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be a function defined in any real numbers 
with 
Prove that on the 
plane, the line 
passes through the fixed point which isn't on the 
axis in regardless of the value of if only if is a linear function in .
foot radius. Bruno completes 
laps every hour, while Brutus completes 
laps every hour. If they start at the same point but run in opposite directions, how far along the track’s circumference (in feet) from the starting point are they when they meet for the sixth time? Note: Do not count the moment they start running as a meeting point.
. I couldn't solve this and was wondering if any of you can find the pattern
. I couldn't solve this and was wondering if any of you can find the pattern
.
, and thus the next term is 
![$ a,b \in [0 ,1] . $](http://latex.artofproblemsolving.com/9/1/9/91980d5ca5d97fe4b61797e6fbb378211674d9cd.png)
Prove that
.
.
and 
Prove that
and 
.
Prove that
,![\[
1, 2, 5, 40, 1280, ?
\]](http://latex.artofproblemsolving.com/a/b/d/abdadcc9d1511e0bc50630d0764932c6723f6123.png)
![\[
a_n = \frac{381}{8}n^{4} - \frac{1885}{4}n^{3} + \frac{13103}{8}n^{2} - \frac{9313}{4}n + 1115
\]](http://latex.artofproblemsolving.com/3/a/5/3a5bf7857ee63dfefc85efdd2f73062889f38dd4.png)
:![\[
a_1 = \frac{381}{8}(1)^4 - \frac{1885}{4}(1)^3 + \frac{13103}{8}(1)^2 - \frac{9313}{4}(1) + 1115 = 1
\]](http://latex.artofproblemsolving.com/a/4/b/a4b8690544780a4953ff30b7e9bfd650d5c05c64.png)
:![\[
a_2 = \frac{381}{8}(2)^4 - \frac{1885}{4}(2)^3 + \frac{13103}{8}(2)^2 - \frac{9313}{4}(2) + 1115 = 2
\]](http://latex.artofproblemsolving.com/4/0/1/401ac0e6b9e04a3bbb18b93adf3afd46aa401c2a.png)
:![\[
a_3 = \frac{381}{8}(3)^4 - \frac{1885}{4}(3)^3 + \frac{13103}{8}(3)^2 - \frac{9313}{4}(3) + 1115 = 5
\]](http://latex.artofproblemsolving.com/7/f/c/7fc785dbf6d9a0992d3bb5e2011bc4db67ef1c02.png)
:![\[
a_4 = \frac{381}{8}(4)^4 - \frac{1885}{4}(4)^3 + \frac{13103}{8}(4)^2 - \frac{9313}{4}(4) + 1115 = 40
\]](http://latex.artofproblemsolving.com/2/1/b/21b4e2e4d19b6d5c983417f70abf485ad3eb155a.png)
:![\[
a_5 = \frac{381}{8}(5)^4 - \frac{1885}{4}(5)^3 + \frac{13103}{8}(5)^2 - \frac{9313}{4}(5) + 1115 = 1280
\]](http://latex.artofproblemsolving.com/b/d/9/bd9f2528ec68ad7182630c86181261f2eeabad17.png)
:![\[
a_6 = \frac{381}{8}(6)^4 - \frac{1885}{4}(6)^3 + \frac{13103}{8}(6)^2 - \frac{9313}{4}(6) + 1115
\]](http://latex.artofproblemsolving.com/4/a/8/4a8c57a4df9d6246743b76544bd06d424c99c27f.png)
![\[
a_6 = \frac{381}{8} \times 1296 - \frac{1885}{4} \times 216 + \frac{13103}{8} \times 36 - \frac{9313}{4} \times 6 + 1115
\]](http://latex.artofproblemsolving.com/b/a/d/bade9652d9835650aa5c7ae840e9564696946715.png)
![\[
a_6 = 69420
\]](http://latex.artofproblemsolving.com/a/9/c/a9cc1cb8407e92daee7dff378e21ddbde47fd9fd.png)
![\[
\boxed{69420}
\]](http://latex.artofproblemsolving.com/2/f/1/2f16609b47f00089d42077ac2f80a849752c8766.png)
.
= (next Catalan number)
= (next Catalan number)
(next Catalan number), 26 + 14 = 40
(next Catalan number), 1238 + 42 = 1280
