AoPS Academy
, your right-neighbor’s is 
, and the next person over is 
, then for every trio in the circle they see
be an acute triangle with 
. Denote by 
and 
points on the segment 
such that 
. 
is a point on segment 
. 
intersects 
and 
at 
and 
, respectively. The angle bisectors of 
and 
intersect at 
. If 
and 
, prove that 
is cyclic.
, let 
denote the sum of the digits of . Determine the value of 
.
be a scalene triangle with incircle 
.
be the circle tangent to and passing through 
and 
, and denote by 
, 
the second intersection points of and 
, , resp. Define 
, 
, 
, 
cyclically.
be the circle internally tangent to 
, 
, 
. is tangent to 
.
, where 
and 
are relatively prime positive integers. What is 
?
, given that 
are positive integers? that make this possible?
, the difference is 
(the "discriminant" is exactly greater than the area).
more than the first term, and the fourth term is 
more than the second term. Find the first term.
such that 
.
of fair coins. Bob flips 
fair coins. Denote by 
the probability that Alice flips at least as many heads as Bob does given that she has exactly coins. Find, with proof, a closed form for 
.
kawaii if it satisfies 
(
is the number of positive factors of , while 
is the number of integers not more than that are relatively prime with ). Find all that is kawaii.
such that 
coprime, 
and 
.
works, now we show other 
,
.
be the two smallest prime factors of 
![\[\frac{\varphi(n)}n = \prod_{\text{prime }p\mid n} \left(1-\frac1p\right)\le \left(1-\frac1{q_1}\right)\left(1-\frac1{q_2}\right)=1+\frac1{q_1q_2}-\left(\frac1{q_1}+\frac1{q_2}\right){\overset{\text{AM-GM}}{\le}} \left(1 - \frac1{\sqrt{q_1q_2}}\right)^2\le\left(1-\frac1{\sqrt{n}}\right)^2\]](http://latex.artofproblemsolving.com/7/3/2/7322fddf32a558fede16a67bdb13ca22ab4850e6.png)
contradiction. Otherwise, 
,
,
which gives 
.