[/quote]
,
and some point 
on 
.
intersects the circumcircle of 
at 
and the perpendicular from 
intersects the circumcircle of 
at 
.
does not depend on the choice of 
grid consisting of 2025 unit squares. Within each unit square is a car, facing either up, down, left, or right. If the square in front of a car in the direction it is facing is empty, it can choose to move forward. Each car wishes to exit the 
cars from the 
coins, all of which show 
on one side and 
for which Solveig can keep Peer from winning.
be an angle in the interval 
.
is irrational, and that 
and ![$ \cos[(k + 1)\theta ]$](http://latex.artofproblemsolving.com/b/3/a/b3a14c57a1a8b5a6585195c394181c5124f8c471.png)
are both rational for some positive integer 
,
.
be three positive reals such that 
.![\[ \frac { x^5-x^2 }{x^5+y^2+z^2} + \frac {y^5-y^2}{x^2+y^5+z^2} + \frac {z^5-z^2}{x^2+y^2+z^5} \geq 0 . \]](http://latex.artofproblemsolving.com/a/1/4/a14adf0f1e35df57c734c7df701d5a67da22dc3c.png)
be a triangle with altitude 
.
at 
,
and 
.
and 
and 
such that quadrilateral 
is a rhombus.
be an integer. Two players, Alice and Bob, play the following game on the complete graph 
:
,![\[
\left( \frac{1}{4} - \varepsilon \right) n^2 + n.
\]](http://latex.artofproblemsolving.com/a/9/c/a9cca00e87c643af32eb75a2501b59060f3504fb.png)
such that the following inequality holds for all 
:![\[
ab + bc + ca \leq \frac{(a + b + c)^2}{3} + k \cdot \max \{ (a - b)^2, (b - c)^2, (c - a)^2 \} \leq a^2 + b^2 + c^2
\]](http://latex.artofproblemsolving.com/1/c/e/1ce66163068b499383adb4a4ecec86c5f4edaec3.png)
be positive real numbers such that : 
. Prove that : 
where 
are lengths of parallel sides and 
are other two sides.
,
,
,
,
in some order.
board that is entirely covered by dominoes, with no overlap. He jumps along the vertical lines of the board, always going from the midpoint of the vertical segment of a 
square to another midpoint of the vertical segment, according to the rules:
When the domino is horizontal, the cricket jumps to the opposite vertical segment (such as from 
to 
)
When the domino is vertical downwards in relation to its position, the cricket jumps diagonally downwards (such as from 
to 
When the domino is vertically upwards relative to its position, the cricket jumps diagonally upwards (such as from 
)
board.
Prove that
candies. A starts first. On a move, if in the box there are 
so that 
and 
,
Y by
Circles 
and 
with centers 
and 
respectively intersect at point 
.The extension of the segment 
beyond the point intersects the circle at the point 
, and the extension of the segment 
beyond the point intersects the circle at the point 
. The straight line passing through parallel to 
, intersects the circles and for the second time at the points 
and 
, respectively. Rays 
and 
intersect at point 
. Find the angle between lines 
and .
and 
with centers 
and 
respectively intersect at point 
.The extension of the segment 
beyond the point intersects the circle at the point 
, and the extension of the segment 
beyond the point intersects the circle at the point 
. The straight line passing through parallel to 
, intersects the circles and for the second time at the points 
and 
, respectively. Rays 
and 
intersect at point 
. Find the angle between lines 
and .
