[/quote]
,
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.
,![\[
\sum_{\text{cyc}} \sqrt{1 + \left(x\sqrt{1 + y^2} + y\sqrt{1 + x^2}\right)^2} \geq \sum_{\text{cyc}} xy + 2\sum_{\text{cyc}} x
\]](http://latex.artofproblemsolving.com/9/5/5/955863a0cf8747ae45b736b0631f243d3908eb84.png)
such that there exist positive integers 
for which the following holds 
for all 
.
is a perfect cube
of a prime number 
and a positive integer 
for which 
is an integer.
is the centre of the excircle opposite the vertex 
This excircle is tangent to the side 
at 
and 
,
and 
meet at 
,
and 
meet at 
Let 
be the point of intersection of the lines 
and 
be the point of intersection of the lines 
and 
Prove that 
is the circle that is tangent to the line segment 
,
.
such that if 
and 
are distinct rational numbers satisfying 
or 
,
?
and 
.
be a positive integer. Consider the following equation: ![\[ \{x\}+\{2x\}+ \dots + \{nx\} = \lfloor x \rfloor + \lfloor 2x \rfloor + \dots + \lfloor 2nx \rfloor\]](http://latex.artofproblemsolving.com/8/2/9/8299b3ceb1650418c0a373536cb1d2931900efd9.png)
a) For 
,
.
real solutions.
and 
intersect at two points 
and 
.
be the line tangent to these circles at 
and 
,
be the line parallel to 
on 
on 
and 
meet at 
;
and 
;
and 
.
.
.
are there such that 
and 
is an integer.
Y by Adventure10, Mango247
Let 
be an equilateral triangle with center 
and side equal to 
. Let 
be a point on the 
side such that 
and let 
be a point on the side 
such that 
. Calculate the measure of the internal angles of the triangle 
.
be an equilateral triangle with center 
and side equal to 
. Let 
be a point on the 
side such that 
and let 
be a point on the side 
such that 
. Calculate the measure of the internal angles of the triangle 
.This post has been edited 1 time. Last edited by parmenides51, Jul 23, 2021, 9:27 PM
and 
.
.
