and 
.
Where 
satisfies 
.
,
.
and a constant 
,
and 
are also friendly polynomials. Prove that 
.
lying outside the circle 
,
and 
.
is chosen on the segment 
.
and 
be the midpoints of segments 
and 
respectively. The circumcircle of triangle 
intersects 
)
passes through the centroid of triangle 
.
Y by
Let 
be the Euler's totient function.
1. For any given integer
, does there exist 
such that for any 
, 
and 
, 
is a non-negative power of 
?
2. For integer
, are there integers 
and 
satisfying:
![\[
\varphi(k_i) \in \left ( \frac{x}{a} ,x \right ], i = 1,2; \quad \varphi(k_1) \neq \varphi(k_2).
\]](//latex.artofproblemsolving.com/6/c/b/6cbe59f15b51439d1202267f521b573a52f1fe67.png)
And these two different 
correspond to the same 
and 
as described in (1), yet 
.
3. Define
as the number of elements in set 
. For integer , let 
and 
. Compare 
with 
.
be the Euler's totient function.1. For any given integer
, does there exist 
such that for any 
, 
and 
, 
is a non-negative power of 
?2. For integer
, are there integers 
and 
satisfying:![\[
\varphi(k_i) \in \left ( \frac{x}{a} ,x \right ], i = 1,2; \quad \varphi(k_1) \neq \varphi(k_2).
\]](http://latex.artofproblemsolving.com/6/c/b/6cbe59f15b51439d1202267f521b573a52f1fe67.png)
And these two different 
correspond to the same 
and 
as described in (1), yet 
.3. Define
as the number of elements in set 
. For integer , let 
and 
. Compare 
with 
.This post has been edited 1 time. Last edited by steven_zhang123, Mar 29, 2025, 11:37 PM
