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2001 IMC
1
IMC 2001 Problem 7
IMC 2001 Problem 7
Source: IMC 2001 Day 2 Problem 1
October 30, 2020
Polynomials
Problem Statement
Let
r
,
s
≥
1
r, s \geq 1
r
,
s
≥
1
be integers and
a
0
,
a
1
,
.
.
.
,
a
r
−
1
,
b
0
,
b
1
,
.
.
.
,
b
s
−
1
a_{0}, a_{1}, . . . , a_{r-1}, b_{0}, b_{1}, . . . , b_{s-1}
a
0
,
a
1
,
...
,
a
r
−
1
,
b
0
,
b
1
,
...
,
b
s
−
1
be real non-negative numbers such that
(
a
0
+
a
1
x
+
a
2
x
2
+
.
.
.
+
a
r
−
1
x
r
−
1
+
x
r
)
(
b
0
+
b
1
x
+
b
2
x
2
+
.
.
.
+
b
s
−
1
x
s
−
1
+
x
s
)
=
1
+
x
+
x
2
+
.
.
.
+
x
r
+
s
−
1
+
x
r
+
s
(a_0+a_1x+a_2x^2+. . .+a_{r-1}x^{r-1}+x^r)(b_0+b_1x+b_2x^2+. . .+b_{s-1}x^{s-1}+x^s) =1 +x+x^2+. . .+x^{r+s-1}+x^{r+s}
(
a
0
+
a
1
x
+
a
2
x
2
+
...
+
a
r
−
1
x
r
−
1
+
x
r
)
(
b
0
+
b
1
x
+
b
2
x
2
+
...
+
b
s
−
1
x
s
−
1
+
x
s
)
=
1
+
x
+
x
2
+
...
+
x
r
+
s
−
1
+
x
r
+
s
. Prove that each
a
i
a_i
a
i
and each
b
j
b_j
b
j
equals either
0
0
0
or
1
1
1
.
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