[/quote]
,
which satisfy:
is inscribed in circle 
R
and 
intersect at 
is chosen on the diagonal 
so that 
is chosen on the segment 
so that 
Prove that line 
touches 
stations in the sky, and any two stations are connected by exactly one bridge; every bridge has the same length and does not intersect with any other. 
,
rectangular board, using 
markers which are white on one side and black on the other. Initially, each square of the board contains a marker with its white side up, except for one corner square, which contains a marker with its black side up. In each move, one may take away one marker with its black side up, but must then turn over all markers which are in squares having an edge in common with the square of the removed marker. Determine all pairs 
of positive integers such that all markers can be removed from the board.
,
,
,
on a side of 
and its corresponding vertex bisects the perimeter, we write 
.
be points on sides 
,
,
,
,
,
,
,
.
.
be positive real numbers, and denote 
.
of positive integers, such that 
,
,![$$ a_{n+1}=\sum_{i=2}^{n+1} \lfloor \sqrt[i]{a_n} \rfloor. $$](http://latex.artofproblemsolving.com/e/4/7/e47c377c88129c1dde6ed238ae59593f3026663b.png)
Show that for any prime 
where 
is a permutation of 
,
be a set with 
)
of three-element subsets of 
,
,
.
,
,
.
and 
.
.
and orthocenter 
,
intersects 
,
and parallel to 
intersects 
.
similarly. Let 
intersects 
at 
,
intersects 
at 
,
intersects 
at 
.
.
on 
chosen from the three colors, it is possible to recolor at most 
Y by
Given a prime number 
, a set is said to be -good if the set contains exactly three elements 
and 
.
Find all prime number such that 
can be partitioned into several -good sets.
Proposed by capoouo
, a set is said to be -good if the set contains exactly three elements 
and 
.Find all prime number
such that 
can be partitioned into several -good sets.Proposed by capoouo
This post has been edited 1 time. Last edited by capoouo, Apr 21, 2024, 5:42 PM
Reason: remove awkward line-breaks; add "proposed by ..."
Reason: remove awkward line-breaks; add "proposed by ..."
.
rewrite into 
.
as 
,
.
runs through 
we achieve 
sets. The contrary is obvious.
,
,
?
.
doesn't satisfy as to partition 
into several 
.
.
.
.
be primitive root of 
as 
.
,
,
through 
and we achieves 
