to the cross product 
reveals that a negative determinant [of 
] corresponds to a matrix which reverses the direction of the cross product of two vectors.[/quote]
and 
,
and 
although its supposed to be a quick comparison.
chance of landing heads when flipped. Krithik is playing a game where he starts with 
point. Every turn, he flips the coin, and if it lands heads, he gains 
points. Find the expected number of turns until Krithik wins.
be an isosceles triangle with 
Let 
be a point on the circumcircle of 
Let 
intersect the extension of 
at 
Let 
be the midpoint of segment 
and let 
be the intersection of 
and 
Let the extension of 
intersect 
and the circumcircle of 
and 
respectively. Given that 
and 
compute the length of 
Y by Mango247, Mango247
Farey's calculator always displays exactly 
digits after the decimal point, and has one operation: 
Pressing the button will output the square root of the input, with everything beyond the digits truncated and forgotten by the calculator. Farey enters
![\[ 1.000000\underline{abcd}\]](//latex.artofproblemsolving.com/f/c/c/fcc8ba83d5ffaaeea6609e8d984f31d2a5701ae6.png)
where 
, 
, 
, and 
are digits, and continues pressing the 
operator until the calculator displays 
. Find the unique four-digit number 
such that the number of times the rightmost digit displayed is even is maximized.
Aaron Guo
digits after the decimal point, and has one operation: 
Pressing the button will output the square root of the input, with everything beyond the digits truncated and forgotten by the calculator. Farey enters![\[ 1.000000\underline{abcd}\]](http://latex.artofproblemsolving.com/f/c/c/fcc8ba83d5ffaaeea6609e8d984f31d2a5701ae6.png)
where 
, 
, 
, and 
are digits, and continues pressing the 
operator until the calculator displays 
. Find the unique four-digit number 
such that the number of times the rightmost digit displayed is even is maximized.Aaron Guo
This post has been edited 1 time. Last edited by I-_-I, Jul 23, 2022, 11:59 PM
?
.
,
instead of just 
,
is really small and like a x2 pattern, but we must not neglect the 
.
which we can write an explicit formula for, 
,
.
or whatever then 
;
would be less than 
.
