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Contests
National and Regional Contests
France Contests
French Mathematical Olympiad
1997 French Mathematical Olympiad
Problem 4
Problem 4
Part of
1997 French Mathematical Olympiad
Problems
(1)
function on a triangle
Source: French MO 1997 P4
4/10/2021
In a triangle
A
B
C
ABC
A
BC
, let
a
,
b
,
c
a,b,c
a
,
b
,
c
be its sides and
m
,
n
,
p
m,n,p
m
,
n
,
p
be the corresponding medians. For every
α
>
0
\alpha>0
α
>
0
, let
λ
(
α
)
\lambda(\alpha)
λ
(
α
)
be the real number such that
a
α
+
b
α
+
c
α
=
λ
(
α
)
α
(
m
α
+
n
α
+
p
α
)
α
.
a^\alpha+b^\alpha+c^\alpha=\lambda(\alpha)^\alpha\left(m^\alpha+n^\alpha+p^\alpha\right)^\alpha.
a
α
+
b
α
+
c
α
=
λ
(
α
)
α
(
m
α
+
n
α
+
p
α
)
α
.
(a) Compute
λ
(
2
)
\lambda(2)
λ
(
2
)
. (b) Find the limit of
λ
(
α
)
\lambda(\alpha)
λ
(
α
)
as
α
\alpha
α
approaches
0
0
0
. (c) For which triangles
A
B
C
ABC
A
BC
is
λ
(
α
)
\lambda(\alpha)
λ
(
α
)
independent of
α
\alpha
α
?
function
geometry
Triangle