MathDB
Problems
Contests
National and Regional Contests
Vietnam Contests
Vietnam Team Selection Test
2023 Vietnam Team Selection Test
2
2
Part of
2023 Vietnam Team Selection Test
Problems
(1)
Mutually non-generable functions
Source: Vietnam TST 2023 P2
4/13/2023
Given three functions
P
(
x
)
=
(
x
2
−
1
)
2023
,
Q
(
x
)
=
(
2
x
+
1
)
14
,
R
(
x
)
=
(
2
x
+
1
+
2
x
)
34
.
P(x) = (x^2-1)^{2023}, Q(x) = (2x+1)^{14}, R(x) = \left(2x+1+\frac 2x \right)^{34}.
P
(
x
)
=
(
x
2
−
1
)
2023
,
Q
(
x
)
=
(
2
x
+
1
)
14
,
R
(
x
)
=
(
2
x
+
1
+
x
2
)
34
.
Initially, we pick a set
S
S
S
containing two of these functions, and we perform some operations on it. Allowed operations include:- We can take two functions
p
,
q
∈
S
p,q \in S
p
,
q
∈
S
and add one of
p
+
q
,
p
−
q
p+q, p-q
p
+
q
,
p
−
q
, or
p
q
pq
pq
to
S
S
S
. - We can take a function
p
∈
S
p \in S
p
∈
S
and add
p
k
p^k
p
k
to
S
S
S
for
k
k
k
is an arbitrary positive integer of our choice. - We can take a function
p
∈
S
p \in S
p
∈
S
and choose a real number
t
t
t
, and add to
S
S
S
one of the function
p
+
t
,
p
−
t
,
p
t
p+t, p-t, pt
p
+
t
,
p
−
t
,
pt
. Show that no matter how we pick
S
S
S
in the beginning, there is no way we can perform finitely many operations on
S
S
S
that would eventually yield the third function not in
S
S
S
.
algebra