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Korea Junior Mathematics Olympiad
2003 Korea Junior Math Olympiad
2
2
Part of
2003 Korea Junior Math Olympiad
Problems
(1)
Quadratic equation two integer solution for (a-b)^2<=8sqrt(ab)
Source: 2003 Korea Junior Math Olympiad
6/28/2024
a
,
b
a, b
a
,
b
are odd numbers that satisfy
(
a
−
b
)
2
≤
8
a
b
(a-b)^2 \le 8\sqrt {ab}
(
a
−
b
)
2
≤
8
ab
. For
n
=
a
b
n=ab
n
=
ab
, show that the equation
x
2
−
2
(
[
n
]
+
1
)
x
+
n
=
0
x^2-2([\sqrt n]+1)x+n=0
x
2
−
2
([
n
]
+
1
)
x
+
n
=
0
has two integral solutions.
[
r
]
[r]
[
r
]
denotes the biggest integer, not strictly bigger than
r
r
r
.
Integer
quadratics equation
algebra
number theory
quadratics