be a scalene triangle with point 
is an equilateral . Let 
and 
at 
and 
respectively and 
and 
at 
and 
respectively .
,
on 
satisfies 
and 
.
be the point where the perpendicular from 
to line 
meets the angle bisector of 
.
holds.
be a positive integer such that 
.
.
be lines in the plane such that no two of them are parallel and no three of them are concurrent. Let 
be the intersection point of lines 
.
on 
and 
on 
such that 
Interior points.
)
.
and 
Prove that
![\[\lfloor x \rfloor\]](http://latex.artofproblemsolving.com/d/3/f/d3f8facf62413e1f17c89b9c183a8735c2defe8b.png)
denote the greatest integer less than or equal to x . What is the sum of the squares of the real numbers x for which ![\[ x^2 - 20\lfloor x \rfloor + 19 = 0 \]](http://latex.artofproblemsolving.com/f/a/9/fa96f2d6217a16029e5f03499e0a83deffdfa28a.png)
Prove that
of the pizza, where 
is a positive integer. Find 
![\[\mathcal{P}(x) = x^{108}+x^{102}+x^{96}+2x^{54}+3x^{36}+4x^{24}+5x^{18}+6\]](http://latex.artofproblemsolving.com/e/b/2/eb28918359848dd07784a3c2c6a34b934743c016.png)
be 
.![\[\dfrac{r_1^6+r_2^6+\dots+r_{108}^6}{r_1^6r_2^6+r_1^6r_3^6+\dots+r_{107}^6r_{108}^6}\]](http://latex.artofproblemsolving.com/3/1/5/315341a7adf35bc04e40ee0685a011c9f7b76115.png)
without Newton Sums.
Is there an elegant way to prove this?
can be partitioned into 
pairs such that the sum of the elements in each pair is a power of 
.
is stylish because the set 
can be partitioned as 
,
,
.
to the cross product 
reveals that a negative determinant [of 
] corresponds to a matrix which reverses the direction of the cross product of two vectors.[/quote]
and 
,
and 
although its supposed to be a quick comparison.
chance of landing heads when flipped. Krithik is playing a game where he starts with 
point. Every turn, he flips the coin, and if it lands heads, he gains 
points. Find the expected number of turns until Krithik wins.
Y by
Let 
be the feet of perpendiculars drawn on sides 
, 
, 
from 
respectively . Let 
be the intersection point of straight lines 
and 
. Let 
be the intersection point of the circumscribed circle of triangle 
with side , that is not 
. Let 
be the intersection point of the circumscribed circle of triangle 
with side 
that is not 
. In this case, show that straight line 
passes through the midpoint of line segment 
.
be the feet of perpendiculars drawn on sides 
, 
, 
from 
respectively . Let 
be the intersection point of straight lines 
and 
. Let 
be the intersection point of the circumscribed circle of triangle 
with side , that is not 
. Let 
be the intersection point of the circumscribed circle of triangle 
with side 
that is not 
. In this case, show that straight line 
passes through the midpoint of line segment 
.This post has been edited 3 times. Last edited by parmenides51, Nov 28, 2022, 6:22 AM
,
is the external bisector of 
are concyclic, then 
.
.
