of natural numbers with 
elements with 
of 
if the product of the elements of 
be positive reals with sum 1. Show that for any 
,
satisfying 
for all 
,
.
,
,
are pairwise distinct real numbers satisfying 
Prove that
Let 
Prove that
chessboard such that no two pawns attack each other?
,
and 
meet at a point 
.
and the points 
lie on the same straight line.
be real numbers such that 
Prove that
.
and 
is an integer.
be reals such that 
. Prove that
Let 
. Prove that
and 
for each positive integer 
?
and ![$e\in [-2,2]$](http://latex.artofproblemsolving.com/b/b/7/bb78c9dedb18af5e6a904f5492917921e41a7ebc.png)
such that 
Find the value of 
be a positive integer such that 
,![\[
A = \{a_1, a_2, a_3, \ldots, a_m\}
\]](http://latex.artofproblemsolving.com/1/7/9/179687a134d5495ffff0b44fe763a2bb8b3ddd9c.png)
consist of distinct positive integers not exceeding 2025. Suppose that for every 
,
,
,
as well. Prove that![\[
\frac{a_1 + a_2 + a_3 + \cdots + a_m}{m} \geq 1013.
\]](http://latex.artofproblemsolving.com/c/0/d/c0d190f78d94deac1b947ba57a2de51d0b06f0bd.png)
for which I can find nonnegative integers 
such that for all indices 
,
divides 
?
...")
integers - take the set of all integers with a digit sum of 
in base 
to the list.
Y by Adventure10
Let 
be an acutangle triangle and 
any point on the 
side. Let 
be the symmetrical of in
and 
is the symmetrical relative to 
. 
straight 
intersects straight at 
, while straight 
intersects the line in 
. Prove that the points 
and are on the same circumference.
be an acutangle triangle and 
any point on the 
side. Let 
be the symmetrical of in
and 
is the symmetrical relative to 
. 
straight 
intersects straight at 
, while straight 
intersects the line in 
. Prove that the points 
and are on the same circumference.This post has been edited 1 time. Last edited by Zelderis, Nov 23, 2019, 1:16 PM
Similarily 
with radius 
gives that 
as desired.
