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
.





















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
.



































Solution
![[asy]
/* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki, go to User:Azjps/geogebra */
import graph; size(15cm);
real labelscalefactor = 0.5; /* changes label-to-point distance */
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */
pen dotstyle = black; /* point style */
real xmin = -7.976002749378333, xmax = 34.86531199295848, ymin = -10.481189671238441, ymax = 12.239223899756556; /* image dimensions */
/* draw figures */
draw(circle((4.02,0.64), 4.671316730858656));
draw((xmin, -1.4436090225563913*xmin + 27.16711976319434)--(xmax, -1.4436090225563913*xmax + 27.16711976319434)); /* line */
draw((0.18,-2.02)--(17.401986181172095,2.0454555016526608));
draw((17.401986181172095,2.0454555016526608)--(5.17527077355485,5.166206959449599));
draw((5.17527077355485,5.166206959449599)--(24.587486816833866,-8.327598047573366));
draw((0.18,-2.02)--(24.587486816833866,-8.327598047573366));
draw((0.18,-2.02)--(13.720717655728016,7.359767959436592));
draw((7.86,3.3)--(8.644181009288122,-0.021928994182223827));
draw((0.8275136207952526,4.050165790484705)--(24.587486816833866,-8.327598047573366));
draw((0.18,-2.02)--(5.17527077355485,5.166206959449599));
/* dots and labels */
dot((7.86,3.3),dotstyle);
label("$B$", (6.338195399361469,3.090290649029001), NE * labelscalefactor);
dot((0.18,-2.02),dotstyle);
label("$A$", (-0.5150645633606531,-2.5813727683961987), NE * labelscalefactor);
dot((13.720717655728016,7.359767959436592),dotstyle);
label("$P$", (13.866653387967544,7.681637225039878), NE * labelscalefactor);
dot((8.644181009288122,-0.021928994182223827),dotstyle);
label("$C$", (8.90394789772049,-1.5685757295702702), NE * labelscalefactor);
dot((17.401986181172095,2.0454555016526608),linewidth(3.pt) + dotstyle);
label("$D$", (17.917841543271265,1.9762139063204798), NE * labelscalefactor);
dot((5.17527077355485,5.166206959449599),linewidth(3.pt) + dotstyle);
label("$E$", (5.257878557947144,5.487243640917033), NE * labelscalefactor);
dot((24.587486816833866,-8.327598047573366),linewidth(3.pt) + dotstyle);
label("$F$", (25.074940617641165,-8.489355494880781), NE * labelscalefactor);
dot((6.090428041178006,-3.547424951721978),linewidth(3.pt) + dotstyle);
label("$G$", (5.865556781242702,-4.6069668460480555), NE * labelscalefactor);
dot((0.8275136207952526,4.050165790484705),linewidth(3.pt) + dotstyle);
label("$G'$", (0.2614131664058928,4.440686700796906), NE * labelscalefactor);
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
/* end of picture */
[/asy]](http://latex.artofproblemsolving.com/0/c/6/0c60f19dac196db75f17aa7938bb3bf472112cf8.png)
Let




![\[\angle{G'AB}=\angle{G'CB}=\angle{BAG}\]](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)


![\[\angle{DFE}=\angle{PFE}\stackrel{AEPF \text{cyclic}}{=}\angle{BAE}=\angle{DEB},\]](http://latex.artofproblemsolving.com/c/5/1/c51263bb41ac43d7f258345954acd490af1a6a8f.png)

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.


![\[a_n < \frac{a_0+a_1+a_2+\cdots+a_n}{n} \leq a_{n+1}.\]](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.)
Archives












Shouts
Submit
270 shouts
Contributors
62861 • aftermaths • agbdmrbirdyface • blue8931 • bluephoenix • cjquines0 • DeathLlama9 • doitsudoitsu • FlyingCucumber • hwl0304 • Irrational_phi • jdeaks1000 • MATH1945 • Mathaddict11 • MathAwesome123 • mathguy623 • mathmaster2000 • MathStudent2002 • monkey8 • pandyhu2001 • pinetree1 • rkm0959 • SantaDragon • shiningsunnyday • skipiano • wu2481632 • zephyrcrush78
Tags
About Owner
- Posts: 1350
- Joined: Dec 19, 2014
Blog Stats
- Blog created: Jun 11, 2016
- Total entries: 193
- Total visits: 30959
- Total comments: 579
Search Blog