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Problems
Contests
International Contests
IMO Longlists
1972 IMO Longlists
35
35
Part of
1972 IMO Longlists
Problems
(1)
Inequality for a, b, c, d being reals and am+b=-cm+d=m
Source:
12/6/2010
(
a
)
(a)
(
a
)
Prove that for
a
,
b
,
c
,
d
∈
R
,
m
∈
[
1
,
+
∞
)
a, b, c, d \in\mathbb{R}, m \in [1,+\infty)
a
,
b
,
c
,
d
∈
R
,
m
∈
[
1
,
+
∞
)
with
a
m
+
b
=
−
c
m
+
d
=
m
am + b =-cm + d = m
am
+
b
=
−
c
m
+
d
=
m
,
(
i
)
a
2
+
b
2
+
c
2
+
d
2
+
(
a
−
c
)
2
+
(
b
−
d
)
2
≥
4
m
2
1
+
m
2
,
and
(i)\sqrt{a^2 + b^2}+\sqrt{c^2 + d^2}+\sqrt{(a-c)^2 + (b-d)^2}\ge \frac{4m^2}{1+m^2},\text{ and}
(
i
)
a
2
+
b
2
+
c
2
+
d
2
+
(
a
−
c
)
2
+
(
b
−
d
)
2
≥
1
+
m
2
4
m
2
,
and
(
i
i
)
2
≤
4
m
2
1
+
m
2
<
4.
(ii) 2 \le \frac{4m^2}{1+m^2} < 4.
(
ii
)
2
≤
1
+
m
2
4
m
2
<
4.
(
b
)
(b)
(
b
)
Express
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
as functions of
m
m
m
so that there is equality in
(
i
)
.
(i).
(
i
)
.
inequalities
function
inequalities unsolved