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.
be a subset of 
. Suppose there is a positive integer 
such that for any integer 
, one can find positive integers 
so that 
and all the digits in the decimal representations of (expressed without leading zeros) are in . Find the smallest possible value of 
.
, let 
denote the number of elements in the set .
be the set of all functions![\[ f : \{1, 2, \ldots, n\} \to \{1, 2, \ldots, k\} \quad \text{with } n \geq 3,\; k \geq 2 \]](http://latex.artofproblemsolving.com/b/1/b/b1bf08934378882b9cd0b7d983f83b15453acca9.png)
satisfying the condition:![\[ f(i) \ne f(i+1) \quad \text{for every } i,\; 1 \leq i \leq n-1. \]](http://latex.artofproblemsolving.com/f/d/1/fd11b44d7fc482b5c6018e19161f6e30b7835c2d.png)
Show that![\[ |F| = k(k-1)^{n-1}. \]](http://latex.artofproblemsolving.com/a/8/8/a88a07984bad88a854de4dad8c24424aaad38365.png)
denote the number of functions in satisfying 
.
, show that![\[ c(n, k) = k(k-1)^{n-1} - c(n-1, k). \]](http://latex.artofproblemsolving.com/8/7/2/87285a27401b9853e72b45249196954082d9ec30.png)
,![\[ c(n, k) = (k-1)^n - (-1)^n(k-1). \]](http://latex.artofproblemsolving.com/2/f/c/2fcbcb385fcc1b3956232dadfd7ec4a31f4d71d6.png)
, 
be the orthocenter of it and 
be any point on the side 
. The points 
are on the segments 
, respectively, such that the points 
and 
are cyclic. The segments 
and 
intersect at 
is a point on 
such that 
is tangent to the circumcircle of triangle 
at 
and 
intersect at 
. Prove that the points 
and lie on the same line.
, point 
lies on side 
and point 
lies on side 
. Let 
and 
be points on segments 
and 
, respectively, such that 
is parallel to 
. Let 
be a point on line 
, such that lies strictly between and , and 
. Similarly, let 
be the point on line 
, such that lies strictly between and , and 
.
, and are concyclic.
-liter capacity stand at the vertices of a regular pentagon. Cinderella and her wicked Stepmother go through a sequence of rounds: At the beginning of every round, the Stepmother takes one liter of water from the nearby river and distributes it arbitrarily over the five buckets. Then Cinderella chooses a pair of neighbouring buckets, empties them to the river and puts them back. Then the next round begins. The Stepmother goal's is to make one of these buckets overflow. Cinderella's goal is to prevent this. Can the wicked Stepmother enforce a bucket overflow?
and 
such that 
is divisible by 
?
such that for all 
,![\[f(x+yf(x))+y = xy + f(x+y).\]](http://latex.artofproblemsolving.com/c/f/3/cf3d20a041c27244e90876119b4568b7a3e13c03.png)
is acute triangle. 
is orthocenter, 
is the midpoint of , 
is the midpoint of smaller arc . Point 
is on 
such that, 
is perpendicular to 
. Prove that: 
is paralelogram(Synthetic sol needed).
, 
.
, 
be positive integers. Julia and Florian play a game on a 
board. Julia cells. such that Florian has a strategy to always deduce the entire tiling.
be an equilateral triangle of side length . Point 
is chosen on side such that the length of 
is 
. Likewise points 
and 
are chosen on sides 
and . with 
. Line segments are drawn from points 
to their corresponding opposite vertices. The intersections of these line segments form a triangle, labeled 
. Prove that 
is an equilateral triangle with side length 
.
Find the maximum positive integer 
such that for every subset 
with n elements, there always exist two elements a, b in T such that:
Let 
be such that any positive integer 
where the non-negative integers 
have all their digits in 
Find the smallest possible number of elements in 
?