such that, for all positive integers 
and 
,
,![\[
\sum_{\text{cyc}} \sqrt{1 + \left(x\sqrt{1 + y^2} + y\sqrt{1 + x^2}\right)^2} \geq \sum_{\text{cyc}} xy + 2\sum_{\text{cyc}} x
\]](http://latex.artofproblemsolving.com/9/5/5/955863a0cf8747ae45b736b0631f243d3908eb84.png)
be an integer, and let 
and 
be sequences of non-negative integers. For each integer 
,
and 
,
is the integer such that 
is divisible by 
equals the following arithmetic mean: ![\[a_i = \frac{1}{2b_i+1} \sum_{s=-b_i}^{b_i} a_{i+s}.\]](http://latex.artofproblemsolving.com/1/1/7/117ca4963c8e443fd3cbfb55a907bee4be295151.png)
Suppose that neither 
nor 
of the numbers 
are zero.
Y by
Given an acute-angled triangle 
with 
.Inside the triangle , take point 
so that 
.On the straight lines 
and 
, points 
and 
are taken 
and 
, respectively. Let 
be the midpoint of 
. In this case, show that 
.
with 
.Inside the triangle , take point 
so that 
.On the straight lines 
and 
, points 
and 
are taken 
and 
, respectively. Let 
be the midpoint of 
. In this case, show that 
.This post has been edited 1 time. Last edited by parmenides51, Nov 28, 2022, 6:57 AM
