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.
, points 
and 
lie on 
and 
respectively such that 
. Points 
and 
are the reflections of and with respect to the midpoints of sides and , respectively. Point 
lies on segment 
such that the circumcenter of triangle 
coincides with the circumcenter of triangle . passes through the midpoint of segment 
.
-th move, Jumpy swaps the positions of the two walnuts adjacent to walnut . such that, on the -th move, Jumpy swaps some walnuts 
and 
such that 
.
written on the first page and 
empty pages. Suppose in each of the next seconds, Alice can flip to the next page, and write down the sum or product of two numbers (possibly the same) which are already written in her grimoire.
be the largest possible number such that for any 
, Alice can write down the number on the last page of her grimoire. Prove that there exists a positive integer 
such that for all 
, we have that ![\[n^{0.99n}\leqslant F(n)\leqslant n^{1.01n}.\]](http://latex.artofproblemsolving.com/5/7/e/57e2b7e3d97564665d8de28a3afd74bdcf74ba5b.png)
of positive integers for which there exist positive integers 
and such that
holds for all integers 
(Note that 
denotes the greatest common divisor of integers 
and 
)
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.
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?
be a triangle where 
, with circumcenter and incenter 
. The angle bisector of 
intersects the circumcircle of in 
and . Let 
be the projection of onto , and 
be the projection of onto 
. Prove that 
bisects 
.
be a triangle where , with circumcenter and incenter . The angle bisector of intersects the circumcircle of in and . Let be the projection of onto , and be the projection of onto . Prove that bisects .
with no obtuse angles.![$$\frac1{\sqrt3}\cdot\left(\cot A+\cot B+\cot C\right)+\left(2-\sqrt 3\right)\sqrt[3]{\cot A\cot B\cot C}\geq\frac2{\sqrt3}.$$](http://latex.artofproblemsolving.com/d/1/4/d14154281d7ccf60a2dcfe1d937b974ae1d74f97.png)
such that : 
such that : 
Prove that
