[/quote]
,
with orthocenter 
.
to meet 
at 
.
meets the midpoint of 
.
,
.
intersect its circumcircle 
at 
.
determine a triangle 
,
is relative to 
and so on. Prove that lines 
are concurrent at a point 
and 
.
be a cyclic quadrilateral whose opposite sides are not parallel. Suppose points 
lie in the interiors of segments 
respectively, such that 
Let 
intersect 
at 
,
intersect 
at 
.
and 
are either parallel or coincide.
and 
.
Where 
such that :
.
be reals such that 
Prove that 
Let 
Prove that 
be a fixed integer. There are 
beads on a circular necklace. You wish to paint the beads using 
colors, such that among any 
consecutive beads every color appears at least once. Find the largest value of 
for which this task is 
possible.
This easy via the binomial theorem for the quantity is just 
,
.
such that 
.
as 
.
Y by
Three grasshoppers play the following leapfrog game. They start off at three vertices of a square. At each step, a grasshopper leaps over another one and lands at the point symmetric to the point where it was. That is, if a grasshopper at a point 
leaps over a grasshopper at a point 
, then it lands at a point 
where is the midpoint of 
. Is it possible for one of them to reach the fourth vertex of the square?
leaps over a grasshopper at a point 
, then it lands at a point 
where is the midpoint of 
. Is it possible for one of them to reach the fourth vertex of the square?This post has been edited 1 time. Last edited by parmenides51, Jun 21, 2022, 10:05 AM
.
units away horizontally and 
vertically from another grasshopper, it must move 
units horizontally and 
vertically. Thus, the parity of the x-coordinate and y-coordinate must remain the same throughout this process. However, since the parity of both x and y coordinates for 
is odd, for a grasshopper to reach here, both its initial x and y coordinates must be odd. Since 0 is even, all 3 grasshoppers clearly have at least 1 coordinate that is even and thus cannot reach the fourth vertex.
