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Kvant 2019
M2546
Nice identities with sums of powers
Nice identities with sums of powers
Source: Kvant Magazine No. 2 2019 M2546
March 14, 2023
algebra
Kvant
Problem Statement
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be real numbers
a
+
b
+
c
=
0
a + b +c = 0
a
+
b
+
c
=
0
. Show that[*]
a
2
+
b
2
+
c
2
2
⋅
a
3
+
b
3
+
c
3
3
=
a
5
+
b
5
+
c
5
5
\displaystyle \frac{a^2 + b^2 + c^2}{2} \cdot \frac{a^3 + b^3 + c^3}{3} = \frac{a^5 + b^5 + c^5}{5}
2
a
2
+
b
2
+
c
2
⋅
3
a
3
+
b
3
+
c
3
=
5
a
5
+
b
5
+
c
5
. [*]
a
2
+
b
2
+
c
2
2
⋅
a
5
+
b
5
+
c
5
5
=
a
7
+
b
7
+
c
7
7
\displaystyle \frac{a^2 + b^2 + c^2}{2} \cdot \frac{a^5 + b^5 + c^5}{5} = \frac{a^7 + b^7 + c^7}{7}
2
a
2
+
b
2
+
c
2
⋅
5
a
5
+
b
5
+
c
5
=
7
a
7
+
b
7
+
c
7
. [I]Folklore[/I]
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