AoPS Academy
and 
If each person remains there for 
minutes, what's the probability that all three people meet each other?
with 
, inscribed in circle 
.
be the orthocenter of triangle 
, and 
be the midpoint of 
., parallel to 
, intersects lines 
and 
at points 
and 
, respectively.
be the circumcenter of triangle 
. Prove that: Points 
are collinear.
of integers satisfying![\[x_0 = 0, |x_n| = |x_{n-1} + 1|\]](http://latex.artofproblemsolving.com/4/8/e/48e7b6f98f2062879377a9a2e507a63eac9269a8.png)
.
.
therefore u can use 1 and -1 alternating coming to the minimum of 0, because there is sequence is periodic with the period 4, and the first 4 
besides 
are 0 and they repeat themselves 250 times. also 0 is the smallest number possible, since 
when is positive. In other words, you can't go from 1 to -1.
. 24 more numbers. Then, you go
. Prove that
and solve for the area of a hexagon in that 
square.
,
and 
and so forth.
.
and this means that 
or 
since we want the minimum. Then 
,
,
,
.