[/quote]
,
= 2∠AMP.
be an arbitrary prime number.we have a deck of cards which a number is written on the back of each of them. such that for every 
we have at most one card with number 
written on its back and we also have exactly one card with number zero on its back.we want to design a game in which we need to decide for every two cards 
witch one wins from the other one using the following rules.
:
wins from 
and also 
, then 
:
, then in case of existing of both cards 
and 
module 
| 
with 
are positive integers and 
)
lines in the plane such that no two of them are parallel and no three of them are concurrent. We color at least 
regions in black from all limited regions which have constructed in the plane. We call a triangle constructed by three lines “black-less” if there exist exactly one black region inside that. Prove that there exist at least 
“black-lass” triangles in plane.
,![\[
1^2 + 2^2 + 3^2 + \ldots + k^2 = \frac{k(k + 1)(2k + 1)}{6}
\]](http://latex.artofproblemsolving.com/b/5/d/b5d49e25bd8d8a211b72372ff967558e1a8365ce.png)
Find the smallest positive integer 
is divisible by 200.
Let ![$[n]$](http://latex.artofproblemsolving.com/0/d/e/0deff9086fd7840ef23860f5bc924f1d4d2d2310.png)
denote the set of natural numbers less than or equal to 
![$f(m,n) = \sum_{(x_1,x_2,x_3, \cdots, x_m) \in [n]^{m}} \frac{x_1}{x_1 + x_2 + x_3 + \cdots + x_m} \binom{n}{x_1} \binom{n}{x_2} \binom{n}{x_3} \cdots \binom{n}{x_m} 2^{\left(\sum_{i=1}^{m} x_i\right)}$](http://latex.artofproblemsolving.com/f/d/f/fdf7b2fcfb960e93d59f3f84cef4372262c69499.png)
be a sequence of real positive numbers such that for all 
:
Prove that for all large enough natural 
we have that 
.
,
.
at point 
at 
is tangent to 
,
.
and let 
be reals such that 
are parallel. Angle bisectors of 
and 
meet at 
.
and 
meet at 
.
is cyclic
.
Let 
.
Y by
Two adjacent sides of a quadrilateral 
are equal, 
, and the other two are not, 
. Also 
. A circle is drawn through points 
and 
, which intersects for the second time the segment 
at point 
, and the line 
at point 
. Find the length of the segment 
if it is known that 
.
are equal, 
, and the other two are not, 
. Also 
. A circle is drawn through points 
and 
, which intersects for the second time the segment 
at point 
, and the line 
at point 
. Find the length of the segment 
if it is known that 
.This post has been edited 1 time. Last edited by parmenides51, Nov 21, 2020, 1:28 PM
