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Argentina Contests
Argentina National Olympiad
2022 Argentina National Olympiad
6
6
Part of
2022 Argentina National Olympiad
Problems
(1)
P(x)=a_{2n}x^{2n}+a_{2n-1}x^{2n-1}+...+a_1x+a_0
Source: 2022 Argentina OMA Finals L3 p6
3/25/2024
For every positive integer
n
n
n
, we consider the polynomial of real coefficients, of
2
n
+
1
2n+1
2
n
+
1
terms,
P
(
x
)
=
a
2
n
x
2
n
+
a
2
n
−
1
x
2
n
−
1
+
.
.
.
+
a
1
x
+
a
0
P(x)=a_{2n}x^{2n}+a_{2n-1}x^{2n-1}+...+a_1x+a_0
P
(
x
)
=
a
2
n
x
2
n
+
a
2
n
−
1
x
2
n
−
1
+
...
+
a
1
x
+
a
0
where all coefficients are real numbers satisfying
100
≤
a
i
≤
101
100 \le a_i \le 101
100
≤
a
i
≤
101
for
0
≤
i
≤
2
n
0 \le i \le 2n
0
≤
i
≤
2
n
. Find the smallest possible value of
n
n
n
such that the polynomial can have at least one real root.
algebra
polynomial