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.
there exists an -digit number divisible by 
all of whose digits are odd.
+1 w
+2 w
be non-zero distinct real numbers so that there exist functions 
so that:
and large enough 
.
so that:
and large enough .
be a quadrilateral with both an incircle and a circumcircle. Let 
and 
be the incenter and circumcenter of , respectively. Let 
be the intersection of lines 
and 
, and let 
be the intersection of lines 
and 
. Let 
and 
be the intersections of the line 
with lines and , respectively. Prove that the circumcircle of 
, the circumcircle of 
, and the line 
are concurrent.
such that 
.
vertices is written a real number. Prove that
(with possibly repeating vertices) in which the sequence of numbers is increasing.
be positive integer such that ![\[ \text{LCM}(x+2, y+2)+\text{LCM}(x, y)=2\text{LCM}(x+1, y+1).\]](http://latex.artofproblemsolving.com/7/3/3/7338e4d6058ededa9b8cd9fcc945ae362db4bad4.png)
Prove that is divisible by .
distinct prime numbers such that such that 
(
) chessboard. Initially, Alice’s queen is placed at the bottom-left corner, and Bob’s queen is at the bottom-right corner. All the other squares on the board are covered by neutral pieces. does Alice have a winning strategy?
with side length 
is drawn on a plane. For a positive real 
, we place three blue equilateral triangle shaped paper with side length on a plane to cover completely. Find the minimum value of . As shown in the picture, it doesn't matter if the blue papers overlap each other or stick out from . Folding or tearing the paper is not allowed.
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.
and a positive integer is given. and for each Then for every integer the conditionis satisfied. Prove that this sequence must be eventually constant.
, prove 
.
. Also 
. We have 
. We will proceed by using induction. Suppose that 
holds, we will prove 
. On the other hand, for all 
Thus, we obtain the following
implying that the sequence 
must be eventually constant as desired.
such that
for any 
,
![\[a_{n+1}=\max_{0\leq k<n}(f_n(k))=f_n(g(n))\]](http://latex.artofproblemsolving.com/8/b/1/8b11c506616b015473f027f4874c01d88634da4e.png)
Step 
,
,
will hold.
: