[/quote]
,
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.
and 
,
.
.
be an isosceles triangle with 
Let 
be a point on the circumcircle of 
Let 
intersect the extension of 
at 
Let 
be the midpoint of segment 
and let 
be the intersection of 
and 
Let the extension of 
intersect 
and the circumcircle of 
and 
respectively. Given that 
and 
compute the length of 
and 
and 
are on the same side of 
be midpoint of segment 
be midpoint of segment 
with 
and 
that verify the following equation for all 
:
be positive real integer such that 
Prove that
such that 
and 
is a prime number
be a positive integer, and let its positive divisors be![\[
d_1 < d_2 < \cdots < d_k.
\]](http://latex.artofproblemsolving.com/6/3/e/63ec82aba0d6c4eece89af6a1247a89498616281.png)
Define 
to be the number of ordered pairs 
with 
such that 
.
.![\[
n = p_1^{e_1} p_2^{e_2} \cdots p_k^{e_k},
\]](http://latex.artofproblemsolving.com/2/7/2/272dc2cc33ebbf695311cb61567fa0297e1299b6.png)
where the 
are distinct primes and the 
are positive integers.
be the circumcenter of an acute-angled triangle 
.
meets the side 
and 
are 
and 
and 
beyond 
and 
.
is a rectangle if and only if 
.
satisfying
Prove
beautiful divisors and difference of any 
that verify the following equation for all 
:
.
such that no sum of three distinct elements of 
.
.
Y by
Circles 
and 
with centres 
and 
, respectively, intersect at points 
and 
so that the centre of each circle lies outside the other circle. Line 
intersects circle again at point 
and line 
intersects circle again at point 
. Prove that the points 
and are concyclic
and 
with centres 
and 
, respectively, intersect at points 
and 
so that the centre of each circle lies outside the other circle. Line 
intersects circle again at point 
and line 
intersects circle again at point 
. Prove that the points 
and are concyclic
kite,
,
(by central angle theorem)
is cyclic, likewise we get 
is cyclic, and we're done.
