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Problems
Contests
National and Regional Contests
France Contests
French Mathematical Olympiad
1997 French Mathematical Olympiad
1997 French Mathematical Olympiad
Part of
French Mathematical Olympiad
Subcontests
(5)
Problem 5
1
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locus of points with angle relationship
Given two distinct points
A
,
B
A,B
A
,
B
in the plane, for each point
C
C
C
not on the line
A
B
AB
A
B
, we denote by
G
G
G
and
I
I
I
the centroid and incenter of the triangle
A
B
C
ABC
A
BC
, respectively.(a) For
0
<
α
<
π
0<\alpha<\pi
0
<
α
<
π
, let
Γ
\Gamma
Γ
be the set of points
C
C
C
in the plane such that
∠
(
C
A
→
,
C
B
→
)
=
α
+
2
k
π
\angle\left(\overrightarrow{CA},\overrightarrow{CB}\right)=\alpha+2k\pi
∠
(
C
A
,
CB
)
=
α
+
2
kπ
as an oriented angle, where
k
∈
Z
k\in\mathbb Z
k
∈
Z
. If
C
C
C
describes
Γ
\Gamma
Γ
, show that points
G
G
G
and
I
I
I
also descibre arcs of circles, and determine these circles. (b) Suppose that in addition
π
3
<
α
<
π
\frac\pi3<\alpha<\pi
3
π
<
α
<
π
. For which positions of
C
C
C
in
Γ
\Gamma
Γ
is
G
I
GI
G
I
minimal? (c) Let
f
(
α
)
f(\alpha)
f
(
α
)
denote the minimal
G
I
GI
G
I
from the part (b). Give
f
(
α
)
f(\alpha)
f
(
α
)
explicitly in terms of
a
=
A
B
a=AB
a
=
A
B
and
α
\alpha
α
. Find the minimum value of
f
(
α
)
f(\alpha)
f
(
α
)
for
α
∈
(
π
3
,
π
)
\alpha\in\left(\frac\pi3,\pi\right)
α
∈
(
3
π
,
π
)
.
Problem 4
1
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function on a triangle
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
?
Problem 3
1
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orthogonal projection of unit cube onto plane
Let
C
C
C
be a unit cube and let
p
p
p
denote the orthogonal projection onto the plane. Find the maximum area of
p
(
C
)
p(C)
p
(
C
)
.
Problem 2
1
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maximal volume of a cylinder in region
A region in space is determined by a sphere with center
O
O
O
and radius
R
R
R
, and a cone with vertex
O
O
O
which intersects the sphere in a circle of radius
r
r
r
. Find the maximum volume of a cylinder contained in this region, having the same axis as the cone.
Problem 1
1
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labelling vertices of 1997-gon
Each vertex of a regular
1997
1997
1997
-gon is labeled with an integer, so that the sum of the integers is
1
1
1
. We write down the sums of the first
k
k
k
integers read counterclockwise, starting from some vertex
(
k
=
1
,
2
,
…
,
1997
)
(k=1,2,\ldots,1997)
(
k
=
1
,
2
,
…
,
1997
)
. Can we always choose the starting vertex so that all these sums are positive? If yes, how many possible choices are there?