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Random concyclicity in a square config
Maths_VC 0
Source: Serbia JBMO TST 2025, Problem 1
Let 
be a random point on the smaller arc 
of the circumcircle of square 
, and let 
be the intersection point of segments 
and 
. The feet of the tangents from point 
to the circumcircle of the triangle 
are 
and 
, where 
is the center of the square. Prove that points 
, 
, and lie on a single circle.
be a random point on the smaller arc 
of the circumcircle of square 
, and let 
be the intersection point of segments 
and 
. The feet of the tangents from point 
to the circumcircle of the triangle 
are 
and 
, where 
is the center of the square. Prove that points 
, 
, and lie on a single circle.
0 replies
Circle is tangent to circumcircle and incircle
ABCDE 74
N
by zuat.e
Source: 2016 ELMO Problem 6
Elmo is now learning olympiad geometry. In triangle 
with 
, let its incircle be tangent to sides 
, 
, and at , 
, and 
, respectively. The internal angle bisector of 
intersects lines 
and 
at 
and 
, respectively. Let 
and 
be distinct points on side such that 
. Finally, let 
be the circumcircle of 
.
(a) Help Elmo show that is tangent to the circumcircle of 
.
(b) Help Elmo show that is tangent to the incircle of .
James Lin
with 
, let its incircle be tangent to sides 
, 
, and at , 
, and 
, respectively. The internal angle bisector of 
intersects lines 
and 
at 
and 
, respectively. Let 
and 
be distinct points on side such that 
. Finally, let 
be the circumcircle of 
.(a) Help Elmo show that
is tangent to the circumcircle of 
.(b) Help Elmo show that
is tangent to the incircle of .James Lin
74 replies
Division on 1989
mistakesinsolutions 3
N
by reni_wee
Prove that for positive integer 
greater than 
is divisible by 
greater than 
is divisible by 
3 replies
Easy Geo Regarding Euler Line
USJL 12
N
by Ilikeminecraft
Source: 2021 Taiwan TST Round 2 Independent Study 1-G
Let be a convex quadrilateral with pairwise distinct side lengths such that 
. Let 
be the circumcenters of 
, respectively. Show that 
, the Euler line of 
and the Euler line of 
are concurrent.
(Remark: The Euler line of a triangle is the line on which its circumcenter, centroid, and orthocenter lie.)
Proposed by usjl
be a convex quadrilateral with pairwise distinct side lengths such that 
. Let 
be the circumcenters of 
, respectively. Show that 
, the Euler line of 
and the Euler line of 
are concurrent.(Remark: The Euler line of a triangle is the line on which its circumcenter, centroid, and orthocenter lie.)
Proposed by usjl
12 replies
3^n + 61 is a square
VideoCake 24
N
by maromex
Source: 2025 German MO, Round 4, Grade 11/12, P6
Determine all positive integers 
such that 
is the square of an integer.
such that 
is the square of an integer.
24 replies
satisfying
Prove
Lines AD, BE, and CF are concurrent
orl 49
N
by Ilikeminecraft
Source: IMO Shortlist 2000, G3
Let be the circumcenter and 
the orthocenter of an acute triangle . Show that there exist points , , and on sides , , and respectively such that ![\[ OD + DH = OE + EH = OF + FH\]](//latex.artofproblemsolving.com/a/8/7/a874af4cb802b9b341b055140a08e096aaf2112e.png)
and the lines 
, 
, and 
are concurrent.
be the circumcenter and 
the orthocenter of an acute triangle . Show that there exist points , , and on sides , , and respectively such that ![\[ OD + DH = OE + EH = OF + FH\]](http://latex.artofproblemsolving.com/a/8/7/a874af4cb802b9b341b055140a08e096aaf2112e.png)
and the lines 
, 
, and 
are concurrent.
49 replies
Is this F.E.?
Jackson0423 1
N
by jasperE3
Let the set
be finite.Find all functions
satisfying the following condition for all real numbers 
:![\[
f(x - 1 - f(x)) = f(x) - x - 1.
\]](http://latex.artofproblemsolving.com/3/a/5/3a51479b61a5c025ed81f539e964ce96b5857f0f.png)
1 reply
IMO Shortlist 2014 N2
hajimbrak 31
N
by Sakura-junlin
Determine all pairs 
of positive integers such that ![\[\sqrt[3]{7x^2-13xy+7y^2}=|x-y|+1.\]](//latex.artofproblemsolving.com/e/c/7/ec778edf97100f3b19350d8a817317ebf1f391bd.png)
Proposed by Titu Andreescu, USA
of positive integers such that ![\[\sqrt[3]{7x^2-13xy+7y^2}=|x-y|+1.\]](http://latex.artofproblemsolving.com/e/c/7/ec778edf97100f3b19350d8a817317ebf1f391bd.png)
Proposed by Titu Andreescu, USA
31 replies
