positive integers forming a perfect square
by cielblue, May 2, 2025, 8:25 PM
Find all positive integers 
such that 
is a perfect square.
such that 
is a perfect square.A nice and easy gem off of StackExchange
by NamelyOrange, May 2, 2025, 8:13 PM
Define 
as the set of all numbers of the form 
for some nonnegative 
and 
. Find (with proof) all pairs 
such that 
and 
.
as the set of all numbers of the form 
for some nonnegative 
and 
. Find (with proof) all pairs 
such that 
and 
.Hard diophant equation
by MuradSafarli, May 2, 2025, 6:12 PM
Find all positive integers 
such that the equation
is satisfied.
such that the equation
is satisfied.
Euler's function
by luutrongphuc, May 2, 2025, 3:52 PM
Find all real numbers 
such that for every positive real 
, there exists an integer 
satisfying
![\[
\frac{\varphi(n!)}{n^\alpha\,(n-1)!} \;>\; c.
\]](//latex.artofproblemsolving.com/4/4/4/4443cf7166cc65e67e2e504a826c4ac5f91aab2b.png)
such that for every positive real 
, there exists an integer 
satisfying![\[
\frac{\varphi(n!)}{n^\alpha\,(n-1)!} \;>\; c.
\]](http://latex.artofproblemsolving.com/4/4/4/4443cf7166cc65e67e2e504a826c4ac5f91aab2b.png)
, such that for all 
:![\[ f(x+f(y))=\dfrac{f(x)}{1+f(xy)}\]](http://latex.artofproblemsolving.com/1/4/1/1418fad9fd00b31c38bfbc5657e2a322d66ef450.png)
,![\[\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}\geq 4+\frac{8(a-c)^2}{(a+b+c+d)^2}.\]](http://latex.artofproblemsolving.com/7/b/6/7b6e32d32d89ca058a06974e7276f0a5874d90a7.png)
Austrian Regional MO 2025 P1
by BR1F1SZ, Apr 18, 2025, 4:25 PM
Let 
be a positive integer. Furthermore, let ![$x_1, x_2,\ldots, x_n \in [0, 2]$](//latex.artofproblemsolving.com/b/2/a/b2a6731961601353e26224c285a32923fbb54eb3.png)
be real numbers subject to 
. Prove the inequality
When does equality hold?
(Walther Janous)
be a positive integer. Furthermore, let ![$x_1, x_2,\ldots, x_n \in [0, 2]$](http://latex.artofproblemsolving.com/b/2/a/b2a6731961601353e26224c285a32923fbb54eb3.png)
be real numbers subject to 
. Prove the inequality
When does equality hold?(Walther Janous)
This post has been edited 1 time. Last edited by BR1F1SZ, Apr 18, 2025, 4:26 PM
An almost identity polynomial
by nAalniaOMliO, Mar 28, 2025, 8:28 PM
Let be a positive integer and 
be a polynomial with integer coefficients such that 
.
Prove that
is divisible by 
.
be a positive integer and 
be a polynomial with integer coefficients such that 
.Prove that
is divisible by 
.This post has been edited 1 time. Last edited by nAalniaOMliO, Mar 29, 2025, 1:34 PM
at everystep a, b, c are replaced by a+\gcd(b,c), b+\gcd(a,c), c+\gcd(a,b)
by NJAX, May 31, 2024, 12:21 PM
Three positive integers are written on the board. In every minute, instead of the numbers 
, Elbek writes 
. Prove that there will be two numbers on the board after some minutes, such that one is divisible by the other.
Note.
- Greatest common divisor of numbers 
and 
Proposed by Sergey Berlov, Russia
, Elbek writes 
. Prove that there will be two numbers on the board after some minutes, such that one is divisible by the other.Note.
- Greatest common divisor of numbers 
and 
Proposed by Sergey Berlov, Russia
This post has been edited 1 time. Last edited by NJAX, May 31, 2024, 12:34 PM
Increments and Decrements in Square Grid
by ike.chen, Jul 9, 2023, 4:36 AM
In each square of a garden shaped like a 
board, there is initially a tree of height 
. A gardener and a lumberjack alternate turns playing the following game, with the gardener taking the first turn:
. Determine the largest 
such that the gardener can ensure there are eventually majestic trees on the board, no matter how the lumberjack plays.
board, there is initially a tree of height 
. A gardener and a lumberjack alternate turns playing the following game, with the gardener taking the first turn:
- The gardener chooses a square in the garden. Each tree on that square and all the surrounding squares (of which there are at most eight) then becomes one unit taller.
- The lumberjack then chooses four different squares on the board. Each tree of positive height on those squares then becomes one unit shorter.
. Determine the largest 
such that the gardener can ensure there are eventually majestic trees on the board, no matter how the lumberjack plays.This post has been edited 3 times. Last edited by ike.chen, Dec 20, 2023, 5:34 AM
Reason: Typo
Reason: Typo
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