[/quote]
,
,![\[
\sum_{\text{cyc}} \sqrt{1 + \left(x\sqrt{1 + y^2} + y\sqrt{1 + x^2}\right)^2} \geq \sum_{\text{cyc}} xy + 2\sum_{\text{cyc}} x
\]](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.
is given on the plane. Let the points 
,
and 
be on the sides 
,
and 
,
,
and 
)
,
and 
are taken such that 
is an equilateral hexagon. Suppose that the circumcenters of 
are coincident. Then determine the least possible value of:
Note: 
area of 
such that ![\[f(x)(f(yf(x)+1))=f(x)+f(y)\]](http://latex.artofproblemsolving.com/1/0/6/1069c29ad9eeb7fe8597d7bac806006a12ff50c9.png)
for all positive reals 
and 
and also satisfies the property that ![\[\mathbb{R}^+\subseteq\frac{\text{Im}(f)}{\text{Im}(f)},\]](http://latex.artofproblemsolving.com/1/0/2/102f23eabefae24b158fbd7516b58b1e879c1b74.png)
or in other words, the set of positive reals is a subset of the set ![\[\left\{\frac{x}{y}\mid x,y\in\text{Im}(f)\right\}\]](http://latex.artofproblemsolving.com/f/d/4/fd4e9cf0bfc63603529ded7679e2811560219338.png)
and 
be two circles with respective centres 
and 
intersecting in two distinct points 
is an obtuse angle. Let the circumcircle of 
intersect 
and 
.
intersect 
intersect 
are concyclic.
from the set of positive integers to the set of positive integers such that, for all positive integers 
and 
,![\[ a, f(b) \text{ and } f(b + f(a) - 1).\]](http://latex.artofproblemsolving.com/4/a/b/4ab97da27e05d0f9e7762f74d0d0bd2dc8b9842f.png)
and 
.![$S=\sqrt[3]{\frac{a^3+b^3+c^3}{3}}+\frac{a+b+c}{9}$](http://latex.artofproblemsolving.com/9/d/9/9d93ff9d9f7355e452f1007aeb87423e3491353a.png)
and 
be the incircle and circumcircle of the acute triangle 
,
so that all of its sides are tangent to 
and 
,
and 
,
and 
intersects on 
be a cyclic quadrilateral with circumcentre 
and 
meet at point 
lie on a circle with centre 
are collinear.
such that, for all positive integers 
and 
be a positive integer. We say that 
such that:
for which![\[
a \mid s^b - l.
\]](http://latex.artofproblemsolving.com/5/d/9/5d9db7678c54d1c6a909b4aca39a329cadd71e2c.png)
2. For every proper divisor 
of 
and 
)![\[
a' \mid s^b - l.
\]](http://latex.artofproblemsolving.com/4/b/2/4b26b9bd3da8958287703c3f575bf5d5da7fd477.png)
Determine all bao-good positive integers 
such that![\[(a^2 + b^2 + c^2)^2 = 3(a^3b + b^3c + c^3a) + 1. \]](http://latex.artofproblemsolving.com/9/e/4/9e400fd807a9535ede983b75dcc307d8b75a35b6.png)
.
Y by
There is a tent in the shape of a hemisphere on the plane. Three balls are placed inside the tent. The first two have a radius of 
, touch each other, as well as the roof and the diameter of the tent base. The third ball of radius 
touches the other two, the roof and the base of the tent. Find .
, touch each other, as well as the roof and the diameter of the tent base. The third ball of radius 
touches the other two, the roof and the base of the tent. Find .This post has been edited 1 time. Last edited by parmenides51, Jan 24, 2021, 9:51 PM
