Found problems: 397
2025 Bulgarian Spring Mathematical Competition, 10.3
In the cell $(i,j)$ of a table $n\times n$ is written the number $(i-1)n + j$. Determine all positive integers $n$ such that there are exactly $2025$ rows not containing a perfect square.
2020 Greece Team Selection Test, 4
Let $a$ and $b$ be two positive integers. Prove that the integer
\[a^2+\left\lceil\frac{4a^2}b\right\rceil\]
is not a square. (Here $\lceil z\rceil$ denotes the least integer greater than or equal to $z$.)
[i]Russia[/i]
2024 Tuymaada Olympiad, 3
All perfect squares, and all perfect squares multiplied by two, are written in a row in increasing order. let $f(n)$ be the $n$-th number in this sequence. (For instance, $f(1)=1,f(2)=2,f(3)=4,f(4)=8$.) Is there an integer $n$ such that all the numbers
\[f(n),f(2n),f(3n),\dots,f(10n^2)\]
are perfect squares?
2020 Iran MO (2nd Round), P5
Call a pair of integers $a$ and $b$ square makers , if $ab+1$ is a perfect square.
Determine for which $n$ is it possible to divide the set $\{1,2, \dots , 2n\}$ into $n$ pairs of square makers.
1998 Singapore Team Selection Test, 3
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.
1977 All Soviet Union Mathematical Olympiad, 244
Let us call "fine" the $2n$-digit number if it is exact square itself and the two numbers represented by its first $n$ digits (first digit may not be zero) and last $n$ digits (first digit may be zero, but it may not be zero itself) are exact squares also.
a) Find all two- and four-digit fine numbers.
b) Is there any six-digit fine number?
c) Prove that there exists $20$-digit fine number.
d) Prove that there exist at least ten $100$-digit fine numbers.
e) Prove that there exists $30$-digit fine number.
2016 District Olympiad, 2
If $ a,n $ are two natural numbers corelated by the equation $ \left\{ \sqrt{n+\sqrt n}\right\} =\left\{ \sqrt a\right\} , $ then $
1+4a $ is a perfect square. Justify this statement. Here, $ \{\} $ is the usual fractionary part.
1940 Moscow Mathematical Olympiad, 062-
Find a four-digit number that is perfect square and such that the first two digits are the same and the last two as well.
2010 Contests, 3
Consider a triangle $XYZ$ and a point $O$ in its interior. Three lines through $O$ are drawn, parallel to the respective sides of the triangle. The intersections with the sides of the triangle determine six line segments from $O$ to the sides of the triangle. The lengths of these segments are integer numbers $a, b, c, d, e$ and $f$ (see figure).
Prove that the product $a \cdot b \cdot c\cdot d \cdot e \cdot f$ is a perfect square.
[asy]
unitsize(1 cm);
pair A, B, C, D, E, F, O, X, Y, Z;
X = (1,4);
Y = (0,0);
Z = (5,1.5);
O = (1.8,2.2);
A = extension(O, O + Z - X, X, Y);
B = extension(O, O + Y - Z, X, Y);
C = extension(O, O + X - Y, Y, Z);
D = extension(O, O + Z - X, Y, Z);
E = extension(O, O + Y - Z, Z, X);
F = extension(O, O + X - Y, Z, X);
draw(X--Y--Z--cycle);
draw(A--D);
draw(B--E);
draw(C--F);
dot("$A$", A, NW);
dot("$B$", B, NW);
dot("$C$", C, SE);
dot("$D$", D, SE);
dot("$E$", E, NE);
dot("$F$", F, NE);
dot("$O$", O, S);
dot("$X$", X, N);
dot("$Y$", Y, SW);
dot("$Z$", Z, dir(0));
label("$a$", (A + O)/2, SW);
label("$b$", (B + O)/2, SE);
label("$c$", (C + O)/2, SE);
label("$d$", (D + O)/2, SW);
label("$e$", (E + O)/2, SE);
label("$f$", (F + O)/2, NW);
[/asy]
2008 Singapore Junior Math Olympiad, 5
Determine all primes $p$ such that $5^p + 4 p^4$ is a perfect square, i.e., the square of an integer.
2001 All-Russian Olympiad Regional Round, 9.8
Sasha wrote a non-zero number on the board and added it to it on the right, one non-zero digit at a time, until he writes out a million digits. Prove that an exact square has been written on the board no more than $100$ times.
2021 Malaysia IMONST 2, 2
Can we find positive integers $a$ and $b$ such that both $(a^2 + b)$ and $(b^2 + a)$ are perfect squares?
1995 Austrian-Polish Competition, 7
Consider the equation $3y^4 + 4cy^3 + 2xy + 48 = 0$, where $c$ is an integer parameter. Determine all values of $c$ for which the number of integral solutions $(x,y)$ satisfying the conditions (i) and (ii) is maximal:
(i) $|x|$ is a square of an integer;
(ii) $y$ is a squarefree number.
2012 Danube Mathematical Competition, 1
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.
2010 All-Russian Olympiad Regional Round, 11.4
We call a triple of natural numbers $(a, b, c)$ [i]square [/i] if they form an arithmetic progression (in exactly this order), the number $b$ is coprime to each of the numbers $a$ and $c$, and the number $abc$ is a perfect square. Prove that for any given a square triple, there is another square triple that has at least one common number with it.
2011 Korea Junior Math Olympiad, 3
Let $x, y$ be positive integers such that $gcd(x, y) = 1$ and $x + 3y^2$ is a perfect square. Prove that $x^2 + 9y^4$ can't be a perfect square.
1992 Tournament Of Towns, (348) 6
Consider the sequence $a(n)$ defined by the following conditions: $$a(1) = 1\,\,\,\, a(n + 1) = a(n) + [\sqrt{a(n)}] \,\,\, , \,\,\,\, n = 1,2,3,...$$
Prove that the sequence contains an infinite number of perfect squares. (Note: $[x]$ means the integer part of $x$, that is the greatest integer not greater than $x$.)
(A Andjans)
2003 Belarusian National Olympiad, 2
Let $P(x) =(x+1)^p (x-3)^q=x^n+a_1x^{n-1}+a_2x^{n-2}+...+a_{n-1}x+a_n$ where $p$ and $q$ are positive integers
a) Given $a_1=a_2$, prove that $3n$ is a perfect square.
b) Prove that there exist infinitely many pairs $(p, q)$ of positive integers p and q such that the equality $a_1=a_2$ is valid for the polynomial $P(x)$.
(D. Bazylev)
2017 Saudi Arabia JBMO TST, 6
Find all pairs of prime numbers $(p, q)$ such that $p^2 + 5pq + 4q^2$ is a perfect square.
2018 Polish Junior MO Second Round, 5
Each integer has been colored in one of three colors. Prove that exist two different numbers of the same color, whose difference is a perfect square.
1984 IMO Longlists, 25
Prove that the product of five consecutive positive integers cannot be the square of an integer.
2011 Bundeswettbewerb Mathematik, 2
Proove that if for a positive integer $n$ , both $3n + 1$ and $10n + 1$ are perfect squares , then $29n + 11$ is not a prime number.
2009 Hanoi Open Mathematics Competitions, 3
Let $a, b,c$ be positive integers with no common factor and satisfy the conditions $\frac1a +\frac1b=\frac1c$
Prove that $a + b$ is a square.
2021 Malaysia IMONST 1, 5
How many integers $n$ (with $1 \le n \le 2021$) have the property that $8n + 1$ is a perfect square?
2004 Germany Team Selection Test, 3
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]