One more problem defined only with lines
by Assassino9931, Apr 27, 2025, 10:31 PM
Let 
be a triangle and the points 
and 
on 
, 
and 
on 
, and 
and 
on 
be such that 
and 
. The intersections of 
with 
and 
are 
and 
respectively, and the intersections of 
with 
and 
are 
and 
, respectively. Prove that the lines 
and 
are concurrent.
be a triangle and the points 
and 
on 
, 
and 
on 
, and 
and 
on 
be such that 
and 
. The intersections of 
with 
and 
are 
and 
respectively, and the intersections of 
with 
and 
are 
and 
, respectively. Prove that the lines 
and 
are concurrent.
be fixed points on this order on a line. Let 
be a variable circle through 
and 
at the points 
and 
.
and 
and 
with 
passes through a fixed point independent of 
be the set of all lines in the plane and 
be a set of 
points in the plane. For a line 
let 
be the number of distinct points among the orthogonal projections of the points from 
.
be an integer. Alice and Bob play the following game on the vertices of a regular 
-
rounds: In Alice’s turn on the 
-
position clockwise or anticlockwise. If at the end of any person’s turn the two tokens are on the same vertex, then Alice wins the game, otherwise Bob wins. Decide for each value of 
such that if 
denotes the number of positive integers less than or equal to 
be an integer and denote 
.
we denote Im 
,
and 
.
I
+ 
.
is not a perfect cube of an integer.
such that for all 
,![\[f(x+yf(x))+y = xy + f(x+y).\]](http://latex.artofproblemsolving.com/c/f/3/cf3d20a041c27244e90876119b4568b7a3e13c03.png)
with 
.![\[
(x^{2}-yz)(y^{2}-zx)(z^{2}-xy).
\]](http://latex.artofproblemsolving.com/a/8/7/a87fae27584012a867bc0a90190c0317f9f7ab07.png)
,
,
?
the sum of all positive divisors of 
April 2025
November 2024