[/quote]
,
and 
Prove that
numbers 
,
where 
is the number of times the number 
appears in the list. So, for example, for the list 
,
.
numbers. Then he asked him to apply the program to the resulting list, and so on, until a number greater than or equal to 
appears in the list. Find the largest value of 
,
such that 
Prove that 
.
meets 
at points 
,
is 
.
intersects line 
at 
,
intersects line 
,
intersect at 
.
is tangent to 
.
be pairwise coprime positive integers with 
being prime and 
.![$I = [0, a_1 a_2 \cdots a_n ]$](http://latex.artofproblemsolving.com/a/2/b/a2bce96b048c9b8fc94926db80aba37fd5037b4a.png)
of the real line, mark all integers that are divisible by at least one of the numbers 
. These points split 
into a number of smaller segments. Prove that the sum of the squares of the lengths of these segments is divisible by 
such that for all 
be real numbers such that 
Prove that
In the 
plane with the origin 
,
intersects the lines 
,
at the points 
,
be positive real numbers.![$$\sqrt[3]{\left(\frac{a^{2022}-a}{b}+\frac{2021}{a^{\frac{a}{b}}}+1\right)\left(\frac{b^{2022}-b}{c}+\frac{2021}{b^{\frac{b}{c}}}+1\right)\left(\frac{c^{2022}-c}{a}+\frac{2021}{c^{\frac{c}{a}}}+1\right)}$$](http://latex.artofproblemsolving.com/c/4/9/c4923b16be8a8faf86422477a6ed6b4b60eb2552.png)
be non-negative real numbers such that 
.![\[
P = \frac{a}{a^2 + 2} + \frac{b}{b^2 + 2} + \frac{c}{c^2 + 2}.
\]](http://latex.artofproblemsolving.com/d/b/e/dbea7db769c4996fc081ee75837e177b298cffcb.png)
Y by Adventure10, Mango247
It is considered a rectangle 
whose sides measure 
and 
.
Four right isosceles triangles are constructed externally to the rectangle:
rectangle in 
, 
rectangle in 
, 
rectangle in 
, 
rectangle in 
Let
be the foot of the perpendicular to the line 
passing through
.
Show that perimeter
whose sides measure 
and 
.Four right isosceles triangles are constructed externally to the rectangle:
rectangle in 
, 
rectangle in 
, 
rectangle in 
, 
rectangle in 
Let
be the foot of the perpendicular to the line 
passing through
.Show that perimeter
This post has been edited 1 time. Last edited by parmenides51, May 17, 2019, 8:26 PM
Reason: name edit
Reason: name edit
