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10
SMT 2023 Algebra #10
SMT 2023 Algebra #10
Source:
May 3, 2023
Problem Statement
Suppose that
p
(
x
)
,
q
(
x
)
p(x),q(x)
p
(
x
)
,
q
(
x
)
are monic polynomials with nonnegative integer coefficients such that
1
5
x
≥
1
q
(
x
)
−
1
p
(
x
)
≥
1
3
x
2
\frac{1}{5x}\ge\frac{1}{q(x)}-\frac{1}{p(x)}\ge\frac{1}{3x^2}
5
x
1
≥
q
(
x
)
1
−
p
(
x
)
1
≥
3
x
2
1
for all integers
x
≥
2
x\ge2
x
≥
2
. Compute the minimum possible value of
p
(
1
)
⋅
q
(
1
)
p(1)\cdot q(1)
p
(
1
)
⋅
q
(
1
)
.
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