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1968 IMO Shortlist
6
The equation has at least n-1 roots
The equation has at least n-1 roots
Source:
September 23, 2010
algebra
polynomial
equation
roots
Real Roots
IMO Shortlist
Problem Statement
If
a
i
(
i
=
1
,
2
,
…
,
n
)
a_i \ (i = 1, 2, \ldots, n)
a
i
(
i
=
1
,
2
,
…
,
n
)
are distinct non-zero real numbers, prove that the equation
a
1
a
1
−
x
+
a
2
a
2
−
x
+
⋯
+
a
n
a
n
−
x
=
n
\frac{a_1}{a_1-x} + \frac{a_2}{a_2-x}+\cdots+\frac{a_n}{a_n-x} = n
a
1
−
x
a
1
+
a
2
−
x
a
2
+
⋯
+
a
n
−
x
a
n
=
n
has at least
n
−
1
n - 1
n
−
1
real roots.
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