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 
.Yay for inequality
by shiningsunnyday, Apr 14, 2017, 9:10 AM
2013 USAMO 4 wrote:
Find all real numbers 
satisfying ![\[\min(\sqrt{x+xyz},\sqrt{y+xyz},\sqrt{z+xyz})=\sqrt{x-1}+\sqrt{y-1}+\sqrt{z-1}.\]](//latex.artofproblemsolving.com/2/6/e/26edbd87f49eb68ec9b662b602db712e5151dc9c.png)
satisfying ![\[\min(\sqrt{x+xyz},\sqrt{y+xyz},\sqrt{z+xyz})=\sqrt{x-1}+\sqrt{y-1}+\sqrt{z-1}.\]](http://latex.artofproblemsolving.com/2/6/e/26edbd87f49eb68ec9b662b602db712e5151dc9c.png)
![\[a_1b_1 + a_2b_2 + \ldots + a_n b_n \le \sqrt{a_1^2+a_2^2+ \ldots + a_n^2} \cdot \sqrt{b_1^2+b_2^2+ \ldots + b_n^2}.\]](http://latex.artofproblemsolving.com/5/4/f/54fd6110e36d0f1a2fc5ba503c59c41f0d2b88cd.png)
WLOG let 
so that ![\[\min(\sqrt{x+xyz},\sqrt{y+xyz},\sqrt{z+xyz})=\sqrt{x+xyz} = \sqrt{(x-1+1)(1+yz)}\]](http://latex.artofproblemsolving.com/5/2/a/52a2b979127ca95766c2404e3380546c317ae1c8.png)
, and from Cauchy, ![\[\sqrt{(x-1+1)(1+yz)} \ge \sqrt{x-1}+\sqrt{yz}=\sqrt{x-1}+\sqrt{(y-1 +1)(1+ z-1)} \ge \sqrt{x-1}+\sqrt{y-1}+\sqrt{z-1}.\]](http://latex.artofproblemsolving.com/d/7/a/d7a084b72655f0eafc291733da4e2758f05ceeea.png)
In order for the original equation to be true, equality must be present in both inequalities, giving us ![\[x-1 = \frac{1}{yz}\]](http://latex.artofproblemsolving.com/d/3/b/d3b3cd562d28b9398742a479b5d56c84561d394d.png)
![\[y-1 = \frac{1}{z-1},\]](http://latex.artofproblemsolving.com/1/3/c/13c1c6cfa9e0f7b8e2ae8062a216e36ce14153b5.png)
which gives us the triplet ![\[(x, y, z) = \left(\frac{y^2+y-1}{y^2}, y, \frac{y}{y-1} \right)\]](http://latex.artofproblemsolving.com/3/5/c/35c03ce49e20f2eef7a81031dea41728542052d9.png)
for 
![\[y \ge x \Rightarrow y \ge \frac{y^2+y-1}{y^2} \Rightarrow y^3 -y^2-y+1 = (y+1)(y-1)^2 \ge 0\]](http://latex.artofproblemsolving.com/7/4/6/7468b2a9d10d21dd1938e77aabb8cbb1fcf2e174.png)
![\[z \ge x \Rightarrow \frac{y}{y-1} \ge \frac{y^2+y-1}{y^2} \Rightarrow y^3 \ge y^3-2y+1 \Rightarrow 2y \ge 1.\]](http://latex.artofproblemsolving.com/c/a/5/ca5f502f389874abbe54c7fe899c78db5a177375.png)
Also, 
as otherwise that would make 
undefined. Specifically, the original condition breaks down to ![\[\sqrt{1+zx} = \sqrt{x-1}+\sqrt{z-1}\]](http://latex.artofproblemsolving.com/6/a/5/6a50a0a4977c1de1a5330dff7a5dbbf70e6feaed.png)
but as we previously had gotten ![\[\sqrt{1+zx} > \sqrt{zx} = \sqrt{(z-1+1)(1+x-1)} \ge \sqrt{z-1}+\sqrt{x-1},\]](http://latex.artofproblemsolving.com/9/d/d/9dd08f065e62a2add79fe682b9542c2bf5aaa52a.png)
contradiction. Finally, we check that ![\[\frac{y}{y-1} = 1 + \frac{1}{y-1} > 1\]](http://latex.artofproblemsolving.com/5/8/2/582df949d003d20ef57befe9a550e354f748063b.png)
![\[\frac{y^2+y-1}{y^2} = 1 + \frac{y-1}{y^2} > 1.\]](http://latex.artofproblemsolving.com/f/a/1/fa123f6d78ac187d316df4f5b2a9630b3a708415.png)
Therefore, we conclude 
for 
generates our solution set.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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