Perfect Squares, Infinite Integers and Integers
by steven_zhang123, Mar 16, 2025, 12:06 PM
For which integer
, are there infinitely many positive integers
such that
is a perfect square? (Here
denotes the integer part of the real number
?





Unsolved Diophantine(I think)
by Nuran2010, Mar 14, 2025, 4:41 PM
Find all solutions for the equation
where
is a positive integer and
is a prime.(Don't get mad at me,I've used the search function and did not see a correct and complete solution anywhere.)



Another NT FE
by nukelauncher, Sep 22, 2020, 11:58 PM
Find all functions
such that
divides
for all positive integers
and
with
.






Variable point on the median
by MarkBcc168, Jun 11, 2019, 12:23 AM
Let
be a scalene triangle with circumcircle
. Let
be the midpoint of
. A variable point
is selected in the line segment
. The circumcircles of triangles
and
intersect
again at points
and
, respectively. The lines
and
intersect (a second time) the circumcircles to triangles
and
at
and
, respectively. Prove that as
varies, the circumcircle of
passes through a fixed point
distinct from
.





















IMO 2014 Problem 1
by Amir Hossein, Jul 8, 2014, 12:17 PM
Let
be an infinite sequence of positive integers. Prove that there exists a unique integer
such that
![\[a_n < \frac{a_0+a_1+a_2+\cdots+a_n}{n} \leq a_{n+1}.\]](//latex.artofproblemsolving.com/9/4/f/94fc5a51b7e68588123c5b527fe75183bc4c4937.png)
Proposed by Gerhard Wöginger, Austria.


![\[a_n < \frac{a_0+a_1+a_2+\cdots+a_n}{n} \leq a_{n+1}.\]](http://latex.artofproblemsolving.com/9/4/f/94fc5a51b7e68588123c5b527fe75183bc4c4937.png)
Proposed by Gerhard Wöginger, Austria.
This post has been edited 1 time. Last edited by v_Enhance, Nov 5, 2023, 5:17 PM
Reason: missing < sign
Reason: missing < sign
Sequences and limit
by lehungvietbao, Jan 3, 2014, 10:32 AM
Let
be two positive sequences defined by
and
for all
.
Prove that they are converges and find their limits.


![\[ \begin{cases} {{x}_{n+1}}{{y}_{n+1}}-{{x}_{n}}=0 \\ x_{n+1}^{2}+{{y}_{n}}=2 \end{cases} \]](http://latex.artofproblemsolving.com/2/0/1/2012dc6ff3a41478547cea4aa9b6bccbe17a3623.png)

Prove that they are converges and find their limits.
IMO 2012 P5
by mathmdmb, Jul 11, 2012, 7:03 PM
Let
be a triangle with
, and let
be the foot of the altitude from
. Let
be a point in the interior of the segment
. Let
be the point on the segment
such that
. Similarly, let
be the point on the segment
such that
. Let
be the point of intersection of
and
.
Show that
.
Proposed by Josef Tkadlec, Czech Republic















Show that

Proposed by Josef Tkadlec, Czech Republic
This post has been edited 3 times. Last edited by Eternica, Jun 19, 2024, 10:03 AM
One secuence satisfying condition
by hatchguy, Sep 4, 2011, 12:18 AM
Prove that there exists only one infinite secuence of positive integers
with
,
and
for all positive integers
.





Can this sequence be bounded?
by darij grinberg, Jan 19, 2005, 11:00 AM
Let
,
,
, ... be an infinite sequence of real numbers satisfying the equation
for all
, where
and
are two different positive reals.
Can this sequence
,
,
, ... be bounded?
Proposed by Mihai Bălună, Romania







Can this sequence



Proposed by Mihai Bălună, Romania
This post has been edited 1 time. Last edited by djmathman, Sep 27, 2015, 2:12 PM
To share with readers my favorite problem I came across today :) (Shout for contrib.)
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