[/quote]
,
and the recursive relation
for all natural number 
.
.
.
be a triangle with 
let 
be the circumcircle of 
and let 
be its radius. Point 
is chosen on 
such taht 
and point 
is the foot of the perpendicular from 
.
mets 
.
is chosen on line 
such that 
and 
lie on a line in that order. Finally, let 
be a point satisfying 
and 
.
in the plane is assigned a real number 
It is known that 
whenever 
is the centroid of 
Prove that 
for all points 
such that 
for all 
be a positive integer. A board with a format 
is divided in 
(or 
)
,
,
with 
or 
for each 
,
,
,
,
are constructed by the following rules: ![\[a_0 = b_0 = 1, \quad a_1 = b_1 = 7,\]](http://latex.artofproblemsolving.com/d/f/d/dfd49833078b6b676cd261cc358e8114b2c4b7f4.png)
![\[\begin{array}{lll}
a_{i+1} =
\begin{cases}
2a_{i-1} + 3a_i, \\
3a_{i-1} + a_i,
\end{cases} &
\begin{array}{l}
\text{if } \varepsilon_i = 0, \\
\text{if } \varepsilon_i = 1, \end{array}
& \text{for each } i = 1, \dots, n - 1, \\[15pt]
b_{i+1}=
\begin{cases}
2b_{i-1} + 3b_i, \\
3b_{i-1} + b_i,
\end{cases} &
\begin{array}{l}
\text{if } \varepsilon_{n-i} = 0, \\
\text{if } \varepsilon_{n-i} = 1, \end{array}
& \text{for each } i = 1, \dots, n - 1.
\end{array}\]](http://latex.artofproblemsolving.com/8/9/0/8908021c60992e39acdb4fcf6b587f96b88539e0.png)
Prove that 
.
and 
squares of the 
be a real or complex inner product space. Suppose that 
and that 
Then
and 
intersect at the point 
.
?
,
.
are chosen from 0 to 1. What is the probability that their positive difference is more than 
?
?
families, there are 
children respectively. If a random child from any of the families is chosen, what is the probability that the child has 
siblings? Express your answer as a common fraction.[/quote]
,
people, exactly 
such that 
.
,
such that 
.
.
has 
,
,
.
.
be an integer, and let 
be positive integers. Show that there exist positive integers 
satisfying the following three conditions:
for 
the remainders of 
denotes the integer part of real number 
Y by aidan0626, e_is_2.71828, Exponent11, PikaPika999
i don't even know if this is solvable
Prove that there are finite/infinite powers of 2 where all the digits are also powers of 2. (For example,
and 
are numbers that work, but 
and 
don't work.)
Prove that there are finite/infinite powers of 2 where all the digits are also powers of 2. (For example,
and 
are numbers that work, but 
and 
don't work.)This post has been edited 1 time. Last edited by ceilingfan404, Apr 20, 2025, 7:55 PM
)
-
.
,
,
.
for some 
,
,
always have a digit equal to 
would be broken down into 
and since a, b,c are powers of 2, you could just look at the 5s?) idk how to continue though.
,
in base 
?
