[/quote]
,
be a triangle with altitude 
.
-
at 
,
and 
.
and 
on lines 
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 
candies, the player chooses positive integer 
so that 
and 
,
.
such that 
and 
.
be a trapezoid with parallel sides 
.
and 
lie on the line segments 
and 
,
.
and 
on the line segment 
satisfying ![\[\angle{APB} = \angle{BCD}\qquad\text{and}\qquad \angle{CQD} = \angle{ABC}.\]](http://latex.artofproblemsolving.com/b/1/a/b1ae208955104081bafe221832509d9dd20eeb83.png)
Prove that the points 
and 
are concyclic.
,
with center 
.
the midpoint of side 
intersects the circle 
be a point on the circle 
and 
.
.
.
and from 
to 
and 
as the reflections of 
and 
be the circumcircle of 
.
the second intersection of line 
with 
the intersection of ray 
with 
is the circumcenter of 
can be inscribed within a circle.
Y by Adventure10, Mango247
We have a 
billiards table, with a single ball at is center. The ball is sent in a straight line, and after 29 meters it stops at a corner of the table. How many times did the ball bounce on the borders of the table?
Note: The ball bounces obeying the law of reflection, that is, the angle of incidence between the path of the ball and the border of the table is equal to the angle of reflection.
billiards table, with a single ball at is center. The ball is sent in a straight line, and after 29 meters it stops at a corner of the table. How many times did the ball bounce on the borders of the table?Note: The ball bounces obeying the law of reflection, that is, the angle of incidence between the path of the ball and the border of the table is equal to the angle of reflection.
feet) be 
,
feet) be 
.![\[29^{2}= 16 \cdot (2a+1)^{2}+(2b+1)^{2}\]](http://latex.artofproblemsolving.com/a/8/c/a8ca720d4649abf4126360ffcc99138f4fd28404.png)
in non-negative integers. We test the few possible values of 
.
.
