[/quote]
,
be positive reals. Prove that there is a cyclic permutation 
of 
such that the inequality:
holds for all positive real numbers 
and 
such that the decimal representation of the number:
ends in 
digits 
.
,
is acute, 
,
.
be the intersection of the tangent to the circumcircle of 
and the perpendicular bisector of segment 
.
,
.
.
be a set of integers containing the numbers 
and 
.
belongs to 
be real numbers such that 
and 
Prove that
coins are distributed among them, where each person has at least 
coins. Each person is then assigned a random index number in 
such that no two people have the same number. Then every minute, if 
is the number of minutes passed, the person with index number congruent to 
be a convex cyclic quadrilateral with circumcircle 
.
produced beyond 
meet 
produced beyond 
,
.
be a line through 
and 
at 
and 
respectively, so that 
(respectively 
)
)
and 
are points on the arc 
of 
so that 
are tangent to 
are collinear.
.
Where 
.![$\sqrt[3]{ab}+ \sqrt[3]{cd} < \sqrt[3]{(a+b+c)(b+c+d)}$](http://latex.artofproblemsolving.com/0/5/0/0502ba65f0545b3af84c41b2549d6256550b473b.png)
.
Y by
If the expression for the nth term of the infinite sequence 
is ![$\left[\sqrt{\alpha n} + \frac{1}{\beta}\right]$](//latex.artofproblemsolving.com/1/2/b/12b45d9d0f5268002da2adae8dac73390a106d96.png)
(Here ![$\left[.\right]$](//latex.artofproblemsolving.com/3/c/a/3ca27308567a179890adebe5721a274c19ace40f.png)
denotes GIF) then
Let 
and 
then the number of numbers out of the first 
natural numbers which are divisible by 
or 
is
None of these
Let 
and 
. The number of divisors of the number 
which are of the form 
is equal to
None of these
is ![$\left[\sqrt{\alpha n} + \frac{1}{\beta}\right]$](http://latex.artofproblemsolving.com/1/2/b/12b45d9d0f5268002da2adae8dac73390a106d96.png)
(Here ![$\left[.\right]$](http://latex.artofproblemsolving.com/3/c/a/3ca27308567a179890adebe5721a274c19ace40f.png)
denotes GIF) then
Let 
and 
then the number of numbers out of the first 
natural numbers which are divisible by 
or 
is
None of these
Let 
and 
. The number of divisors of the number 
which are of the form 
is equal to
None of theseThis post has been edited 1 time. Last edited by P162008, Apr 28, 2025, 2:51 AM
Reason: Typo
Reason: Typo
