by Assassino9931, Apr 27, 2025, 10:21 PM 
be an acute triangle and let 
be an interior point for it such that 
. Prove that
When does equality hold?by egxa, Apr 27, 2025, 1:57 PM 
cities in a country, where 
is an integer. Some pairs of cities are connected by direct (two-way) flights. For two cities 
and 
we define:
A 
between and as a sequence of distinct cities 
, 
, such that there are direct flights between 
and 
for every 
;
A 
between and as a path between and such that no other path between and has more cities;
A 
between and as a path between and such that no other path between and has fewer cities. and in the country, there exist a long path and a short path between them that have no cities in common (except and ). Let 
be the total number of pairs of cities in the country that are connected by direct flights. In terms of , find all possible values by skellyrah, Apr 24, 2025, 8:32 PM 
by David-Vieta, Sep 8, 2024, 5:06 AM . Consider a 
grid, a set 
of squares is called connected if for any points 
in , there exists an integer 
and 
squares 
in such that and shares a common side (
).
satisfying that however the squares are colored black or white, there always exists a connected set for which the absolute value of the difference between the number of black and white squares is at least .by Lukaluce, Jul 2, 2022, 9:11 PM 
be a convex quadrilateral with 
. Let 
be the point of intersection of 
with 
and let 
be the midpoint of 
. On the extension of 
, beyond the point 
, we pick a point 
such that 
. Let 
and 
be the projections of and respectively on 
. The circumcircle of the triangle 
meets again at the point 
. If 
is the point of intersection of with , prove that the lines 
and 
intersect on the line 
.by Jalil_Huseynov, May 17, 2022, 6:44 PM 
of positive integers such that 
is multiple of 
and 
is multiple of 
.by bgn, Apr 5, 2017, 11:56 AM let 
be the -excenter. Let 
be an arbitrary circle that passes through 
and intersects the extensions of sides 
(extended from 
) at 
respectively. Let 
be points on segments 
respectively such that 
and 
.Lines 
intersect at . Lines 
intersect at .
are collinear.by utkarshgupta, Apr 16, 2016, 5:54 AM be a quadrilateral inscribed in a circle . The lines 
and meet at , the lines and meet at 
, and the diagonals 
and 
meet at 
. Let be the midpoint of the segment 
, and let be the common point of the segment 
and the circle . Prove that the circumcircle of the triangle 
and are tangent to one another.
be the center of 
.
and 
.
be the reflection of in 
is a straight line (radical axis are concurrent).
is the polar od , 
and 
are straight lines..
and 
are cyclic quadrilaterals.
lies on the polar of ,
is a straight line; is the midpoint of 
and 
is the reflection of about , is the reflection of about 
is a cyclic pentagon.
and centre .
and 
maps to .
are tangent to 
;
and 
are tangent to 
.by juckter, Jun 22, 2014, 4:09 PM be an acute triangle and 
its circumcircle. Let be the line tangent to at . Let and be the intersections of the circumference with center and radius with lines and , respectively. Prove the orthocenter of lies on line 
.by orl, Jun 26, 2005, 12:16 PM . Let be the circumcenter of this triangle . Let 
, , be the feet of the altitudes of triangle from the vertices , , 
, respectively. Denote by 
, 
, 
the midpoints of these altitudes 
, 
, 
, respectively. The incircle of triangle has center and touches the sides , 
, at the points , , , respectively. Prove that the four lines 
, 
, 
and 
are concurrent. (When the point concides with , we consider the line as an arbitrary line passing through .)Stay insane,Coz it's your will, labour and pain,which takes you to the top of the mountain.
September 2017
February 2017
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be non-negative number. Prove that 