[/quote]
,
equals
and so on to create any number!
,
such that 
and 
be arbitrary points on the side 
of triangle ABC such that 
.
and 
with the incircle form a quadrilateral 
.
)
)
defined in the set of rational numbers and taking their values in the same set such that the equation 
holds for all rational numbers 
and 
.
be an integer, and let 
denote the set of all permutations of 
.
be a positive integer. Consider an 
grid, where initially each cell of the grid is filled with a permutation from 
and choose a row or column of the grid, then for each cell in the selected row (or column), replace the current permutation 
in the cell with 
.
be the smallest integer such that for any initial configuration of permutations in the grid, if it is possible to transform all permutations into the identity permutation through a finite number of operations, then this can be achieved in no more than 
such that for any integer 
,![\[c_1 \cdot \frac{n^2}{\ln n} < f(n) < c_2 \cdot \frac{n^2}{\ln n} \]](http://latex.artofproblemsolving.com/6/2/b/62b19e28fb8a57e3a891953abb47f552c26800bf.png)
are positive real numbers and 
.
,
and 
.
and ![$ [a,b,\ldots,g]$](http://latex.artofproblemsolving.com/f/d/1/fd1b562f41252c79cea0de72b652676f10413837.png)
denote the greatest common divisor and least common multiple, respectively, of the positive integers 
.
and ![$ [6,15]=30$](http://latex.artofproblemsolving.com/2/3/7/2374c21eed2f701dc46ffef813a7fffdd7be32bc.png)
.![\[ \frac{[a,b,c]^2}{[a,b][b,c][c,a]}=\frac{(a,b,c)^2}{(a,b)(b,c)(c,a)}.\]](http://latex.artofproblemsolving.com/d/e/c/decd37a88a59d84485c88227cc915a1927c18faf.png)
Y by
Amanda has 
red marbles, 
yellow marbles, 
green marbles, and 
blue marbles. If Amanda takes the marbles one by one without replacing them until the 
marble. Then the probability that the 
marble is red is?
red marbles, 
yellow marbles, 
green marbles, and 
blue marbles. If Amanda takes the marbles one by one without replacing them until the 
marble. Then the probability that the 
marble is red is?This post has been edited 2 times. Last edited by Ro.Is.Te., May 23, 2025, 11:51 AM
Reason: LaTeX
Reason: LaTeX
?
marble. Then the probability that the 
marble is red is?
marbles in total and and 
