MathDB
Problems
Contests
National and Regional Contests
France Contests
French Mathematical Olympiad
1990 French Mathematical Olympiad
1990 French Mathematical Olympiad
Part of
French Mathematical Olympiad
Subcontests
(5)
Problem 5
1
Hide problems
maximum area involving A-excircle
In a triangle
A
B
C
ABC
A
BC
,
Γ
\Gamma
Γ
denotes the excircle corresponding to
A
A
A
,
A
′
,
B
′
,
C
′
A',B',C'
A
′
,
B
′
,
C
′
are the points of tangency of
Γ
\Gamma
Γ
with
B
C
,
C
A
,
A
B
BC,CA,AB
BC
,
C
A
,
A
B
respectively, and
S
(
A
B
C
)
S(ABC)
S
(
A
BC
)
denotes the region of the plane determined by segments
A
B
′
,
A
C
′
AB',AC'
A
B
′
,
A
C
′
and the arc
C
′
A
′
B
′
C'A'B'
C
′
A
′
B
′
of
Γ
\Gamma
Γ
.Prove that there is a triangle
A
B
C
ABC
A
BC
of a given perimeter
p
p
p
for which the area of
S
(
A
B
C
)
S(ABC)
S
(
A
BC
)
is maximal. For this triangle, give an approximate measure of the angle at
A
A
A
.
Problem 4
1
Hide problems
tetrahedron inscribed in cube, maximum area
(a) What is the maximum area of a triangle with vertices in a given square (or on its boundary)? (b) What is the maximum volume of a tetrahedron with vertices in a given cube (or on its boundary)?
Problem 3
1
Hide problems
diophantine equation involving reciprocals of squares
(a) Find all triples of integers
(
a
,
b
,
c
)
(a,b,c)
(
a
,
b
,
c
)
for which
1
4
=
1
a
2
+
1
b
2
+
1
c
2
\frac14=\frac1{a^2}+\frac1{b^2}+\frac1{c^2}
4
1
=
a
2
1
+
b
2
1
+
c
2
1
. (b) Determine all positive integers
n
n
n
for which there exist positive integers
x
1
,
x
2
,
…
,
x
n
x_1,x_2,\ldots,x_n
x
1
,
x
2
,
…
,
x
n
such that
1
=
1
x
1
2
+
1
x
2
2
+
…
+
1
x
n
2
1=\frac1{x_1^2}+\frac1{x_2^2}+\ldots+\frac1{x_n^2}
1
=
x
1
2
1
+
x
2
2
1
+
…
+
x
n
2
1
.
Problem 2
1
Hide problems
game, painting tetrahedron faces
A game consists of pieces of the shape of a regular tetrahedron of side
1
1
1
. Each face of each piece is painted in one of
n
n
n
colors, and by this, the faces of one piece are not necessarily painted in different colors. Determine the maximum possible number of pieces, no two of which are identical.
Problem 1
1
Hide problems
sequences, u_(2n)=u_n and u_(2n+1)=1=u_n
Let the sequence
u
n
u_n
u
n
be defined by
u
0
=
0
u_0=0
u
0
=
0
and
u
2
n
=
u
n
u_{2n}=u_n
u
2
n
=
u
n
,
u
2
n
+
1
=
1
−
u
n
u_{2n+1}=1-u_n
u
2
n
+
1
=
1
−
u
n
for each
n
∈
N
0
n\in\mathbb N_0
n
∈
N
0
. (a) Calculate
u
1990
u_{1990}
u
1990
. (b) Find the number of indices
n
≤
1990
n\le1990
n
≤
1990
for which
u
n
=
0
u_n=0
u
n
=
0
. (c) Let
p
p
p
be a natural number and
N
=
(
2
p
−
1
)
2
N=(2^p-1)^2
N
=
(
2
p
−
1
)
2
. Find
u
N
u_N
u
N
.