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.
such that 
divides 
.
+1 w
workers stand in a row. Every contiguous group of at least 
workers is a 
. The chief wants to assign each brigade a leader (which is a member of the brigade) so that each worker’s number of assignments is divisible by 
. Prove that the number of such ways to assign the leaders is divisible by .
be a fixed acute-angled triangle. Consider some points 
and 
lying on the sides 
and 
, respectively, and let 
be the midpoint of 
. Let the perpendicular bisector of intersect the line 
at 
, and let the perpendicular bisector of 
intersect the lines and at 
and 
, respectively. We call the pair 
, if the quadrilateral 
is cyclic.
and 
are interesting. Prove that 
and 
be four prime numbers such that 
Prove that the sum of the four prime numbers is divisible by 
.
and 
and one starting number from the set 
used. Alice chooses independently her game number and Bob chooses the starting number. The other number is given to Bob. Then Alice adds her game number to the starting number, Bob adds his game number to the result, Alice adds her number of games to the result, etc. The game continues until the number 
is reached or exceeded. wins. If is exceeded, the game ends in a draw.
Show that Bob cannot win. What starting number does Bob have to choose to prevent Alice from winning?
is given. Over the side draw an equilateral triangle 
on the outside. The midpoint of the segment 
is 
and the midpoint of the side 
is 
. Prove that 
.
and 
be integers with 
. Find all pairs 
such that 
be an acute triangle. Let 
be the reflection of point 
over the line . Let 
and be the circumcenter and the orthocenter of triangle , respectively, and let be the midpoint of segment 
. Let 
and 
be the points where the reflection of line 
with respect to line 
intersects the circumcircle of triangle , where point lies on the arc not containing . If 
is a point on the line 
such that 
, prove that 
.
be a regular hexagon with sidelength s. The points 
and 
are on the diagonals 
and 
, respectively, such that 
. Prove that the three points 
, and are on a line.
be nonzero real numbers with 
Determine all possible values of 
of positive integers such that
, and 
be the angles of a triangle, and let 
, and 
. If 
, show that![\[s(s - a)(s - b)(s -c) \geq 0.\]](http://latex.artofproblemsolving.com/1/3/4/1344a0cbbd4e1aa4bcf6a1e777501c62daff32b3.png)
has 
. Point is the foot of the perpendicular from to . Points and lie on sides and 
, respectively, such that lies inside triangle 
and ![\[
\angle CHS - \angle CSB = 90^{\circ}, \quad \angle THC - \angle DTC = 90^{\circ}. \]](http://latex.artofproblemsolving.com/f/8/c/f8c35912c531eb2dae8f62b74666b7757d2b9ba3.png)
Prove that line is tangent to the circumcircle of triangle 
.
. For each 
, define the 
. So, 
is the family of all the nonempty sets of 
. As there's only one empty set, we have by the Inclusion-Exclusion Principle:
This easy via the binomial theorem for the quantity is just 
,