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China Team Selection Test
2024 China Team Selection Test
10
difference of roots
difference of roots
Source: 2024 CTST P10
March 12, 2024
algebra
polynomial
2024 CTST
Problem Statement
Let
M
M
M
be a positive integer.
f
(
x
)
:
=
x
3
+
a
x
2
+
b
x
+
c
∈
Z
[
x
]
f(x):=x^3+ax^2+bx+c\in\mathbb Z[x]
f
(
x
)
:=
x
3
+
a
x
2
+
b
x
+
c
∈
Z
[
x
]
satisfy
∣
a
∣
,
∣
b
∣
,
∣
c
∣
≤
M
.
|a|,|b|,|c|\le M.
∣
a
∣
,
∣
b
∣
,
∣
c
∣
≤
M
.
x
1
,
x
2
x_1,x_2
x
1
,
x
2
are different roots of
f
.
f.
f
.
Prove that
∣
x
1
−
x
2
∣
>
1
M
2
+
3
M
+
1
.
|x_1-x_2|>\frac 1{M^2+3M+1}.
∣
x
1
−
x
2
∣
>
M
2
+
3
M
+
1
1
.
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