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All-Russian Olympiad Regional Round
2000 All-Russian Olympiad Regional Round
11.1
11.1
Part of
2000 All-Russian Olympiad Regional Round
Problems
(1)
prod (x-a_1)=prod (x+a_1) - All-Russian MO 2000 Regional (R4) 11.1
Source:
9/26/2024
Prove that it is possible to choose different real numbers
a
1
,
a
2
,
.
.
.
,
a
10
a_1, a_2, . . . , a_{10}
a
1
,
a
2
,
...
,
a
10
that the equation
(
x
−
a
1
)
(
x
−
a
2
)
.
.
.
.
(
x
−
a
10
)
=
(
x
+
a
1
)
(
x
+
a
2
)
.
.
.
(
x
+
a
10
)
(x - a_1)(x -a_2).... (x -a_{10}) = (x + a_1)(x + a_2) ...(x + a_{10})
(
x
−
a
1
)
(
x
−
a
2
)
....
(
x
−
a
10
)
=
(
x
+
a
1
)
(
x
+
a
2
)
...
(
x
+
a
10
)
will have exactly
5
5
5
different real roots.
algebra
polynomial