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.![\[\frac {x^{2}}{\left(x - 1\right)^{2}} + \frac {y^{2}}{\left(y - 1\right)^{2}} + \frac {z^{2}}{\left(z - 1\right)^{2}} \geq 1\]](http://latex.artofproblemsolving.com/f/a/6/fa6c46c12e7e6bf0b04f6f9a910df6c2b452fe6a.png)
for all real numbers 
, 
, 
, each different from 
, and satisfying 
., , , each different from , and satisfying .
be a prime and suppose 
is prime. Prove that 
is prime
This is a special case of Pocklington's theorem. A proof of this special case is required.
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?
, 
.
, and they will meet near the school. Find Vanya’s speed of walking.
, and they will meet near the school. Find Vanya’s speed of walking.
, and they will meet near the school. Find Vanya’s speed of walking.
the speed of Anya and 
the speed of Vanya. We have two equations :
,
