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Spain Mathematical Olympiad
2023 Spain Mathematical Olympiad
4
4
Part of
2023 Spain Mathematical Olympiad
Problems
(1)
Roots of polynomial composition
Source: Spain MO 2023 P4
3/12/2023
Let
x
1
≤
x
2
≤
x
3
≤
x
4
x_1\leq x_2\leq x_3\leq x_4
x
1
≤
x
2
≤
x
3
≤
x
4
be real numbers. Prove that there exist polynomials of degree two
P
(
x
)
P(x)
P
(
x
)
and
Q
(
x
)
Q(x)
Q
(
x
)
with real coefficients such that
x
1
x_1
x
1
,
x
2
x_2
x
2
,
x
3
x_3
x
3
and
x
4
x_4
x
4
are the roots of
P
(
Q
(
x
)
)
P(Q(x))
P
(
Q
(
x
))
if and only if
x
1
+
x
4
=
x
2
+
x
3
x_1+x_4=x_2+x_3
x
1
+
x
4
=
x
2
+
x
3
.
algebra
polynomial
Spain