AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
be a scalene triangle with point 
and 
on the plane such that 
is an equilateral . Let 
intersect 
and 
at 
and 
respectively and 
intersect 
and 
at 
and 
respectively .
Prove that: 
is the sum of 
square numbers
Let 
and 
is the sum of square numbers. Prove that: 
is the sum of square numbers
real number on the board such that each number is more than 
.
on the board and write 
.
and 
.
?
be triangle and point 
be 
altitude of 
.
be a point on and such that 
and 
. Point 
is the intersection of 
and 
. Point be intersection of 
and 
. Line 
intersect circumcircle of again at .
.
bisect .
, define a sequence of integers 
by letting 
and 
for 
. Let 
. Determine the largest possible 
such that, for some positive integer , the numbers 
are all prime.
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 
such that 
for all real number 
and 
be a sequence of perfect squares such that 
can be obtained by concatenating a digit to the right of 
. Determine all such sequences that are of maximum length.
be unit vectors on a plane, where is an odd number. Prove that there exist 
for 
such that 
be 
and let the orthocenter of be 
. Then prove that the isogonal conjugate of wrt lies on the Euler line 
of .
be and let the orthocenter of be . Then prove that the isogonal conjugate of wrt lies on the Euler line of .
with circumcenter 
orthic triangle 
tangential triangle 
Let 
be the isogonal conjugate (WRT ) of the isotomic conjugate of 
WRT and 
be the medial triangle of Then is the homothetic center of 
be the midpoint of 
and 
be the intersection of 
with 
respectively, then lie on the X -midline 
of Let 
be the excentral triangle of 
then it's clear that 
and 
are homothetic, so 
are concurrent.
lie on the circle 
with center and diameter 
so 
and 
are antiparallel WRT 
and 
lies on 
Similarly, we can prove lies on 
so is the homothetic center of 
with orthocenter 
circumcenter 
Let 
be the isotomic conjugate of WRT and 
be the circum-rectangular hyperbola of passing through 
Then 
is a tangent of 
be the orthic triangle of and 
be the Nagel point of 
Since is the crosspoint of and 
WRT 
so 
is tangent to the Jerabek hyperbola 
of at On the other hand, by Lemma we get passes through the isogonal conjugate of WRT so is tangent to at 
be the isotomic conjugate of the circumcenter of WRT and let be the circum-rectangular hyperbola of passing through Notice the tangent of at passes through the isogonal conjugate 
of WRT 
so by Corollary we conclude that lies on the Euler line of 
, given a point 
, define it's isotomic conjugate 
, and isogonal conjugate of be 
be circumcenter, centroid, orthocenter respectively.
tangent to the Jerabek hyperbola 
of 
lies on , 
are collinear, 
and the image of isotomic conjugate of is 
are collinear. 
so 
is tangent to . Let the center of be 
, the midpoint of 
be 
, We want to proof 
are collinear, it's well-known that this line pass through and Tarry point (the fourth intersection of and Kiepert hyperbola of .)
be 
, lies on so lies on . Since both 
are isotomic conjugate, so 
pass through the isotomic conjugate of the infinitely point on , consider the image of this line after isotomic and isogonal conjugate, it will lead to the fact that are collinear, as desired., is the centroid of orthic triangle wrt 
.
-Orthocenter of . Parallel fron to intersect the 
at 
resp.
cyclically. 
: Inverse of respect to circle 
. Define 
cyclically.
: Circumcenter of 
lies on Euler line of . 
.
a
.
. Prove that
