AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
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 point, and if it lands tails, he loses point. However, after the turn, if he has a negative number of points, his point counter resets to . Krithik wins when he earns 
points. Find the expected number of turns until Krithik wins.
and 
, Jaqueline has a ruler that allows tracing the line 
. Jaqueline also has a special object that allows the construction of a circle of diameter . Also, always when two circles (or a circle and a line, or two lines) intersect, she can mark the points of the intersection with a pencil and trace more lines and circles using these dispositives by the points marked. Initially, she has an acute scalene triangle 
. Show that Jaqueline can construct the incenter of .
be an isosceles triangle with 
Let 
be a point on the circumcircle of on minor arc 
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 at 
and 
respectively. Given that 
and 
compute the length of 
be triangle with point 
and on 
and 
are on the same side of 
be midpoint of segment and 
be midpoint of segment 
be intersection of 
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 
.
. when![\[
n = p_1^{e_1} p_2^{e_2} \cdots p_k^{e_k},
\]](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 with 
. The line 
meets the side at . The circumcenters of the triangles 
and 
are 
and , respectively. Extend the sides 
and 
beyond 
, and choose on the respective extensions points and such that 
and 
. Prove that the quadrilateral 
is a rectangle if and only if 
.
satisfying
Prove
is a positive integer. Call all positive divisors of which are different from and beautiful divisors.We call a special number when it has at least 
beautiful divisors and difference of any beautiful divisors divides as well. Find all special numbers.
that verify the following equation for all 
:
. Let be a subset of 
such that no sum of three distinct elements of is divisible by 
. What is the maximum value of 
.
