A lot of tangent circle
by ItzsleepyXD, Mar 23, 2025, 7:53 AM
Let 
be a triangle with circumcircle 
and circumcenter 
. Let 
and 
represent the 
-excircle and -excenter, respectively. Denote by 
the circle tangent to 
, 
, and on the arc not containing , and similarly for 
. Let the tangency points of 
with line be 
, respectively. Let 
be the intersection point of 
and . Define 
as the point on segment 
such that 
. Suppose that 
intersects again at 
. Let 
be the touch point of the -mixtilinear incircle and , and let 
be the antipode of with respect to . Let 
be the intersection of 
and 
.
Show that the line
is the radical axis of and .
be a triangle with circumcircle 
and circumcenter 
. Let 
and 
represent the 
-excircle and -excenter, respectively. Denote by 
the circle tangent to 
, 
, and on the arc not containing , and similarly for 
. Let the tangency points of 
with line be 
, respectively. Let 
be the intersection point of 
and . Define 
as the point on segment 
such that 
. Suppose that 
intersects again at 
. Let 
be the touch point of the -mixtilinear incircle and , and let 
be the antipode of with respect to . Let 
be the intersection of 
and 
.Show that the line
is the radical axis of and .Circles and Chords
by steven_zhang123, Mar 23, 2025, 7:29 AM
(1) Let , 
and 
be points on circle divided into three equal parts. Construct three equal circles 
, 
, and 
tangent to internally at points , , and respectively. Let 
be any point on arc 
, and draw tangents 
, 
, and 
to circles , , and respectively. Prove that 
.
(2) Let
, 
, 
, 
be points on circle divided into 
equal parts. Construct equal circles , , , 
tangent to internally at , , , . Let be any point on circle , and draw tangents 
, 
, , 
to circles , , , . If the sum of 
of , , , equals the sum of the remaining 
(where 
), find all such .
, 
and 
be points on circle divided into three equal parts. Construct three equal circles 
, 
, and 
tangent to internally at points , , and respectively. Let 
be any point on arc 
, and draw tangents 
, 
, and 
to circles , , and respectively. Prove that 
.(2) Let
, 
, 
, 
be points on circle divided into 
equal parts. Construct equal circles , , , 
tangent to internally at , , , . Let be any point on circle , and draw tangents 
, 
, , 
to circles , , , . If the sum of 
of , , , equals the sum of the remaining 
(where 
), find all such .
permutations of sets
by cloventeen, Mar 23, 2025, 2:36 AM
Find the number of permutations of the set 
with the set 
such that each element in the permutations has at most one immediate neighbor greater than itself.
with the set 
such that each element in the permutations has at most one immediate neighbor greater than itself.number theory question?
by jag11, Mar 22, 2025, 10:41 PM
This post has been edited 1 time. Last edited by jag11, Yesterday at 10:41 PM
Reason: edit
Reason: edit
Integer FE
by GreekIdiot, Mar 22, 2025, 8:53 PM
Let 
denote the set of positive integers
Find all
such that for all 
it holds that 
denote the set of positive integersFind all
such that for all 
it holds that 
Eventually constant sequence with condition
by PerfectPlayer, Mar 18, 2025, 4:27 AM
A positive real number sequence 
and a positive integer 
is given.
Let
and for each 
![\[f_n(k)=\frac{a_n+\dots+a_{k+1}}{n-k}-\frac{a_k+\dots+a_1}{k}\]](//latex.artofproblemsolving.com/0/4/1/041f80cf63f17c0f62c166021af474a5d305eb87.png)
Then for every integer 
the condition
![\[a_{n+1}=\max_{0\leq k<n}(f_n(k))\]](//latex.artofproblemsolving.com/b/1/f/b1f5aff787de8fc9c113cf8441a6b2900a19f24f.png)
is satisfied. Prove that this sequence must be eventually constant.
and a positive integer 
is given.Let
and for each 
![\[f_n(k)=\frac{a_n+\dots+a_{k+1}}{n-k}-\frac{a_k+\dots+a_1}{k}\]](http://latex.artofproblemsolving.com/0/4/1/041f80cf63f17c0f62c166021af474a5d305eb87.png)
Then for every integer 
the condition![\[a_{n+1}=\max_{0\leq k<n}(f_n(k))\]](http://latex.artofproblemsolving.com/b/1/f/b1f5aff787de8fc9c113cf8441a6b2900a19f24f.png)
is satisfied. Prove that this sequence must be eventually constant.triangles with equal areas
by mathuz, Dec 28, 2023, 7:41 AM
Let 
be a trapezoid with 
. A point 
is chosen inside the trapezoid, and a point 
is chosen inside the triangle 
such that 
, 
. Prove that triangles 
and 
have equal areas.
be a trapezoid with 
. A point 
is chosen inside the trapezoid, and a point 
is chosen inside the triangle 
such that 
, 
. Prove that triangles 
and 
have equal areas.This post has been edited 1 time. Last edited by mathuz, Dec 28, 2023, 7:42 AM
EM // AC wanted, isosceles trapezoid related
by parmenides51, Dec 18, 2020, 6:46 PM
Given is an isosceles trapezoid with 
and 
. The projection from 
on 
is 
. The midpoint of the diagonal 
is 
. Prove that 
is parallel to 
.
(Karl Czakler)
with 
and 
. The projection from 
on 
is 
. The midpoint of the diagonal 
is 
. Prove that 
is parallel to 
.(Karl Czakler)
This post has been edited 1 time. Last edited by parmenides51, May 9, 2024, 11:56 PM
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