[/quote]
,
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 
of convex polygons (in the plane). It turns out that every point of the plane lies in the same number of elements of 
as elements of 
.
.
.
meets 
at points 
,
to 
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 
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)
![$$(ab+bc+ca)\left[\frac{1}{(a+b)^{2}}+\frac{1}{(b+c)^{2}}+\frac{1}{(c+a)^{2}}\right]\ge \frac{9}{4}+k\cdot\frac{a(a-b)(a-c)+b(b-a)(b-c)+c(c-a)(c-b)}{(a+b+c)^{3}}$$](http://latex.artofproblemsolving.com/f/6/e/f6ed10f7fff1cc94edd8f451e75718a0916a8bfa.png)
holds for all 
and 
Prove that
Y by Adventure10, Mango247
Line segment 
is perpendicular to the plane of the square 
. Let 
be the foot of the perpendicular from 
to line segment 
. Let 
be the midpoints of 
respectively. Let 
be on line segments 
respectively. Prove that 
is perpendicular to 
.
is perpendicular to the plane of the square 
. Let 
be the foot of the perpendicular from 
to line segment 
. Let 
be the midpoints of 
respectively. Let 
be on line segments 
respectively. Prove that 
is perpendicular to 
.
thus 
.
