[/quote]
,
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 
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 
,
.
Write only the answers to the questions given.
to be more exact:
in years 2002-08 time was 90' for part A and 120' for part B
red balls and 
white balls. Audi took two balls at once from inside the bag. The chance of taking two balls of the same color is ...
unit. The area of the hexagon is ...
and 
.
is ...
so that 
and 
is ...
,
,
,
has an average of 
and a median of 
.
are integers greater than 
which are four consecutive quarters of an arithmetic row with 
.
and 
are squares of two consecutive natural numbers, then the smallest value of 
is ...
,
and 
.
lies on the line segment 
.
and 
.
is ...
meet 
for each natural number 
is ....
provided that the difference of any two numbers at least 
is ...
which satisfies 
are as many as ...
and 
.
.
so that 
and 
.
is ...
consists of 
with 
positive reals is ....
with 
is ....
square units. Two plots are said to be neighbors if they both have one side in common. Initially, there are a total of 
.
so that 
.
Y by
Circles 
and 
with centers 
and 
respectively lie on a plane such that that the circle passes through . The ratio of radius of circle to 
is 
.
a) Prove that the circles and intersect at two distinct points.
b) Let
be these points of intersection. What proportion of the area of circle is is the area of the sector 
?
and 
with centers 
and 
respectively lie on a plane such that that the circle passes through . The ratio of radius of circle to 
is 
.a) Prove that the circles
and intersect at two distinct points.b) Let
be these points of intersection. What proportion of the area of circle is is the area of the sector 
?This post has been edited 2 times. Last edited by parmenides51, Mar 27, 2020, 2:56 AM
,
and radius 
.
,
and radius 
.
and 
.
.
equals 
.
Since 
(it's easy to show) 
which implies that 
Let 
be an exterior angle 
.
(by the sector's formula). By condition: 
because 
and 
.
:
.
.
.
