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.
, define
and
meaning 
is the least common multiple of the numbers 
. Determine whether or not there exists a constant 
such that
for every positive integer .
and 
be positive integers with 
. There are tokens placed initially in the leftmost cell of a horizontal row consisting of cells. A move consists of shifting any token 
, 
, 
, 
, 
, or 
positions to the right. Andrés and Beto alternate turns, starting with Andrés. The winner is the player who moves a token into the rightmost cell. Determine who has a winning strategy in terms of and .
be an acute triangle with circumcenter 
. Draw altitude 
, with 
on side 
. The parallel line to 
passing through intersects line 
at point 
. Prove that point and the midpoints of sides 
and are collinear.
is mundane if there exist positive integers 
and 
less than 
such that 
is a positive integer. Find, with proof, all prime numbers that are not mundane.
be a fixed positive integer. Find the greatest real constant 
that has the following property: Any 
real numbers, not necessarily distinct, that lie in the interval ![$[100,101]$](http://latex.artofproblemsolving.com/8/e/3/8e3fe24e3b899bbf2128eebe08756e1ff963aaa0.png)
, can be partitioned into two groups with sums 
and 
such that
.
, 
and 
.
.
-tromino is a figure made up of three squares, obtained by removing one square from a 
board.
board consisting of 
unit segments. A configuration of several -trominoes is optimal if the -trominoes do not overlap, each one covers exactly three squares of the board, and moreover, no unit segment of the board belongs to two -trominoes. Below is an optimal configuration of 5 -trominoes: for which there exists an optimal configuration of L-trominoes on the board.
such that 
is a divisor of 
.
and natural number ,
is chosen on chord of a circle with center . Let 
be the foot of a perpendicular drawn from point on 
and be the circumcenter of triangle 
. Prove that 
.
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dot((0.8557193713913955,-1.597005112629097),dotstyle);
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is cyclic 
is collinear 
is the midline of 
satisfy that: 
