Found problems: 397
2009 All-Russian Olympiad Regional Round, 9.2
Rational numbers $a$ and $b$ satisfy the equality $$a^3b+ab^3+2a^2b^2+2a + 2b + 1 = 0. $$ Prove that the number $1-ab$ is the square of the rational numbers.
2005 Bosnia and Herzegovina Junior BMO TST, 2
Let n be a positive integer. Prove the following statement:
”If $2 + 2\sqrt{1 + 28n^2}$ is an integer, then it is the square of an integer.”
1997 German National Olympiad, 1
Prove that there are no perfect squares $a,b,c$ such that $ab-bc = a$.
2000 IMO Shortlist, 6
Show that the set of positive integers that cannot be represented as a sum of distinct perfect squares is finite.
2018 May Olympiad, 1
You have a $4$-digit whole number that is a perfect square. Another number is built adding $ 1$ to the unit's digit, subtracting $ 1$ from the ten's digit, adding $ 1$ to the hundred's digit and subtracting $ 1$ from the ones digit of one thousand. If the number you get is also a perfect square, find the original number. It's unique?
2010 Saudi Arabia Pre-TST, 3.2
Prove that among any nine divisors of $30^{2010}$ there are two whose product is a perfect square.
2003 IMO Shortlist, 4
Let $ b$ be an integer greater than $ 5$. For each positive integer $ n$, consider the number \[ x_n = \underbrace{11\cdots1}_{n \minus{} 1}\underbrace{22\cdots2}_{n}5, \] written in base $ b$.
Prove that the following condition holds if and only if $ b \equal{} 10$: [i]there exists a positive integer $ M$ such that for any integer $ n$ greater than $ M$, the number $ x_n$ is a perfect square.[/i]
[i]Proposed by Laurentiu Panaitopol, Romania[/i]
1926 Eotvos Mathematical Competition, 2
Prove that the product of four consecutive natural numbers cannot be the square of an integer.
2011 Argentina National Olympiad, 4
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: $$\frac{1}{a_1b_1}+\frac{1}{a_2b_2}+\frac{1}{a_3b_3}+\ldots +\frac{1}{a_{600}b_{600}}$$
1997 Bundeswettbewerb Mathematik, 4
Prove that if $n$ is a natural number such that both $3n+1$ and $4n+1$ are squares, then $n$ is divisible by $56$.
2014 Singapore Junior Math Olympiad, 2
Let $a$ be a positive integer such that the last two digits of $a^2$ are both non-zero. When the last two digits of $a^2$ are deleted, the resulting number is still a perfect square. Find, with justification, all possible values of $a$.
2018 Junior Balkan Team Selection Tests - Romania, 1
Prove that a positive integer $A$ is a perfect square if and only if, for all positive integers $n$, at least one of the numbers $(A + 1)^2 - A, (A + 2)^2 - A, (A + 3)^2 - A,.., (A + n)^2- A$ is a multiple of $n$.
1972 All Soviet Union Mathematical Olympiad, 161
Find the maximal $x$ such that the expression $4^{27} + 4^{1000} + 4^x$ is the exact square.
1931 Eotvos Mathematical Competition, 2
Let $a^2_1+ a^2_2+ a^2_3+ a^2_4+ a^2_5= b^2$, where $a_1$, $a_2$, $a_3$, $a_4$, $a_5$, and $b$ are integers. Prove that not all of these numbers can be odd.
2009 Dutch Mathematical Olympiad, 2
Consider the sequence of integers $0, 1, 2, 4, 6, 9, 12,...$ obtained by starting with zero, adding $1$, then adding $1$ again, then adding $2$, and adding $2$ again, then adding $3$, and adding $3$ again, and so on. If we call the subsequent terms of this sequence $a_0, a_1, a_2, ...$, then we have $a_0 = 0$, and $a_{2n-1} = a_{2n-2} + n$ , $a_{2n} = a_{2n-1} + n$ for all integers $n \ge 1$.
Find all integers $k \ge 0$ for which $a_k$ is the square of an integer.
1988 IMO Shortlist, 25
A positive integer is called a [b]double number[/b] if its decimal representation consists of a block of digits, not commencing with 0, followed immediately by an identical block. So, for instance, 360360 is a double number, but 36036 is not. Show that there are infinitely many double numbers which are perfect squares.
1999 German National Olympiad, 6b
Determine all pairs ($m,n$) of natural numbers for which $4^m + 5^n$ is a perfect square.
2013 Hanoi Open Mathematics Competitions, 1
How many three-digit perfect squares are there such that if each digit is increased by one, the resulting number is also a perfect square?
(A): $1$, (B): $2$, (C): $4$, (D): $8$, (E) None of the above.
1973 All Soviet Union Mathematical Olympiad, 175
Prove that $9$-digit number, that contains all the decimal digits except zero and does not ends with $5$ can not be exact square.
2008 Tournament Of Towns, 1
An integer $N$ is the product of two consecutive integers.
(a) Prove that we can add two digits to the right of this number and obtain a perfect square.
(b) Prove that this can be done in only one way if $N > 12$
2015 Czech-Polish-Slovak Junior Match, 2
Decide if the vertices of a regular $30$-gon can be numbered by numbers $1, 2,.., 30$ in such a way that the sum of the numbers of every two neighboring to be a square of a certain natural number.
2021 Saudi Arabia Training Tests, 38
Prove that the set of all divisors of a positive integer which is not a perfect square can be divided into pairs so that in each pair, one number is divided by another.
2011 Saudi Arabia Pre-TST, 1.3
Find all positive integers $n$ such that $27^n- 2^n$ is a perfect square.
1990 Bundeswettbewerb Mathematik, 2
The sequence $a_0,a_1,a_2,...$ is defined by $a_0 = 0, a_1 = a_2 = 1$ and $a_{n+2} +a_{n-1} = 2(a_{n+1} +a_n)$ for all $n \in N$. Show that all $a_n$ are perfect squares .
1994 Italy TST, 2
Find all prime numbers $p$ for which $\frac{2^{p-1} -1}{p}$ is a perfect square.