Olympiad problems

2020 Final Mathematical Cup, 1

Let $n$ be a given positive integer. Prove that there is no positive divisor $d$ of $2n^2$ such that $d^2n^2+d^3$ is a square of an integer.

2015 Peru MO (ONEM), 3

Tags: number theory , floor function , Perfect Squares

Let $a_1, a_2, . . . , a_n$ be positive integers, with $n \ge 2$, such that $$ \lfloor \sqrt{a_1 \cdot a_2\cdot\cdot\cdot a_n} \rfloor = \lfloor \sqrt{a_1} \rfloor \cdot \lfloor \sqrt{a_2} \rfloor \cdot\cdot\cdot \lfloor \sqrt{a_n} \rfloor.$$ Prove that at least $n - 1$ of these numbers are perfect squares. Clarification: Given a real number $x$, $\lfloor x\rfloor$ denotes the largest integer that is less than or equal to $x$. For example $\lfloor \sqrt2\rfloor$ and $\lfloor 3\rfloor =3$.

2017 Balkan MO Shortlist, N2

Tags: Perfect Squares , functional equation , functional equation in N , number theory

Find all functions $f :Z_{>0} \to Z_{>0}$ such that the number $xf(x) + f ^2(y) + 2xf(y)$ is a perfect square for all positive integers $x,y$.

2019 Regional Olympiad of Mexico Center Zone, 1

Tags: number theory , Perfect Squares , Perfect Square

Let $a$, $b$, and $c $ be integers greater than zero. Show that the numbers $$2a ^ 2 + b ^ 2 + 3 \,\,, 2b ^ 2 + c ^ 2 + 3\,\,, 2c ^ 2 + a ^ 2 + 3 $$ cannot be all perfect squares.

2023 China Northern MO, 6

Tags: number theory , Perfect Squares , Perfect Square

A positive integer $m$ is called a [i]beautiful [/i] integer if that there exists a positive integer $n$ such that $m = n^2+ n + 1$. Prove that there are infinitely many [i]beautiful [/i] integers with square factors, and the square factors of different beautiful integers are relative prime.

2018 IMAR Test, 4

Tags: number theory , romania , Perfect Squares

Prove that every non-negative integer $n$ is expressible in the form $n=t^2+u^2+v^2+w^2$, where $t,u,v,w$ are integers such that $t+u+v+w$ is a perfect square. [i]* * *[/i]

2020 Junior Balkan Team Selection Tests - Moldova, 4

Tags: Perfect Squares , number theory

A natural number $n$ is called "$k$-squared" if it can be written as a sum of $k$ perfect squares not equal to 0. a) Prove that 2020 is "$2$-squared" , "$3$-squared" and "$4$-squared". b) Determine all natural numbers not equal to 0 ($a, b, c, d ,e$) $a<b<c<d<e$ that verify the following conditions simultaneously : 1) $e-2$ , $e$ , $e+4$ are all prime numbers. 2) $a^2+ b^2 + c^2 + d^2 + e^2$ = 2020.

2010 Puerto Rico Team Selection Test, 5

Tags: number theory , Perfect Squares

Find all prime numbers $p$ and $q$ such that $2p^2q + 45pq^2$ is a perfect square.

2023 AMC 10, 15

Tags: 2023 AMC , 2023 AMC 10B , AMC 10 , AMC , number theory , Perfect Squares

What is the least positive integer $m$ such that $m \cdot 2! \cdot 3! \cdot 4! \cdot 5! \cdots 16!$ is a perfect square? $\textbf{(A) }30\qquad\textbf{(B) }30030\qquad\textbf{(C) }70\qquad\textbf{(D) }1430\qquad\textbf{(E) }1001$

V Soros Olympiad 1998 - 99 (Russia), 9.8

Tags: number theory , Perfect Squares

Find all natural numbers whose decimal notation consists of different digits of the same parity and which are perfect squares.

2015 Greece JBMO TST, 3

Tags: number theory , Perfect Squares , Perfect Square , positive integer

Prove that there is not a positive integer $n$ such that numbers $(n+1)2^n, (n+3)2^{n+2}$ are both perfect squares.

2001 Cuba MO, 3

Tags: number theory , Perfect Squares

Let $n$ be a positive integer. a) Prove that the number $(2n + 1)^3 - (2n - 1)^3$ is the sum of three perfect squares. b) Prove that the number $(2n+1)^3-2$ is the sum of $3n-1$ perfect squares greater than $1$.

1997 Tuymaada Olympiad, 1

Tags: number theory , Product , Perfect Squares , Perfect Square

The product of any three of these four natural numbers is a perfect square. Prove that these numbers themselves are perfect squares.

2023 Czech-Polish-Slovak Junior Match, 1

Tags: number theory , Perfect Squares , Perfect Square , sum of digits

Let $S(n)$ denote the sum of all digits of natural number $n$. Determine all natural numbers $n$ for which both numbers $n + S(n)$ and $n - S(n)$ are square powers of non-zero integers.

2021 Argentina National Olympiad, 5

Tags: number theory , Perfect Squares , contests

Determine all positive integers $n$ such that $$n\cdot 2^{n-1}+1$$ is a perfect square.

1996 IMO Shortlist, 2

Tags: modular arithmetic , number theory , Perfect Squares , IMO , IMO 1996 , quadratic reciprocity , IMO Shortlist

The positive integers $ a$ and $ b$ are such that the numbers $ 15a \plus{} 16b$ and $ 16a \minus{} 15b$ are both squares of positive integers. What is the least possible value that can be taken on by the smaller of these two squares?

2019 Girls in Mathematics Tournament, 1

Tags: number theory , Perfect Squares , Perfect Square , combinatorics

During the factoring class, Esmeralda observed that $1$, $3$ and $5$ can be written as the difference of two perfect squares, as can be seen: $1 = 1^2 - 0^2$ $3 = 2^2 - 1^2$ $5 = 3^2 - 2^2$ a) Show that all numbers written in the form $2 * m + 1$ can be written as a difference of two perfect squares. b) Show how to calculate the value of the expression $E = 1 + 3 + 5 + ... + (2m + 1)$. c) Esmeralda, happy with what she discovered, decided to look for other ways to write $2019$ as the difference of two perfect squares of positive integers. Determine how many ways it can do what you want.

2011 Bosnia And Herzegovina - Regional Olympiad, 4

Tags: number theory , Perfect Squares

For positive integer $n$, prove that at least one of the numbers $$A=2n-1 , B=5n-1, C=13n-1$$ is not perfect square

1986 IMO Shortlist, 5

Tags: quadratics , modular arithmetic , number theory , system of equations , Perfect Squares , IMO , IMO 1986

Let $d$ be any positive integer not equal to $2, 5$ or $13$. Show that one can find distinct $a,b$ in the set $\{2,5,13,d\}$ such that $ab-1$ is not a perfect square.

2000 Romania National Olympiad, 1

Tags: number theory , Perfect Squares

a) Show that the number $(2k + 1)^3 - (2k - 1)^3$, $k \in Z$, is the sum of three perfect squares. b) Represent the number $(2n + 1)^3 -2$, $n \in N^*$, as the sum of $3n- 1$ perfect squares greater than $1$.

2015 India PRMO, 6

Tags: number theory , Digits , sum of digits , Perfect Squares

$6.$ How many two digit positive integers $N$ have the property that the sum of $N$ and the number obtained by reversing the order of the digits of $N$ is a perfect square $?$

2018 Malaysia National Olympiad, B3

Tags: number theory , Perfect Squares

Let $n$ be an integer greater than $1$, such that $3n + 1$ is a perfect square. Prove that $n + 1$ can be expressed as a sum of three perfect squares.

2000 Moldova Team Selection Test, 9

Tags: Sequence , Perfect Squares , Integers , number theory with sequences

The sequence $x_{n}$ is defined by: $x_{0}=1, x_{1}=0, x_{2}=1,x_{3}=1, x_{n+3}=\frac{(n^2+n+1)(n+1)}{n}x_{n+2}+(n^2+n+1)x_{n+1}-\frac{n+1}{n}x_{n} (n=1,2,3..)$ Prove that all members of the sequence are perfect squares.

2012 Danube Mathematical Competition, 1

Tags: Perfect Squares , Perfect Square , number theory

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.

1989 Brazil National Olympiad, 2

Tags: Brazilian Math Olympiad 1989 , Brazilian Math Olympiad , number theory , Perfect Squares , Pell equations

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.