Found problems: 397
1964 All Russian Mathematical Olympiad, 054
Find the smallest exact square with last digit not $0$, such that after deleting its last two digits we shall obtain another exact square.
2010 Dutch Mathematical Olympiad, 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]
2014 Junior Balkan Team Selection Tests - Moldova, 5
Show that for any natural number $n$, the number $A = [\frac{n + 3}{4}] + [ \frac{n + 5}{4} ] + [\frac{n}{2} ] +n^2 + 3n + 3$ is a perfect square. ($[x]$ denotes the integer part of the real number x.)
2019 Regional Olympiad of Mexico Center Zone, 1
Let $a$, $b$, and $c $ be integers greater than zero. Show that the numbers $$2a ^ 2 + b ^ 2 + 3 \,\,, 2b ^ 2 + c ^ 2 + 3\,\,, 2c ^ 2 + a ^ 2 + 3 $$ cannot be all perfect squares.
1981 Swedish Mathematical Competition, 1
Let $N = 11\cdots 122 \cdots 25$, where there are $n$ $1$s and $n+1$ $2$s. Show that $N$ is a perfect square.
2021 Saudi Arabia JBMO TST, 4
Let us call a set of positive integers nice if the number of its elements equals to the average of its numbers. Call a positive integer $n$ an [i]amazing[/i] number if the set $\{1, 2 , . . . , n\}$ can be partitioned into nice subsets.
a) Prove that every perfect square is amazing.
b) Show that there are infinitely many positive integers which are not amazing.
1978 Chisinau City MO, 155
Find the base of the number system less than $100$, in which $2101$ is a perfect square.
1998 Belarus Team Selection Test, 3
a) Let $f(x,y) = x^3 + (3y^2+1)x^2 + (3y^4 - y^2 + 4 y - 1)x + (y^6-y^4 + 2y^3)$. Prove that if for some positive integers $a, b$ the number $f(a, b)$ is a cube of an integer then $f(a, b)$ is also a square of an integer.
b) Are there infinitely many pairs of positive integers $(a, b)$ for which $f(a, b)$ is a square but not a cube ?
2011 QEDMO 9th, 1
Find all integers $n$ for which both $4n + 1$ and $9n + 1$ are perfect squares.
2023 China Northern MO, 6
A positive integer $m$ is called a [i]beautiful [/i] integer if that there exists a positive integer $n$ such that $m = n^2+ n + 1$. Prove that there are infinitely many [i]beautiful [/i] integers with square factors, and the square factors of different beautiful integers are relative prime.
2012 Flanders Math Olympiad, 2
Let $n$ be a natural number. Call $a$ the smallest natural number you need to subtract from $n$ to get a perfect square. Call $b$ the smallest natural number that you must add to $n$ to get a perfect square. Prove that $n - ab$ is a perfect square.
1992 Tournament Of Towns, (333) 1
Prove that the product of all integers from $2^{1917} +1$ up to $2^{1991} -1$ is not the square of an integer.
(V. Senderov, Moscow)
2019 India PRMO, 14
Find the smallest positive integer $n \geq 10$ such that $n + 6$ is a prime and $9n + 7$ is a perfect square.
2021 Polish Junior MO First Round, 1
Is there a six-digit number where every two consecutive digits make up a certain number two-digit number that is the square of an integer? Justify your answer.
2009 Silk Road, 4
Prove that for any prime number $p$ there are infinitely many fours $(x, y, z, t)$ pairwise distinct natural numbers such that the number $(x^2+p t^2)(y^2+p t^2)(z^2+p t^2)$ is a perfect square.
2018 Rio de Janeiro Mathematical Olympiad, 3
Let $n$ be a positive integer. A function $f : \{1, 2, \dots, 2n\} \to \{1, 2, 3, 4, 5\}$ is [i]good[/i] if $f(j+2)$ and $f(j)$ have the same parity for every $j = 1, 2, \dots, 2n-2$.
Prove that the number of good functions is a perfect square.
1987 Mexico National Olympiad, 4
Calculate the product of all positive integers less than $100$ and having exactly three positive divisors. Show that this product is a square.
1991 IMO Shortlist, 14
Let $ a, b, c$ be integers and $ p$ an odd prime number. Prove that if $ f(x) \equal{} ax^2 \plus{} bx \plus{} c$ is a perfect square for $ 2p \minus{} 1$ consecutive integer values of $ x,$ then $ p$ divides $ b^2 \minus{} 4ac.$
2022 Saudi Arabia JBMO TST, 4
Determine the smallest positive integer $a$ for which there exist a prime number $p$ and a positive integer $b \ge 2$ such that $$\frac{a^p -a}{p}=b^2.$$
1998 Romania Team Selection Test, 2
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.
2015 Greece JBMO TST, 3
Prove that there is not a positive integer $n$ such that numbers $(n+1)2^n, (n+3)2^{n+2}$ are both perfect squares.
2013 Hanoi Open Mathematics Competitions, 2
How many natural numbers $n$ are there so that $n^2 + 2014$ is a perfect square?
(A): $1$, (B): $2$, (C): $3$, (D): $4$, (E) None of the above.
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]
1966 Dutch Mathematical Olympiad, 3
How many natural numbers are there whose square is a thirty-digit number which has the following curious property: If that thirty-digit number is divided from left to right into three groups of ten digits, then the numbers given by the middle group and the right group formed numbers are both four times the number formed by the left group?
Russian TST 2020, P3
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]