[/quote]
,
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.
be reals such that 
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 
and radius 
intersects the circle 
.
be a point on the circle 
and 
.
.
.
to 
and 
as the reflections of 
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.
![\[\sqrt{a+h_B}+\sqrt{b+h_C}+\sqrt{c+h_A}=\sqrt{a+h_C}+\sqrt{b+h_A}+\sqrt{c+h_B}\]](http://latex.artofproblemsolving.com/5/d/6/5d6881be29327bcdc3e7bd76c221968578c7f50b.png)
is isosceles.
be all the divisors of 
are all divisors for some 
(
)
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 
.
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 
.
.
is a point on the side extension 
is perpendicular to 
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
Let 
be the longest side of the triangle 
. On the side 
point 
is selected, on the side 
point 
and on the side points 
and 
such that 
, 
, 
. Prove that the points 
lie on the same circle.
be the longest side of the triangle 
. On the side 
point 
is selected, on the side 
point 
and on the side points 
and 
such that 
, 
, 
. Prove that the points 
lie on the same circle.This post has been edited 2 times. Last edited by parmenides51, Mar 10, 2021, 3:51 PM
Reason: latex
Reason: latex
and Angle BKN is 
,
.
therefore angle NML is 
.
,
,
,
,
whilst 
.
,
.
and 
,
Solving this, we get that 
.
.
can be expressed in new terms, like 
.
,
.
and 
,
is thus cyclic as opposite angles sum to 
.
