[/quote]
,
of real numbers such that![\[ a_{m^2 + m + n} = a_{m}^2 + a_m + a_n\]](http://latex.artofproblemsolving.com/b/b/9/bb97053e86e7733fe5c26f7859c87566eb444b52.png)
for all positive integers 
and 
of positive integers satisfying the following two conditions:
,
is in 
,
and 
;
such that 
is not divisible by 
.
such that:
with![\[2x^3zf(z)+yf(y) \ge 3yz^2f(x)\]](http://latex.artofproblemsolving.com/a/b/f/abf10d7e679f6f04db556424a0a0eda26f371fd4.png)
for all 
.
,
,
on 
,
,
be the touch points of the incircle of 
.
intersects 
at 
,
.
intersect again at 
. If 
is the tangent point of the 
−
be a function taking in positive integers and outputting nonnegative integers, defined as follows:
is the number of positive integers 
such that the equation 
has an integer solution 
.
such tha
Adit Aggarwal, India.
such that the following inequality holds for all 
:![\[
ab + bc + ca \leq \frac{(a + b + c)^2}{3} + k \cdot \max \{ (a - b)^2, (b - c)^2, (c - a)^2 \} \leq a^2 + b^2 + c^2
\]](http://latex.artofproblemsolving.com/1/c/e/1ce66163068b499383adb4a4ecec86c5f4edaec3.png)
are smaller than the square root of 
for all proper positive divisors 
of 
and 
for 
.
.
for which 
is divisible by 
.
is divisible by 
satisfy,
Where 
}
be a fixed square and 
a variable point on segment 
.
is constructed such that 
is on segment 
and 
is on segment 
.
and 
.
Y by
For every integer 
, let 
be the set of all positive integers not exceeding 
that are relatively prime to . Consider the polynomial
![\[
P_n(x) = \sum_{k \in S_n} x^{k - 1}
\]](//latex.artofproblemsolving.com/3/a/c/3ac60cb70baf4dfcf297e9af87480a8ee34ba70c.png)
a) Prove that there exists a positive integer
and a polynomial 
with integer coefficients such that
![\[
P_n(x) = (x^{r_n} + 1) Q_n(x).
\]](//latex.artofproblemsolving.com/7/0/e/70e19a9d3e0a7e97abc7aa08d3f23126c14ea7d0.png)
b)Find all integers such that 
is irreducible in ![$\mathbb{Z}[x]$](//latex.artofproblemsolving.com/4/b/7/4b7d768fffbb7a1a6163af392ed348f2d49d8783.png)
.
, let 
be the set of all positive integers not exceeding 
that are relatively prime to . Consider the polynomial![\[
P_n(x) = \sum_{k \in S_n} x^{k - 1}
\]](http://latex.artofproblemsolving.com/3/a/c/3ac60cb70baf4dfcf297e9af87480a8ee34ba70c.png)
a) Prove that there exists a positive integer
and a polynomial 
with integer coefficients such that![\[
P_n(x) = (x^{r_n} + 1) Q_n(x).
\]](http://latex.artofproblemsolving.com/7/0/e/70e19a9d3e0a7e97abc7aa08d3f23126c14ea7d0.png)
b)Find all integers
such that 
is irreducible in ![$\mathbb{Z}[x]$](http://latex.artofproblemsolving.com/4/b/7/4b7d768fffbb7a1a6163af392ed348f2d49d8783.png)
.This post has been edited 2 times. Last edited by luutrongphuc, Apr 5, 2025, 8:27 AM
