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.
such that for any two infinite sequences of positive integers 
and 
, such that 
for all 
, there always exists a real number 
such that 
for all 
.
and 
be two isogonal conjugate with respect to triangle 
. Let 
and 
be two points lying on the circle 
such that 
and 
are perpendicular and parallel to bisector of 
, respectively. Prove that 
and 
bisect two arcs 
containing 
and not containing , respectively, of 
.
such that there exists an integer 
satisfying
, with 
).
and 
be positive integers. Prove that there exists a positive integer 
such that for every odd integer 
, the digits in the base-
representation of 
are all greater than .
denote the set of positive integers
such that for all 
it holds that 
such that
for all 
.
be a positive integer. Megavan and Minivan are playing a game, where Megavan secretly chooses a real number 
in ![$[0, 1]$](http://latex.artofproblemsolving.com/a/b/1/ab178d831a786b92cb4c9ddc2d33578223036f98.png)
. At the start of the game, the only information Minivan has about is in . He needs to now learn about based on the following protocols: at each turn of his, Minivan chooses a number 
and submits to Megavan, where Megavan replies immediately with one of 
, 
, or 
, subject to two rules:
The answers in the form of and must be truthful; Define the score of a round, known only to Megavan, as follows: 
if the answer is in the form and , and 
if in the form . Then for every positive integer and every consecutive rounds, at least one round has score no more than 
.
such that 
and 
. Let 
be the minimum number of queries that Minivan needs in order to guarantee success, regardless of Megavan's strategy. Prove that 
with the set 
such that each element in the permutations has at most one immediate neighbor greater than itself.
with 
and 
. The projection from 
on 
is 
. The midpoint of the diagonal 
is 
. Prove that 
is parallel to 
.
and 
be positive integers with 
. There are cupcakes of different flavors arranged around a circle and people who like cupcakes. Each person assigns a nonnegative real number score to each cupcake, depending on how much they like the cupcake. Suppose that for each person , it is possible to partition the circle of cupcakes into groups of consecutive cupcakes so that the sum of 's scores of the cupcakes in each group is at least 
. Prove that it is possible to distribute the cupcakes to the people so that each person receives cupcakes of total score at least with respect to .
be the orthocenter of acute triangle , let 
be the foot of the altitude from 
to 
, and let be the reflection of across . Suppose that the circumcircle of triangle 
intersects line at two distinct points 
and 
. Prove that is the midpoint of 
.
such that 
.
