AoPS Academy
such that every collection of consecutive positive integers contains an integer whose sum of digits is divisible by 
. for which he can do this, or prove that none exist.
on one side and a 
on the other. Harry has 
of these coins arranged in a line from left to right. He repeatedly performs the following operation: if there are exactly 
coins showing , then he turns over the th coin from the left; otherwise, all coins show and he stops. For example, if 
the process starting with the configuration 
would be 
, which stops after three operations.
, let 
be the number of operations before Harry stops. For example, 
and 
. Determine the average value of over all 
possible initial configurations .
be the foot of the altitude from the vertex 
in the acute-angled 
. Let 
be the midpoint of side 
, and 
be the circumcentre of . Line 
meets line 
at 
. Prove that the points 
, and lie on a circle.
board where we identify the squares with pairs 
where 
and 
denote the row and column number of that square, respectively.
and places a pawn at the bottom left corner (i.e. on 
) and makes the following moves. In his -th move, he moves the pawn from to either 
or 
if is odd and to either 
and 
if is even. Here all the numbers are taken modulo . Find the number of pairs 
that Calvin could have picked such that he can make moves so that the pawn covers all the squares on the board without being on any square twice.
+1 w
be real numbers satisfying 
Prove that 
and determine all cases of equality.
-gon, whose every vertex has a single coin. In a move, he chooses a vertex, removes one coin each from the two adjacent vertices, and adds one coin to the chosen vertex, keeping the remaining coin for himself. He can only make such a move if both adjacent vertices have at least one coin. Griphook stops only when he cannot make any more moves. What is the maximum and minimum number of coins he could have collected?
inscribed in a circle with center . 
is any point not on (O). 
intersect 
at 
. Let 
be the Lemoine points of triangle 
respectively. Prove that 
are collinear.
denote the set of positive integers. Find all functions 
such that ![\[\left\lfloor \frac{f(mn)}{n} \right\rfloor=f(m)\]](http://latex.artofproblemsolving.com/2/3/1/231c0bb0911db66d1a3d2354da7e5dff3004c47f.png)
for all positive integers 
.
such that : 
Prove that
Where 
are are the lengths of the sides of a 
g
then![$$(n-2)\left[\sum_{i=1}^{n}{\frac{a_i^2}{(1-a_i)^2}}-\frac n{(n-1)^2}\right]\geq(2n-1)\left(\sum_{i=1}^{n}{\frac{a_i}{1-a_i}}-\frac n{n-1}\right)^2.$$](http://latex.artofproblemsolving.com/9/3/e/93e5529518807b92cd9e0382f5fb07a293128b04.png)
