let $a, b$ and $c$ be positive integers such that $ab + 1, bc + 1$ and $ca + 1$ are all integer squares. a) Give an example of such numbers $a, b$ and $c$. b) Prove that at least one of the numbers $a, b$ and $c$ is divisible by $4$

2006 Thailand Mathematical Olympiad, 1

Tags: number theory , consecutive , perfect square

Show that the product of three consecutive positive integers is never a perfect square.

2013 Dutch IMO TST, 2

Tags: perfect square , number theory , rational

Determine all integers $n$ for which $\frac{4n-2}{n+5}$ is the square of a rational number.

1989 Austrian-Polish Competition, 9

Tags: number theory , sum of squares , perfect square , odd

Find the smallest odd natural number $N$ such that $N^2$ is the sum of an odd number (greater than $1$) of squares of adjacent positive integers.

2000 Abels Math Contest (Norwegian MO), 1a

Tags: odd , perfect square , number theory

Show that any odd number can be written as the difference between two perfect squares.

1994 Argentina National Olympiad, 2

Tags: algebra , number theory , perfect square

For what positive integer values of $x$ is $x^4 + 6x^3 + 11x^2 + 3x + 31$ a perfect square?

1999 Austrian-Polish Competition, 5

Tags: recurrence relation , sequence , perfect square , number theory

A sequence of integers $(a_n)$ satisfies $a_{n+1} = a_n^3 + 1999$ for $n = 1,2,....$ Prove that there exists at most one $n$ for which $a_n$ is a perfect square.

2015 QEDMO 14th, 10

Tags: perfect square , number theory

Find all prime numbers $p$ for which $p^3- p + 1$ is a perfect square .

1999 Spain Mathematical Olympiad, 2

Tags: perfect square , integer sequence , number theory

Prove that there exists a sequence of positive integers $a_1,a_2,a_3, ...$ such that $a_1^2+a_2^2+...+a_n^2$ is a perfect square for all positive integers $n$.

2022 Saudi Arabia JBMO TST, 1

Tags: number theory , perfect square

The positive $n > 3$ called ‘nice’ if and only if $n +1$ and $8n + 1$ are both perfect squares. How many positive integers $k \le 15$ such that $4n + k$ are composites for all nice numbers $n$?

2011 Argentina National Olympiad, 4

Tags: number theory , perfect square

For each natural number $n$ we denote $a_n$ as the greatest perfect square less than or equal to $n$ and $b_n$ as the least perfect square greater than $n$. For example $a_9=3^2$, $b_9=4^2$ and $a_{20}=4^2$, $b_{20}=5^2$. Calculate: $$\frac{1}{a_1b_1}+\frac{1}{a_2b_2}+\frac{1}{a_3b_3}+\ldots +\frac{1}{a_{600}b_{600}}$$

1984 Bundeswettbewerb Mathematik, 3

Tags: number theory , perfect square

Let $a$ and $b$ be positive integers. Show that if $a \cdot b$ is even, then there are positive integers $c$ and $d$ with $a^2 + b^2 + c^2 = d^2$; if, on the other hand, $a\cdot b$ is odd, there are no such positive integers $c$ and $d$.