[/quote]
,
It is well known that 
.
be positive numbers such that 
.![\[T = \dfrac{bc}{\sqrt{a}+3}+\dfrac{ca}{\sqrt{b}+3}+\dfrac{ab}{\sqrt{c}+3} \]](http://latex.artofproblemsolving.com/6/d/5/6d570d06679f17452790b90171428d645e97fe9f.png)
be the incenter of a triangle 
with 
,
be the midpoint of the arc 
of the circumcircle of the triangle. The perpendicular line to 
passing through 
at point 
.
intersects the circumcircle of triangle 
at point 
.
is tangent to the circumcircle of triangle 
such that there exist infinitely many rational numbers 
satisfying![\[
\left| \frac{q}{p} - \frac{\sqrt{5} - 1}{2} \right| < \frac{\alpha}{p^2}.
\]](http://latex.artofproblemsolving.com/1/1/6/1162c6a7215bff2f9fdaa042a6952f86524e9e92.png)
such that
are all powers of 
.
be a convex quadrilateral. Let 
be the midpoints of 
respectively. If 
concur at a point 
prove that 
,
children standing in a circle. For each girl, it turns out that among the five people following her clockwise, there are more boys than girls. Find the greatest number of girls that can stand in a circle.
so that there exists a unique 
satisfying 
and![\[f(x+g(y))+f(-x-g(-y))=g(x+f(y))+g(-x-f(-y))\]](http://latex.artofproblemsolving.com/d/e/0/de0ea33c3a679ff02762be39e9013b72974e4287.png)
for all 
.
be a triangle with circumcircle 
and incenter 
and let 
be the midpoint of 
.
,
,
are selected on sides 
,
such that 
,
,
.
intersects 
other than 
.
and 
meet on 
interesting if 
is composite and for every divisor 
of 
and 
is also a divisor of 
an isosceles triangle such that 
.
be a point satisfying
.
Y by Adventure10, Mango247
In an infinite grid of regular triangles, Niels and Henrik are playing a game they made up.
Every other time, Niels picks a triangle and writes
in it, and every other time, Henrik picks a triangle where he writes a 
. If one of the players gets four in a row in some direction (see figure), he wins the game.
Determine whether one of the players can force a victory.
Every other time, Niels picks a triangle and writes
in it, and every other time, Henrik picks a triangle where he writes a 
. If one of the players gets four in a row in some direction (see figure), he wins the game.Determine whether one of the players can force a victory.
This post has been edited 1 time. Last edited by parmenides51, Sep 3, 2019, 9:33 AM
regular triangles, and observe that every triangle contributes to exactly 
edge in a hexagon. The strategy for player Henrik is to always play in the triangle sharing an edge with the triangle placed by Niels. This implies that Niels will never get multiple in a row across 
hexagons, so he can never get 
in a row. This is a strategy for Henrik to always draw.
