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, the value of 
is 
. Find all complex numbers .![\begin{align*}
(a+bi)\cdot(a-bi) +[(a+bi)+(a-bi)] =a^2+b^2+2a.
\end{align*}](http://latex.artofproblemsolving.com/4/0/9/409b0aed187f4f2cbc8e3f204f8b5fac9bcb44b0.png)
and 
. when the complex numbers lie on the circle centered at 
with radius 
.
be complex numbers. For a complex number 
where 
, define the norm 
to be the distance of 
from the origin, or 
. Let 
be the minimum value and 
be the maximum value of![\[
\frac{ |a + b| + |b + c| + |c + a| }{ |a| + |b| + |c| }
\]](http://latex.artofproblemsolving.com/1/7/5/1752b55a5bca8490d3dbb398fb2c4e4f9f71f47c.png)
for all complex numbers where 
. Find 
.
. When two complex numbers 
and 
are added, their sum 
is also a vector. Geometrically, this is represented by placing the tail of at the head of or vice versa, following the parallelogram rule. The vector then extends from the origin to the head of the translated , forming a triangle with the origin and the head of .
. To find the minimum value, we can look for configurations where ``cancellation" in the vector sums is maximized. This typically happens when the complex numbers are collinear but point in opposite directions. To do this, let and be at the same point, and let 
be the same distance from the origin as and , but in the opposite direction.
Substituting this in the given equation, we get
is equal to 
.
.
red balls and blue balls. A draw consists of randomly selecting 
of the remaining balls from bag without replacement, and then setting them aside. A draw is called a match if the two balls have the same color. Compute the expected number of draws until either a match occurs or the bag is empty.
are points on a line in that order. A line through 
intersects the circle with diameter 
at 
and 
, the circle with diameter 
at 
, and the circle with diameter 
at 
. Prove that 
= 
. (Note that it doesn't matter if and are swapped.)[/quote]
be an acute triangle such that 
. Let 
a point on such that 
. Let be an internal point of the segment 
. Prove that 
if and only if 
.
Prove that
Let 
Prove that
with a radius of 
.
,
.
.
.