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International Contests
IMO Shortlist
1984 IMO Shortlist
9
9
Part of
1984 IMO Shortlist
Problems
(1)
System has exactly one real solution
Source:
9/8/2010
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be positive numbers with
a
+
b
+
c
=
3
2
\sqrt a +\sqrt b +\sqrt c = \frac{\sqrt 3}{2}
a
+
b
+
c
=
2
3
. Prove that the system of equations
y
−
a
+
z
−
a
=
1
,
\sqrt{y-a}+\sqrt{z-a}=1,
y
−
a
+
z
−
a
=
1
,
z
−
b
+
x
−
b
=
1
,
\sqrt{z-b}+\sqrt{x-b}=1,
z
−
b
+
x
−
b
=
1
,
x
−
c
+
y
−
c
=
1
\sqrt{x-c}+\sqrt{y-c}=1
x
−
c
+
y
−
c
=
1
has exactly one solution
(
x
,
y
,
z
)
(x, y, z)
(
x
,
y
,
z
)
in real numbers.
algebra
system of equations
IMO Shortlist
Pythagorean Theorem
geometry