of positive integers such that 
is an integer.
such that 
has two isogonal conjugate points 
and 
.
intersects circle 
at 
,
intersects circle 
at 
.
and 
are isogonal conjugates in triangle 
,
.
is a nonconstant arithmetic sequence such that 
and the terms 
form a geometric sequence in that order. Find the smallest index 
.
Prove that
Let 
Prove that
.
be positive reals such that 
,
of 
.
![\[f(n) = \sum_{k=0}^{n} \binom{n}{k} a_k, \quad n = 1, 2, 3, \ldots\]](http://latex.artofproblemsolving.com/9/3/7/9377324cb6585daaf2cbec7fed6fce2209835e22.png)
where![\[a_k =
\begin{cases}
3^k, & \text{if } k \text{ is even}, \\
0, & \text{if } k \text{ is odd}.
\end{cases}
\]](http://latex.artofproblemsolving.com/8/b/e/8be520a30bdfd9272163470ee69120c2fd3e9b59.png)
Find the remainder when 
is divided by 
distinct penguin chicks, 
are selected to form the Penchick Manga Club. Then, one of these 
be the number of ways to select the club and assign its president. Compute 
.
is 
where 
are relatively prime positive integers, find 
.
Y by
Let 
be an odd prime number and a sequence 
satisfy 
and 
Prove that always exists positive integer 
satisfying for all positive integers 
, then 
.
P/s:
is "not congruence"
be an odd prime number and a sequence 
satisfy 
and 
Prove that always exists positive integer 
satisfying for all positive integers 
, then 
.P/s:
is "not congruence"This post has been edited 1 time. Last edited by AnhQuang_67, Apr 24, 2025, 2:54 PM
Reason: sorry, my bad
Reason: sorry, my bad
thus 
are not of the same parity there doesnt exist such 
