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Poland Contests
Poland - Second Round
1963 Poland - Second Round
1
a^2 + b^2 + c^2 = (pa + qb + rc)^2 + (qa + rb + pc)^2 + (ra + pb + qc)^2
a^2 + b^2 + c^2 = (pa + qb + rc)^2 + (qa + rb + pc)^2 + (ra + pb + qc)^2
Source: Polish MO Second Round 1963 p1
August 31, 2024
algebra
system of equations
System
Problem Statement
Prove that if the numbers
p
p
p
,
q
q
q
,
r
r
r
satisfy the equality
p
+
q
+
r
=
1
p+q + r=1
p
+
q
+
r
=
1
1
p
+
1
q
+
1
r
=
0
\frac{1}{p} + \frac{1}{q} + \frac{1}{r} = 0
p
1
+
q
1
+
r
1
=
0
then for any numbers
a
a
a
,
b
b
b
,
c
c
c
equality holds
a
2
+
b
2
+
c
2
=
(
p
a
+
q
b
+
r
c
)
2
+
(
q
a
+
r
b
+
p
c
)
2
+
(
r
a
+
p
b
+
q
c
)
2
.
a^2 + b^2 + c^2 = (pa + qb + rc)^2 + (qa + rb + pc)^2 + (ra + pb + qc)^2.
a
2
+
b
2
+
c
2
=
(
p
a
+
q
b
+
rc
)
2
+
(
q
a
+
r
b
+
p
c
)
2
+
(
r
a
+
p
b
+
q
c
)
2
.
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