Olympiad problems

2021 Federal Competition For Advanced Students, P2, 6

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)

2020 Colombia National Olympiad, 1

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

A positive integer is called [i]sabroso [/i]if when it is added to the number obtained when its digits are interchanged from one side of its written form to the other, the result is a perfect square. For example, $143$ is sabroso, since $143 + 341 =484 = 22^2$. Find all two-digit sabroso numbers.

2007 Thailand Mathematical Olympiad, 17

Tags: number theory , Perfect Square

Compute the product of positive integers $n$ such that $n^2 + 59n + 881$ is a perfect square.

2018 Greece National Olympiad, 1

Tags: Sequence , Perfect Square

Let $(x_n), n\in\mathbb{N}$ be a sequence such that $x_{n+1}=3x_n^3+x_n, \forall n\in\mathbb{N}$ and $x_1=\frac{a}{b}$ where $a,b$ are positive integers such that $3\not|b$. If $x_m$ is a square of a rational number for some positive integer $m$, prove that $x_1$ is also a square of a rational number.

2018 Bosnia And Herzegovina - Regional Olympiad, 3

Tags: number theory , prime numbers , Perfect Square

Let $p$ and $q$ be prime numbers such that $p^2+pq+q^2$ is perfect square. Prove that $p^2-pq+q^2$ is prime

2000 Singapore Team Selection Test, 2

Tags: number theory , prime , Perfect Square

Find all prime numbers $p$ such that $5^p + 12^p$ is a perfect square

1998 Akdeniz University MO, 1

Tags: number theory , equation , Perfect Square

Prove that, for $k \in {\mathbb Z^+}$ $$k(k+1)(k+2)(k+3)$$ is not a perfect square.

2012 Bosnia And Herzegovina - Regional Olympiad, 4

Tags: number theory , Perfect Square

Can number $2012^n-3^n$ be perfect square, while $n$ is positive integer

2009 Belarus Team Selection Test, 4

Tags: number theory , Perfect Square , diophantine , Diophantine equation

Let $x,y,z$ be integer numbers satisfying the equality $yx^2+(y^2-z^2)x+y(y-z)^2=0$ a) Prove that number $xy$ is a perfect square. b) Prove that there are infinitely many triples $(x,y,z)$ satisfying the equality. I.Voronovich

2008 Dutch IMO TST, 4

Tags: Perfect Square , number theory , radical

Let $n$ be positive integer such that $\sqrt{1 + 12n^2}$ is an integer. Prove that $2 + 2\sqrt{1 + 12n^2}$ is the square of an integer.

2015 Argentina National Olympiad, 5

Tags: number theory , Perfect Square

Find all prime numbers $p$ such that $p^3-4p+9$ is a perfect square.

2020-21 IOQM India, 6

Tags: number theory , Perfect Square

What is the least positive integer by which $2^5 \cdot 3^6 \cdot 4^3 \cdot 5^3 \cdot 6^7$ should be multiplied so that, the product is a perfect square?

2007 Dutch Mathematical Olympiad, 4

Tags: Perfect Square , number theory , exponential

Determine the number of integers $a$ satisfying $1 \le a \le 100$ such that $a^a$ is a perfect square. (And prove that your answer is correct.)

2012 Estonia Team Selection Test, 1

Tags: Perfect Square , perfect cube , number theory

Prove that for any positive integer $k$ there exist $k$ pairwise distinct integers for which the sum of their squares equals the sum of their cubes.

2007 Estonia Math Open Junior Contests, 10

Tags: combinatorics , Sum , number theory , Perfect Square

Prove that for every integer $k$, there exists a integer $n$ which can be expressed in at least $k$ different ways as the sum of a number of squares of integers (regardless of the order of additions) where the additions are all in different pairs.

1962 Czech and Slovak Olympiad III A, 1

Tags: Perfect Square , prime , algebra , quadratic polynomial

Determine all integers $x$ such that $2x^2-x-36$ is a perfect square of a prime.

2018 Ukraine Team Selection Test, 7

Tags: Perfect Square , number theory , prime , divisor

The prime number $p > 2$ and the integer $n$ are given. Prove that the number $pn^2$ has no more than one divisor $d$ for which $n^2+d$ is the square of the natural number. .

2017 Peru IMO TST, 4

Tags: factorial , number theory , Perfect Square

The product $1\times 2\times 3\times ...\times n$ is written on the board. For what integers $n \ge 2$, we can add exclamation marks to some factors to convert them into factorials, in such a way that the final product can be a perfect square?

1978 Czech and Slovak Olympiad III A, 6

Tags: number theory , algebra , Sequence , Perfect Square

Show that the number \[p_n=\left(\frac{3+\sqrt5}{2}\right)^n+\left(\frac{3-\sqrt5}{2}\right)^n-2\] is a positive integer for any positive integer $n.$ Furthermore, show that the numbers $p_{2n-1}$ and $p_{2n}/5$ are perfect squares $($for any positive integer $n).$

2005 iTest, 34

Tags: number theory , diophantine , Diophantine equation , Perfect Square

If $x$ is the number of solutions to the equation $a^2 + b^2 + c^2 = d^2$ of the form $(a,b,c,d)$ such that $\{a,b,c\}$ are three consecutive square numbers and $d$ is also a square number, find $x$.

1997 IMO Shortlist, 15

Tags: number theory , modular arithmetic , Perfect Square , perfect cube , arithmetic sequence , IMO Shortlist , Romanian TST

An infinite arithmetic progression whose terms are positive integers contains the square of an integer and the cube of an integer. Show that it contains the sixth power of an integer.

1963 Swedish Mathematical Competition., 1

Tags: number theory , Perfect Square

How many positive integers have square less than $10^7$?

2018 Dutch IMO TST, 3

Tags: number theory , Sum , floor function , Perfect Square

Let $n \ge 0$ be an integer. A sequence $a_0,a_1,a_2,...$ of integers is defined as follows: we have $a_0 = n$ and for $k \ge 1, a_k$ is the smallest integer greater than $a_{k-1}$ for which $a_k +a_{k-1}$ is the square of an integer. Prove that there are exactly $\lfloor \sqrt{2n}\rfloor$ positive integers that cannot be written in the form $a_k - a_{\ell}$ with $k > \ell\ge 0$.

2012 Argentina National Olympiad, 2

Tags: number theory , Perfect Square

Determine all natural numbers $n$ for which there are $2n$ distinct positive integers $x_1,…,x_n,y_1,…,y_n$ such that the product $$(11x^2_1+12y^2_1)(11x^2_2+12y^2_2)…(11x^2_n+12y^2_n)$$ is a perfect square.

1964 Dutch Mathematical Olympiad, 3

Tags: number theory , diophantine , Diophantine equation , Perfect Square

Solve $ (n + 1)(n +10) = q^2$, for certain $q$ and maximum $n$.