such that for all 
is great if the following holds: for any point 
on the side 
,
and 
are the feet of the perpendiculars from 
and 
,
lies on the circumcircle of the triangle 
and 
.
board, each square is colored either white or black, with all the squares on the border of the board being black. Additionally, no 
square within the board has all four squares of the same color. Prove that the board contains a 
,
be the circumcenter of triangle 
.
.
be real numbers such that 
Prove that
Let 
Prove that
If a problem has not been solved within 
days, then others are eligible to post a new probkem.
,
)
satisfied 
Find the minimum of 
satisfy that: 
and natural number 
,
are there such that 
and 
?
.
of 
is not empty.
.
Y by
For a subset 
of the set 
, let 
. Find the maximum number of elements in the set 
that satisfies all of the three following conditions:
of the set 
, let 
. Find the maximum number of elements in the set 
that satisfies all of the three following conditions:
- All elements of are subsets of
. - For any two elements
of , 
is not empty. - For any element
of , 
.
This post has been edited 2 times. Last edited by pokmui9909, Mar 30, 2025, 6:02 AM
.
.
of them. Now consider non-byunsing sets. Pair each set with its complement. This is possible because the complement of every non-byunsing set must also be non-byunsing. Only one set of each pair can be an element of 
non-byunsing sets, and only half of them can be an element of 
