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 
?
, 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.)
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 
.
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 
.
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 
.Whoever wrote this... doesn't know what concise means :\
by shiningsunnyday, Apr 11, 2017, 3:46 PM
2004 ISL G2 wrote:
Let be a circle and let 
be a line such that and have no common points. Further, let 
be a diameter of the circle ; assume that this diameter is perpendicular to the line , and the point 
is nearer to the line than the point . Let 
be an arbitrary point on the circle , different from the points and . Let be the point of intersection of the lines 
and . One of the two tangents from the point to the circle touches this circle at a point ; hereby, we assume that the points and lie in the same halfplane with respect to the line . Denote by 
the point of intersection of the lines 
and . Let the line 
intersect the circle at a point 
, different from . Prove that the reflection of the point in the line lies on the line 
.
be a circle and let 
be a line such that and have no common points. Further, let 
be a diameter of the circle ; assume that this diameter is perpendicular to the line , and the point 
is nearer to the line than the point . Let 
be an arbitrary point on the circle , different from the points and . Let be the point of intersection of the lines 
and . One of the two tangents from the point to the circle touches this circle at a point ; hereby, we assume that the points and lie in the same halfplane with respect to the line . Denote by 
the point of intersection of the lines 
and . Let the line 
intersect the circle at a point 
, different from . Prove that the reflection of the point in the line lies on the line 
.![[asy]
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[/asy]](http://latex.artofproblemsolving.com/0/c/6/0c60f19dac196db75f17aa7938bb3bf472112cf8.png)
We show 
is the reflection of 
We'll proceed with straight angle chasing. It suffices to show ![\[\angle{G'AB}=\angle{G'CB}=\angle{BAG}\]](http://latex.artofproblemsolving.com/1/7/9/179c53ddd54aecfb450ca523aa47cd766a93ae02.png)
![\[90-\angle{DCF}=\angle{BAG} \leftrightarrow 90-\angle{BAG}=\angle{PFA}=\angle{DCF} \leftrightarrow \triangle{DCF} \sim \triangle{DFA}\]](http://latex.artofproblemsolving.com/4/3/a/43af76ca0e56f19dc4afd15233ad15fa2fd2c5ef.png)
![\[\frac{DF}{DC}=\frac{AD}{DF} \Rightarrow AD \cdot DC = DF^2.\]](http://latex.artofproblemsolving.com/7/b/5/7b5aad8b4f78989c99e7d393085642ae73997a5c.png)
But by power of a point, 
so it suffices to show 
which follows from a final angle chase: ![\[\angle{DFE}=\angle{PFE}\stackrel{AEPF \text{cyclic}}{=}\angle{BAE}=\angle{DEB},\]](http://latex.artofproblemsolving.com/c/5/1/c51263bb41ac43d7f258345954acd490af1a6a8f.png)
showing 
is isosceles and we're done.Can anyone spot the projective solution (since I found this in a projective chapter in Lemmas).
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.
be an infinite sequence of positive integers. Prove that there exists a unique integer 
such that![\[a_n < \frac{a_0+a_1+a_2+\cdots+a_n}{n} \leq a_{n+1}.\]](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
![\[ \begin{cases} {{x}_{n+1}}{{y}_{n+1}}-{{x}_{n}}=0 \\ x_{n+1}^{2}+{{y}_{n}}=2 \end{cases} \]](//latex.artofproblemsolving.com/2/0/1/2012dc6ff3a41478547cea4aa9b6bccbe17a3623.png)
for all 
.
Prove that they are converges and find their limits.
be two positive sequences defined by 
and![\[ \begin{cases} {{x}_{n+1}}{{y}_{n+1}}-{{x}_{n}}=0 \\ x_{n+1}^{2}+{{y}_{n}}=2 \end{cases} \]](http://latex.artofproblemsolving.com/2/0/1/2012dc6ff3a41478547cea4aa9b6bccbe17a3623.png)
for all 
.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
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
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 .
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
, 
, 
, ... 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
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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