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.
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 .
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 
. be a trapezoid with 
. A point 
is chosen inside the trapezoid, and a point is chosen inside the triangle 
such that 
, 
. Prove that triangles 
and 
have equal areas.
and a positive integer 
is given.
and for each 
![\[f_n(k)=\frac{a_n+\dots+a_{k+1}}{n-k}-\frac{a_k+\dots+a_1}{k}\]](http://latex.artofproblemsolving.com/0/4/1/041f80cf63f17c0f62c166021af474a5d305eb87.png)
Then for every integer 
the condition![\[a_{n+1}=\max_{0\leq k<n}(f_n(k))\]](http://latex.artofproblemsolving.com/b/1/f/b1f5aff787de8fc9c113cf8441a6b2900a19f24f.png)
is satisfied. Prove that this sequence must be eventually constant.
be a triangle with circumcircle 
and circumcenter 
. Let 
and 
represent the 
-excircle and -excenter, respectively. Denote by 
the circle tangent to 
, 
, and on the arc not containing , and similarly for 
. Let the tangency points of 
with line be 
, respectively. Let 
be the intersection point of 
and . Define 
as the point on segment 
such that 
. Suppose that 
intersects again at 
. Let 
be the touch point of the -mixtilinear incircle and , and let 
be the antipode of with respect to . Let 
be the intersection of 
and 
.
is the radical axis of and .
, 
and 
be points on circle divided into three equal parts. Construct three equal circles 
, 
, and 
tangent to internally at points , , and respectively. Let 
be any point on arc 
, and draw tangents 
, 
, and 
to circles , , and respectively. Prove that 
.
, 
, 
, 
be points on circle divided into 
equal parts. Construct equal circles , , , 
tangent to internally at , , , . Let be any point on circle , and draw tangents 
, 
, , 
to circles , , , . If the sum of 
of , , , equals the sum of the remaining 
(where 
), find all such .
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 
.
