A coincidence about triangles with common incenter
by flower417477, Apr 30, 2025, 2:08 PM
have the same incenter 
.Prove that 
is concyclic iff 
is concurrent have the same incenter .Prove that is concyclic iff is concurrentEnjoy with this one
by Halykov, Apr 30, 2025, 2:00 PM
Let 
be a scalene triangle with circumcircle 
and circumcircle 
. Denote 
and 
as the midpoints of 
and 
, respectively. The altitude from 
to intersects again at 
. The line 
meets a second time at 
, and 
intersects at 
. The perpendicular bisector of intersects 
at 
and 
at 
. Let the circumcircle of 
intersect again at 
, and let the line 
intersect at 
.
If
is the reflection of over , prove that the intersection of 
and 
lies on the circumcircle of 
.
be a scalene triangle with circumcircle 
and circumcircle 
. Denote 
and 
as the midpoints of 
and 
, respectively. The altitude from 
to intersects again at 
. The line 
meets a second time at 
, and 
intersects at 
. The perpendicular bisector of intersects 
at 
and 
at 
. Let the circumcircle of 
intersect again at 
, and let the line 
intersect at 
.If
is the reflection of over , prove that the intersection of 
and 
lies on the circumcircle of 
.Conjecture: Intersection of diagonals of cyclic quadrilateral lies on a fixed li
by kieusuong, Apr 30, 2025, 12:28 PM
1 line solution to Inequality
by ItzsleepyXD, Apr 30, 2025, 9:27 AM
Let 
be positive real integer such that 
Prove that
such that 
and 
be positive real integer such that 
Prove that
such that 
and 
a nice problem of nt from PUMaC
by Namisgood, Apr 29, 2025, 9:47 AM
Attachments:
This post has been edited 1 time. Last edited by Namisgood, Yesterday at 9:47 AM
Reason: Typo
Reason: Typo
Solve $\sin(17x)+\sin(13x)=\sin(7x)$
by Speed2001, Apr 29, 2025, 1:06 AM
How to solve the equation:
Approach: I'm trying to factor 
to get 
.
Any hint would be appreciated.
Approach: I'm trying to factor 
to get 
.Any hint would be appreciated.
Difficult combinatorics problem about distinct sums under shifts
by CBMaster, Apr 27, 2025, 5:22 AM
Problem. Let 
be the nonnegative integers in 
where 
. Define 
. Prove that 
.
Bonus problem (Open). Can we prove a tighter result than the one above? That is, is there a function
such that 
where 
, and the statement is still true when 
?
Or, is there a function such that 
where 
, and the statement is still true when ?.
be the nonnegative integers in 
where 
. Define 
. Prove that 
.Bonus problem (Open). Can we prove a tighter result than the one above? That is, is there a function
such that 
where 
, and the statement is still true when 
?Or, is there a function
such that 
where 
, and the statement is still true when ?.This post has been edited 10 times. Last edited by CBMaster, Apr 27, 2025, 5:43 AM
Reason: .
Reason: .
Inequality with condition a+b+c = ab+bc+ca (and special equality case)
by DoThinh2001, May 2, 2019, 11:41 AM
Let 
be real numbers such that 
and 
Prove that
and determine the equality cases.
(Edit: Proposed by sir Leonard Giugiuc, Romania)
be real numbers such that 
and 
Prove that
and determine the equality cases.(Edit: Proposed by sir Leonard Giugiuc, Romania)
This post has been edited 1 time. Last edited by DoThinh2001, May 3, 2019, 10:09 AM
Diophantine equation !
by ComplexPhi, Feb 4, 2015, 2:05 PM
Determine all triples 
satisfying :
![\[n^{2p}=m^2+n^2+p+1\]](//latex.artofproblemsolving.com/7/4/e/74ee592cb8b3f3a3996ea34f40b78fde9d3672fc.png)
where
and 
are integers and 
is a prime number.
satisfying :![\[n^{2p}=m^2+n^2+p+1\]](http://latex.artofproblemsolving.com/7/4/e/74ee592cb8b3f3a3996ea34f40b78fde9d3672fc.png)
where
and 
are integers and 
is a prime number.7^a - 3^b divides a^4 + b^2 (from IMO Shortlist 2007)
by Dida Drogbier, Apr 21, 2008, 2:53 PM
Find all pairs of natural numbers 
such that 
divides 
.
Author: Stephan Wagner, Austria
such that 
divides 
.Author: Stephan Wagner, Austria
The oldest, shortest words — "yes" and "no" — are those which require the most thought.
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