MathDB
Problems
Contests
National and Regional Contests
India Contests
NMTC
2023 NMTC Junior
P3
P3
Part of
2023 NMTC Junior
Problems
(1)
NMTC 2023 FINALS P3 JUNIOR
Source: NMTC 2023 Finals P3 (Junior)
11/5/2023
Let
a
i
(
i
=
1
,
2
,
3
,
4
,
5
,
6
)
a_i (i=1,2,3,4,5,6)
a
i
(
i
=
1
,
2
,
3
,
4
,
5
,
6
)
are reals. The polynomial
f
(
x
)
=
a
1
+
a
2
x
+
a
3
x
2
+
a
4
x
3
+
a
5
x
4
+
a
6
a
5
+
7
x
6
−
4
x
7
+
x
8
f(x)=a_1+a_2x+a_3x^2+a_4x^3+a_5x^4+a_6a^5+7x^6-4x^7+x^8
f
(
x
)
=
a
1
+
a
2
x
+
a
3
x
2
+
a
4
x
3
+
a
5
x
4
+
a
6
a
5
+
7
x
6
−
4
x
7
+
x
8
can be factorized into linear factors
x
−
x
i
x-x_i
x
−
x
i
where
i
∈
1
,
2
,
3
,
.
.
.
,
8
i \in {1,2,3,...,8}
i
∈
1
,
2
,
3
,
...
,
8
. Find the possible values of
a
1
a_1
a
1
.
algebra
polynomial
nmtc