Olympiad problems

2013 India PRMO, 14

Let $m$ be the smallest odd positive integer for which $1+ 2 +...+ m$ is a square of an integer and let $n$ be the smallest even positive integer for which $1 + 2 + ... + n$ is a square of an integer. What is the value of $m + n$?

2019 AMC 10, 11

Tags: AMC , AMC 10 , AMC 10 A , Perfect Squares , perfect cubes , 2019 AMC 10A , 2019 AMC

How many positive integer divisors of $201^9$ are perfect squares or perfect cubes (or both)? $\textbf{(A) } 32 \qquad\textbf{(B) } 36 \qquad\textbf{(C) } 37 \qquad\textbf{(D) } 39 \qquad\textbf{(E) } 41$

2012 Balkan MO Shortlist, N1

Tags: number theory , Sequence , Perfect Squares , consecutive , Perfect Square

A sequence $(a_n)_{n=1}^{\infty}$ of positive integers satisfies the condition $a_{n+1} = a_n +\tau (n)$ for all positive integers $n$ where $\tau (n)$ is the number of positive integer divisors of $n$. Determine whether two consecutive terms of this sequence can be perfect squares.

1996 IMO, 4

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?

2023 Peru MO (ONEM), 1

Tags: number theory , divides , Perfect Squares

We define the set $M = \{1^2,2^2,3^2,..., 99^2, 100^2\}$. a) What is the smallest positive integer that divides exactly two elements of $M$? b) What is the largest positive integer that divides exactly two elements of $M$?

1994 Argentina National Olympiad, 2

Tags: number theory , Perfect Squares , algebra

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

2011 Belarus Team Selection Test, 1

Tags: number theory , Perfect Squares , Perfect Square

Find the least possible number of elements which can be deleted from the set $\{1,2,...,20\}$ so that the sum of no two different remaining numbers is not a perfect square. N. Sedrakian , I.Voronovich

2019 Peru MO (ONEM), 1

Tags: number theory , Perfect Squares , Perfect Square

Determine for what $n\ge 3$ integer numbers, it is possible to find positive integer numbers $a_1 < a_2 < ...< a_n$ such $\frac{1}{a_1}+\frac{1}{a_2}+...+\frac{1}{a_n}=1$ and $a_1 a_2\cdot\cdot\cdot a_n$ is a perfect square.

2018 Istmo Centroamericano MO, 4

Tags: trinomial , quadratics , Perfect Squares , number theory

Let $t$ be an integer. Suppose the equation $$x^2 + (4t - 1) x + 4t^2 = 0$$ has at least one positive integer solution $n$. Show that $n$ is a perfect square.

2018 Azerbaijan BMO TST, 2

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 $x,y$.

2017 CHMMC (Fall), 5

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

Find the number of primes $p$ such that $p! + 25p$ is a perfect square.

1969 IMO Shortlist, 62

Tags: number theory , Perfect Squares , IMO Shortlist , IMO Longlist

Which natural numbers can be expressed as the difference of squares of two integers?

2022 China Team Selection Test, 5

Tags: number theory , Perfect Squares

Show that there exist constants $c$ and $\alpha > \frac{1}{2}$, such that for any positive integer $n$, there is a subset $A$ of $\{1,2,\ldots,n\}$ with cardinality $|A| \ge c \cdot n^\alpha$, and for any $x,y \in A$ with $x \neq y$, the difference $x-y$ is not a perfect square.

2009 Abels Math Contest (Norwegian MO) Final, 1a

Tags: number theory , Perfect Square , Perfect Squares

Show that there exist infinitely many integers that cannot be written as the difference between two perfect squares.

2022 Durer Math Competition Finals, 4

Tags: number theory , Perfect Squares , Divisors

Show that the divisors of a number $n \ge 2$ can only be divided into two groups in which the product of the numbers is the same if the product of the divisors of $n$ is a square number.

2022 Canadian Mathematical Olympiad Qualification, 2

Tags: number theory , Perfect Squares , Perfect Square

Determine all pairs of integers $(m, n)$ such that $m^2 + n$ and $n^2 + m$ are both perfect squares.

Kvant 2023, M2731

Tags: Kvant , number theory , Perfect Squares

There are 2023 natural written in a row. The first number is 12, and each number starting from the third is equal to the product of the previous two numbers, or to the previous number increased by 4. What is the largest number of perfect squares that can be among the 2023 numbers? [i]Based on the British Mathematical Olympiad[/i]

1993 Bundeswettbewerb Mathematik, 3

Tags: number theory , Perfect Squares , Perfect Square

There are pairs of square numbers with the following two properties: (1) Their decimal representations have the same number of digits, with the first digit starting is different from $0$ . (2) If one appends the second to the decimal representation of the first, the decimal representation results another square number. Example: $16$ and $81$; $1681 = 41^2$. Prove that there are infinitely many pairs of squares with these properties.

2003 Junior Tuymaada Olympiad, 2

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

Find all natural $ x $ for which $ 3x+1 $ and $ 6x-2 $ are perfect squares, and the number $ 6x^2-1 $ is prime.

KoMaL A Problems 2017/2018, A. 717

Tags: number theory , prime numbers , Perfect Squares

Let's call a positive integer $n$ special, if there exist two nonnegativ integers ($a, b$), such that $n=2^a\times 3^b$. Prove that if $k$ is a positive integer, then there are at most two special numbers greater then $k^2$ and less than $k^2+2k+1$.

2021 Middle European Mathematical Olympiad, 8

Tags: number theory , Digits , Perfect Squares , memo , MEMO 2021

Prove that there are infinitely many positive integers $n$ such that $n^2$ written in base $4$ contains only digits $1$ and $2$.

2021 Brazil National Olympiad, 3

Tags: number theory , floor function , powers , irrational number , Perfect Squares , Brazilian Math Olympiad , Brazil

Find all positive integers $k$ for which there is an irrational $\alpha>1$ and a positive integer $N$ such that $\left\lfloor\alpha^{n}\right\rfloor$ is a perfect square minus $k$ for every integer $n$ with $n>N$.

2003 Paraguay Mathematical Olympiad, 3

Tags: number theory , Perfect Squares , Perfect Square

Today the age of Pedro is written and then the age of Luisa, obtaining a number of four digits that is a perfect square. If the same is done in $33$ years from now, there would be a perfect square of four digits . Find the current ages of Pedro and Luisa.

2021 Austrian MO National Competition, 6

Tags: number theory , Perfect Square , Perfect Squares

Let $p$ be an odd prime number and $M$ a set derived from $\frac{p^2 + 1}{2}$ square numbers. Investigate whether $p$ elements can be selected from this set whose arithmetic mean is an integer. (Walther Janous)

1994 Spain Mathematical Olympiad, 1

Tags: Arithmetic Progression , number theory , Perfect Squares , Perfect Square

Prove that if an arithmetic progression contains a perfect square, then it contains infinitely many perfect squares.