[/quote]
,
,
and 
.
is prime, 
with 
and 
be a function such that for every regular 
-
we have 
.
for all reals 
.
be a triangle 
)
are the altitudes of the triangle. The bisector of 
intersects 
at 
.
and 
.
passes through the midpoint of 
.
with 
and at most one of them being zero, show that![\[\frac{1}{\max(a^2,b)}+\frac{1}{\max(b^2,c)}+\frac{1}{\max(c^2,a)}\geq\frac{27}{4}\]](http://latex.artofproblemsolving.com/9/e/1/9e14a9ef2d1a0a98006bcf716fa33c3d9ea29bd5.png)
be an integer. Find all sequences 
satisfying the following conditions:![\[ \text{ (a) } a_i \in \left\{0,1\right\} \text{ for all } 1 \leq i \leq n^2 + n;
\]](http://latex.artofproblemsolving.com/3/c/5/3c509ec2e9013e8d3be492c8eb44a7c33841b74e.png)
![\[ \text{ (b) } a_{i + 1} + a_{i + 2} + \ldots + a_{i + n} < a_{i + n + 1} + a_{i + n + 2} + \ldots + a_{i + 2n} \text{ for all } 0 \leq i \leq n^2 - n.
\]](http://latex.artofproblemsolving.com/9/7/d/97d2a467d1c0dc8594ec024c3bb9b8c87ee85b19.png)
and 
.
Equality holds when 
or 
and 
,
cookies from one of the piles, of which he eats 
and puts 
for which Marica has a winning strategy.
be non-negative real numbers such that 
Prove that
be the circumcenter, symmedian point of 
.
at 
,
at 
,
at 
are concurrent.
be distinct natural numbers each of which has exactly 
positive integer factors. For instance, 
has exactly 
.
divides any of the 
'
Y by Adventure10
Three given lines in the plane pass through a point 
.
(a) Prove that there exists a circle that contains in its interior and intersects the three lines at six points 
in that order around the circle such that 
.
(b) Suppose that a circle contains in its interior and intersects the three lines at six points 
in that order around the circle such that . Prove that:
area (hexagon 
) 
area( 
) + area( 
) 
area(
).
.(a) Prove that there exists a circle that contains
in its interior and intersects the three lines at six points 
in that order around the circle such that 
.(b) Suppose that a circle contains
in its interior and intersects the three lines at six points 
in that order around the circle such that . Prove that:
area (hexagon 
) 
area( 
) + area( 
) 
area(
).This post has been edited 1 time. Last edited by parmenides51, Sep 23, 2018, 8:50 PM
Reason: name typo
Reason: name typo
.
are similar. Take a triangle with sides 
)
that is similar to the last six triangles. Then![$$\frac{1}{2}[ABCDEF]\ge [APB]+[CPD]+[EPF]\iff$$](http://latex.artofproblemsolving.com/5/3/7/5374fa075e1d8d5548ff52216873f27d925b9a56.png)
![$$[DPE]+[FPA]+[BPC]\ge [APB]+[CPD]+[EPF]\iff$$](http://latex.artofproblemsolving.com/d/b/4/db4acd41c5b6f280c592882c907fb41ceda69d50.png)
![$$\frac{t^2}{a^2b^2c^2}\left[(a^2-b^2)^2+(b^2-c^2)^2+(c^2-a^2)^2\right]\ge 0.$$](http://latex.artofproblemsolving.com/f/b/e/fbeb3d8ebec047c3015347408eae885f6a3666a8.png)
