Found problems: 397
1992 Tournament Of Towns, (352) 1
Prove that there exists a sequence of $100$ different integers such that the sum of the squares of any two consecutive terms is a perfect square.
(S Tokarev)
1995 Tournament Of Towns, (444) 4
Prove that the number $\overline{40...0}9$ (with at least one zero) is not a perfect square.
(VA Senderov)
2010 All-Russian Olympiad Regional Round, 10.7
Are there three pairwise distinct non-zero integers whose sum is zero and whose sum of thirteenth powers is the square of some natural number?
1992 All Soviet Union Mathematical Olympiad, 570
Define the sequence $a_1 = 1, a_2, a_3, ...$ by $$a_{n+1} = a_1^2 + a_2 ^2 + a_3^2 + ... + a_n^2 + n$$ Show that $1$ is the only square in the sequence.
2016 Costa Rica - Final Round, A2
The initial number of inhabitants of a city of more than $150$ inhabitants is a perfect square. With an increase of $1000$ inhabitants it becomes a perfect square plus a unit. After from another increase of $1000$ inhabitants it is again a perfect square. Determine the quantity of inhabitants that are initially in the city.
2013 Nordic, 1
Let ${(a_n)_{n\ge1}} $ be a sequence with ${a_1 = 1} $ and ${a_{n+1} = \lfloor a_n +\sqrt{a_n}+\frac{1}{2}\rfloor }$ for all ${n \ge 1}$, where ${\lfloor x \rfloor}$ denotes the greatest integer less than or equal to ${x}$. Find all ${n \le 2013}$ such that ${a_n}$ is a perfect square
2017 Rioplatense Mathematical Olympiad, Level 3, 4
Is there a number $n$ such that one can write $n$ as the sum of $2017$ perfect squares and (with at least) $2017$ distinct ways?
1966 Poland - Second Round, 4
Prove that if the natural numbers $ a $ and $ b $ satisfy the equation $ a^2+a = 3b^2 $, then the number $ a+1 $ is the square of an integer.
2021 Durer Math Competition Finals, 2
In a french village the number of inhabitants is a perfect square. If $100$ more people moved in, then the number of people would be $ 1$ bigger than a perfect square. If again $100$ more people moved in, then the number of people would be a perfect square again. How many people lives in the village if their number is the least possible?
2021 Durer Math Competition Finals, 2
Find the number of integers $n$ between $1$ and $2021$ such that $2^n+2^{n+3}$ is a perfect square.
1951 Moscow Mathematical Olympiad, 199
Prove that the sum $1^3 + 2^3 +...+ n^3$ is a perfect square for all $n$.
1999 Bundeswettbewerb Mathematik, 4
A natural number is called [i]bright [/i] if it is the sum of a perfect square and a perfect cube.
Prove that if $r$ and $s$ are any two positive integers, then
(a) there exist infinitely many positive integers $n$ such that both $r+n$ and $s+n$ are [i]bright[/i],
(b) there exist infinitely many positive integers $m$ such that both rm and sm are [i]bright[/i].
2018 Brazil EGMO TST, 1
(a) Let $m$ and $n$ be positive integers and $p$ a positive rational number, with $m > n$, such that $\sqrt{m} -\sqrt{n}= p$. Prove that $m$ and $n$ are perfect squares.
(b) Find all four-digit numbers $\overline{abcd}$, where each letter $a, b, c$ and $d$ represents a digit, such that $\sqrt{\overline{abcd}} -\sqrt{\overline{acd}}= \overline{bb}$.
1988 IMO Longlists, 76
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.
2005 Estonia National Olympiad, 5
Does there exist an integer $n > 1$ such that $2^{2^n-1} -7$ is not a perfect square?
2022 Grand Duchy of Lithuania, 4
Find all triples of natural numbers $(a, b, c)$ for which the number $$2^a + 2^b + 2^c + 3$$ is the square of an integer.
1936 Moscow Mathematical Olympiad, 022
Find a four-digit perfect square whose first digit is the same as the second, and the third is the same as the fourth.
OIFMAT III 2013, 9
Let $ a, b \in Z $, prove that if the expression $ a \cdot 2013^n + b $ is a perfect square for all natural $n$, then $ a $ is zero.
2019 Saudi Arabia Pre-TST + Training Tests, 2.1
Suppose that $a, b, c,d$ are pairwise distinct positive integers such that $a+b = c+d = p$ for some odd prime $p > 3$ . Prove that $abcd$ is not a perfect square.
2021 South East Mathematical Olympiad, 4
For positive integer $k,$ we say that it is a [i]Taurus integer[/i] if we can delete one element from the set $M_k=\{1,2,\cdots,k\},$ such that the sum of remaining $k-1$ elements is a positive perfect square. For example, $7$ is a Taurus integer, because if we delete $3$ from $M_7=\{1,2,3,4,5,6,7\},$ the sum of remaining $6$ elements is $25,$ which is a positive perfect square.
$(1)$ Determine whether $2021$ is a Taurus integer.
$(2)$ For positive integer $n,$ determine the number of Taurus integers in $\{1,2,\cdots,n\}.$
1969 Kurschak Competition, 1
Show that if $2 + 2\sqrt{28n^2 + 1}$ is an integer, then it is a square (for $n$ an integer).
1965 Dutch Mathematical Olympiad, 2
Prove that $S_1 = (n + 1)^2 + (n + 2)^2 +...+ (n + 5)^2$ is divisible by $5$ for every $n$.
Prove that for no $n$: $\sum_{\ell=1}^5 (n+\ell)^2$ is a perfect square.
Let $S_2=(n + 6)^2 + (n + 7)^2 + ... + (n + 10)^2$. Prove that $S_1 \cdot S_2$ is divisible by $150$.