MathDB
Problems
Contests
International Contests
Lusophon Mathematical Olympiad
2015 Lusophon Mathematical Olympiad
4
4
Part of
2015 Lusophon Mathematical Olympiad
Problems
(1)
a^m+a^n=a^p+a^q, a^{3m}+a^{3n}=a^{3p}+a^{3q}, prove mn=pq
Source: Lusophon 2015 CPLP P4
8/29/2018
Let
a
a
a
be a real number, such that
a
≠
0
,
a
≠
1
,
a
≠
−
1
a\ne 0, a\ne 1, a\ne -1
a
=
0
,
a
=
1
,
a
=
−
1
and
m
,
n
,
p
,
q
m,n,p,q
m
,
n
,
p
,
q
be natural numbers . Prove that if
a
m
+
a
n
=
a
p
+
a
q
a^m+a^n=a^p+a^q
a
m
+
a
n
=
a
p
+
a
q
and
a
3
m
+
a
3
n
=
a
3
p
+
a
3
q
a^{3m}+a^{3n}=a^{3p}+a^{3q}
a
3
m
+
a
3
n
=
a
3
p
+
a
3
q
, then
m
⋅
n
=
p
⋅
q
m \cdot n = p \cdot q
m
⋅
n
=
p
⋅
q
.
algebra
system of equations