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PEN O Problems
36
O 36
O 36
Source:
May 25, 2007
induction
inequalities
Problem Statement
Let a and b be non-negative integers such that
a
b
≥
c
2
ab \ge c^{2}
ab
≥
c
2
where
c
c
c
is an integer. Prove that there is a positive integer n and integers
x
1
x_{1}
x
1
,
x
2
x_{2}
x
2
,
⋯
\cdots
⋯
,
x
n
x_{n}
x
n
,
y
1
y_{1}
y
1
,
y
2
y_{2}
y
2
,
⋯
\cdots
⋯
,
y
n
y_{n}
y
n
such that
x
1
2
+
⋯
+
x
n
2
=
a
,
y
1
2
+
⋯
+
y
n
2
=
b
,
x
1
y
1
+
⋯
+
x
n
y
n
=
c
{x_{1}}^{2}+\cdots+{x_{n}}^{2}=a,\;{y_{1}}^{2}+\cdots+{y_{n}}^{2}=b,\; x_{1}y_{1}+\cdots+x_{n}y_{n}=c
x
1
2
+
⋯
+
x
n
2
=
a
,
y
1
2
+
⋯
+
y
n
2
=
b
,
x
1
y
1
+
⋯
+
x
n
y
n
=
c
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