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 .
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
Guessing with intervals
by navi_09220114, Mar 22, 2025, 12:55 PM
Let 
be a positive integer. Megavan and Minivan are playing a game, where Megavan secretly chooses a real number 
in ![$[0, 1]$](//latex.artofproblemsolving.com/a/b/1/ab178d831a786b92cb4c9ddc2d33578223036f98.png)
. At the start of the game, the only information Minivan has about is in . He needs to now learn about based on the following protocols: at each turn of his, Minivan chooses a number 
and submits to Megavan, where Megavan replies immediately with one of 
, 
, or 
, subject to two rules:
The answers in the form of and must be truthful;
Define the score of a round, known only to Megavan, as follows: 
if the answer is in the form and , and 
if in the form . Then for every positive integer 
and every consecutive rounds, at least one round has score no more than 
.
Minivan's goal is to produce numbers
such that 
and 
. Let 
be the minimum number of queries that Minivan needs in order to guarantee success, regardless of Megavan's strategy. Prove that 
Proposed by Anzo Teh Zhao Yang
be a positive integer. Megavan and Minivan are playing a game, where Megavan secretly chooses a real number 
in ![$[0, 1]$](http://latex.artofproblemsolving.com/a/b/1/ab178d831a786b92cb4c9ddc2d33578223036f98.png)
. At the start of the game, the only information Minivan has about is in . He needs to now learn about based on the following protocols: at each turn of his, Minivan chooses a number 
and submits to Megavan, where Megavan replies immediately with one of 
, 
, or 
, subject to two rules:
The answers in the form of and must be truthful; Define the score of a round, known only to Megavan, as follows: 
if the answer is in the form and , and 
if in the form . Then for every positive integer 
and every consecutive rounds, at least one round has score no more than 
.Minivan's goal is to produce numbers
such that 
and 
. Let 
be the minimum number of queries that Minivan needs in order to guarantee success, regardless of Megavan's strategy. Prove that 
Proposed by Anzo Teh Zhao Yang
This post has been edited 2 times. Last edited by navi_09220114, Yesterday at 12:59 PM
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.weird FE on R
by frac, Jan 4, 2025, 12:15 PM
Find all functions 
such that
for all 
.
such that
for all 
.This post has been edited 4 times. Last edited by frac, 3 hours ago
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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