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.
are positive real numbers such that 
prove that
girls on one side of a tennis table, and a queue of boys on the other side. Both the girls and the boys are numbered from 
to in the order they stand. The first game is played by the girl and the boy with the number and then, after each game, the loser goes to the end of their queue, and the winner remains at the table. After a while, it turned out that each girl played exactly one game with each boy. Prove that if is odd, then a girl and a boy with odd numbers played in the last game.
grid. We color the squares with three colors: blue, red, and yellow. Two squares are defined as opposite if they share a vertex but not an edge. A valid coloring is a coloring such that no two squares that are red and blue are opposite each other. Count the number of valid colorings.
be positive real numbers such that 
. Prove that: 
, points 
and 
are in segments 
and 
respectively such that 
. Let 
be the intersection of 
and 
and 
be the intersection of 
and 
. If 
is a cyclic quadrilateral prove that 
boys (with 
) and some girls. Every girl is friend with exactly 
boys and for any 
boys, there is exactly 
girl who is their common friend. Let 
be the greatest number of girls that can be chosen from the camp to form a group such that every boy is friend with at most girl in the group. Prove that 
.
such that 
is a perfect square.
-gon, several diagonals are drawn. Among these diagonals, a diagonal is called good if it intersects exactly one other diagonal drawn (in the interior of the -gon). Find the maximum number of good diagonals.
, where 
is a digit. How many pairs of such numbers are there such that their sum is also a palindrome?
, 
.
, such that 
and 
.
such that 
is prime, 
with 
and ?
,
w![$ \boxed{\ \left(\frac{n}{n+1}\right)^{n+1}<\sqrt [n+1]{(n+1) !}-\sqrt [n]{n!}\ }$](http://latex.artofproblemsolving.com/d/f/2/df2545bd7a1f1759e55190e60648d5cf65a2488a.png)
children standing in a circle. For each girl, it turns out that among the five people following her clockwise, there are more boys than girls. Find the greatest number of girls that can stand in a circle.