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Romania National Olympiad
1997 Romania National Olympiad
3
f(u)+f(v)=7
f(u)+f(v)=7
Source: Romania 1997
August 23, 2005
algebra
polynomial
algebra unsolved
function
Problem Statement
Suppose that
a
,
b
,
c
,
d
∈
R
a,b,c,d\in\mathbb{R}
a
,
b
,
c
,
d
∈
R
and
f
(
x
)
=
a
x
3
+
b
x
2
+
c
x
+
d
f(x)=ax^3+bx^2+cx+d
f
(
x
)
=
a
x
3
+
b
x
2
+
c
x
+
d
such that
f
(
2
)
+
f
(
5
)
<
7
<
f
(
3
)
+
f
(
4
)
f(2)+f(5)<7<f(3)+f(4)
f
(
2
)
+
f
(
5
)
<
7
<
f
(
3
)
+
f
(
4
)
. Prove that there exists
u
,
v
∈
R
u,v\in\mathbb{R}
u
,
v
∈
R
such that
u
+
v
=
7
,
f
(
u
)
+
f
(
v
)
=
7
u+v=7 , f(u)+f(v)=7
u
+
v
=
7
,
f
(
u
)
+
f
(
v
)
=
7
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