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.
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 
.
be a subset of 
such that no sum of three distinct elements of 
.
.
that satisfy the equation 
.
be a circle, and let a line 
intersect 
and 
Circles 
and 
are internally tangent to 
and 
respectively, and both are tangent to 
Similarly, circles 
and 
are externally tangent to 
and 
respectively.
the segment 
has a length of 
and 
find the length of segment 
and 
Prove that
.
Prove that
and 
Prove that
Y by Adventure10
On the side of 
parallelogram 
, the point 
is taken, and on the sides 
and 
the point 
and 
respectively, so that the segments 
and 
are parallel to the diagonals of the parallelogram. Prove that the areas of the triangles 
and 
are equal.
parallelogram 
, the point 
is taken, and on the sides 
and 
the point 
and 
respectively, so that the segments 
and 
are parallel to the diagonals of the parallelogram. Prove that the areas of the triangles 
and 
are equal.
,
to 
be 
,
be 
.
,![$[QCR]=h(rx-x)=xh(r-1)$](http://latex.artofproblemsolving.com/b/2/e/b2e5e7934c5f824d93107c8681f923658d4906a3.png)
,
is 
,![$[PBR]=x(rh-h)=xh(r-1)$](http://latex.artofproblemsolving.com/9/2/9/92949081b306cab500d7105e490e02cb18063539.png)
,
