Nice problem about the Lemoine point of triangle JaBC and OI line
by Ktoan07, Apr 18, 2025, 10:24 AM
Let 
be an acute-angled, non-isosceles triangle with circumcenter 
and incenter 
, such that
![\[
\prod_{\text{cyc}} \left( \frac{1}{a+b-c} + \frac{1}{a+c-b} - \frac{2}{b+c-a} \right) \neq 0,
\]](//latex.artofproblemsolving.com/7/8/1/78125759bf5c73149e4a56538e9b0c44309704a7.png)
where
, 
, and 
.
Let
, 
, and 
be the excenters opposite to vertices 
, 
, and 
, respectively, and let 
, 
, and 
be the Lemoine points of triangles 
, 
, and 
, respectively.
Prove that the circles
, 
, and 
all pass through a common point 
. Moreover, the isogonal conjugate of with respect to lies on the line 
.
be an acute-angled, non-isosceles triangle with circumcenter 
and incenter 
, such that![\[
\prod_{\text{cyc}} \left( \frac{1}{a+b-c} + \frac{1}{a+c-b} - \frac{2}{b+c-a} \right) \neq 0,
\]](http://latex.artofproblemsolving.com/7/8/1/78125759bf5c73149e4a56538e9b0c44309704a7.png)
where
, 
, and 
.Let
, 
, and 
be the excenters opposite to vertices 
, 
, and 
, respectively, and let 
, 
, and 
be the Lemoine points of triangles 
, 
, and 
, respectively.Prove that the circles
, 
, and 
all pass through a common point 
. Moreover, the isogonal conjugate of with respect to lies on the line 
.Attachments:
This post has been edited 1 time. Last edited by Ktoan07, an hour ago
intersect at point 
.
and 
with their common external tangents lie on a circle 
.
and 
with their common external tangents lie on a circle 
.
grid, taking turns. Petya goes first. On his turn, a player writes an uppercase Russian letter in an empty cell (each cell can contain only one letter). When all cells are filled, Petya is declared the winner if there are four consecutive cells horizontally spelling the word ``ПЕТЯ'' (PETYA) from left to right, or four consecutive cells vertically spelling ``ПЕТЯ'' from top to bottom. Can Petya guarantee a win regardless of Vasya's moves?
,
is on segment 
,
is on segment 
such that the circumcircles of triangles 
and 
are tangent to line 
.
,
.
for which there exists an even natural number 
such that the number![\[
(a - 1)(a^2 - 1)\cdots(a^n - 1)
\]](http://latex.artofproblemsolving.com/9/c/a/9caf4eeb82ff46b5ba55ab4b6bc28f0cace586ec.png)
is a perfect square.
points are marked, no three of which are collinear. All possible segments between them are drawn. Grisha assigned to each drawn segment a real number with absolute value no greater than 
.
marked points, he calculated the sum of the numbers on all 
connecting segments. It turned out that the absolute value of each such sum is at least 
pairwise distinct complex numbers are written on a board. It turns out that there are exactly 760 ways to choose two numbers 
from the board such that:![\[
a^2 + b^2 + 1 = 2ab
\]](http://latex.artofproblemsolving.com/8/f/c/8fc5df8dbc8e1647edfaa96dcfa8a0dfdac0c961.png)
Ways that differ by the order of selection are considered the same. Prove that there exist two numbers 
and 
from the board such that:![\[
c^2 + d^2 + 2025 = 2cd
\]](http://latex.artofproblemsolving.com/2/7/0/2706d72797c5e392b3f068a8174bd257e4d93fa1.png)
is given. It is known that triangles 
,
,
,
and 
with integer coefficients is called 
if from the divisibility of both differences 
and 
by 
,
and 
are divisible by 100. Does there exist such an important pair of polynomials 
,
,
and 
is also important?
September 2017
February 2017