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Putnam
1967 Putnam
B2
Putnam 1967 B2
Putnam 1967 B2
Source: Putnam 1967
May 14, 2022
Putnam
inequalities
Problem Statement
Let
0
≤
p
,
r
≤
1
0\leq p,r\leq 1
0
≤
p
,
r
≤
1
and consider the identities
a
)
(
p
x
+
(
1
−
p
)
y
)
2
=
a
x
2
+
b
x
y
+
c
y
2
,
b
)
(
p
x
+
(
1
−
p
)
y
)
(
r
x
+
(
1
−
r
)
y
)
=
α
x
2
+
β
x
y
+
γ
y
2
.
a)\; (px+(1-p)y)^{2}=a x^2 +bxy +c y^2, \;\;\;\, b)\; (px+(1-p)y)(rx+(1-r)y) =\alpha x^2 + \beta xy + \gamma y^2.
a
)
(
p
x
+
(
1
−
p
)
y
)
2
=
a
x
2
+
b
x
y
+
c
y
2
,
b
)
(
p
x
+
(
1
−
p
)
y
)
(
r
x
+
(
1
−
r
)
y
)
=
α
x
2
+
β
x
y
+
γ
y
2
.
Show that
a
)
max
(
a
,
b
,
c
)
≥
4
9
,
b
)
max
(
α
,
β
,
γ
)
≥
4
9
.
a)\; \max(a,b,c) \geq \frac{4}{9}, \;\;\;\; b)\; \max( \alpha, \beta , \gamma) \geq \frac{4}{9}.
a
)
max
(
a
,
b
,
c
)
≥
9
4
,
b
)
max
(
α
,
β
,
γ
)
≥
9
4
.
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