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14
Math Prize 2011 Problem 14
Math Prize 2011 Problem 14
Source:
September 19, 2011
algebra
function
domain
Problem Statement
If
0
≤
p
≤
1
0 \le p \le 1
0
≤
p
≤
1
and
0
≤
q
≤
1
0 \le q \le 1
0
≤
q
≤
1
, define
F
(
p
,
q
)
F(p, q)
F
(
p
,
q
)
by
F
(
p
,
q
)
=
−
2
p
q
+
3
p
(
1
−
q
)
+
3
(
1
−
p
)
q
−
4
(
1
−
p
)
(
1
−
q
)
.
F(p, q) = -2pq + 3p(1-q) + 3(1-p)q - 4(1-p)(1-q).
F
(
p
,
q
)
=
−
2
pq
+
3
p
(
1
−
q
)
+
3
(
1
−
p
)
q
−
4
(
1
−
p
)
(
1
−
q
)
.
Define
G
(
p
)
G(p)
G
(
p
)
to be the maximum of
F
(
p
,
q
)
F(p, q)
F
(
p
,
q
)
over all
q
q
q
(in the interval
0
≤
q
≤
1
0 \le q \le 1
0
≤
q
≤
1
). What is the value of
p
p
p
(in the interval
0
≤
p
≤
1
0 \le p \le 1
0
≤
p
≤
1
) that minimizes
G
(
p
)
G(p)
G
(
p
)
?
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