MathDB
Problems
Contests
National and Regional Contests
France Contests
French Mathematical Olympiad
1998 French Mathematical Olympiad
1998 French Mathematical Olympiad
Part of
French Mathematical Olympiad
Subcontests
(5)
Problem 5
1
Hide problems
existence of subset of points inside triangle
Let
A
A
A
be a set of
n
≥
3
n\ge3
n
≥
3
points in the plane, no three of which are collinear. Show that there is a set
S
S
S
of
2
n
−
5
2n-5
2
n
−
5
points in the plane such that, for each triangle with vertices in
A
A
A
, there exists a point in
S
S
S
which is strictly inside that triangle.
Problem 4
1
Hide problems
intersection of two lines and existence of points
Let there be given two lines
D
1
D_1
D
1
and
D
2
D_2
D
2
which intersect at point
O
O
O
, and a point
M
M
M
not on any of these lines. Consider two variable points
A
∈
D
1
A\in D_1
A
∈
D
1
and
b
∈
D
2
b\in D_2
b
∈
D
2
such that
M
M
M
belongs to the segment
A
B
AB
A
B
.(a) Prove that there exists a position of
A
A
A
and
B
B
B
for which the area of triangle
O
A
B
OAB
O
A
B
is minimal. Construct such points
A
A
A
and
B
B
B
. (b) Prove that there exists a position of
A
A
A
and
B
B
B
for which the area of triangle
O
A
B
OAB
O
A
B
is minimal. Show that for such
A
A
A
and
B
B
B
, the perimeters of
△
O
A
M
\triangle OAM
△
O
A
M
and
△
O
B
M
\triangle OBM
△
OBM
are equal, and that
A
M
tan
1
2
∠
O
A
M
=
B
M
tan
1
2
∠
O
B
M
\frac{AM}{\tan\frac12\angle OAM}=\frac{BM}{\tan\frac12\angle OBM}
t
a
n
2
1
∠
O
A
M
A
M
=
t
a
n
2
1
∠
OBM
BM
. Construct such points
A
A
A
and
B
B
B
.
Problem 3
1
Hide problems
image of floor function
Let
k
≥
2
k\ge2
k
≥
2
be an integer. The function
f
:
N
→
N
f:\mathbb N\to\mathbb N
f
:
N
→
N
is defined by
f
(
n
)
=
n
+
⌊
n
+
n
k
k
⌋
.
f(n)=n+\left\lfloor\sqrt[k]{n+\sqrt[k]n}\right\rfloor.
f
(
n
)
=
n
+
⌊
k
n
+
k
n
⌋
.
Determine the set of values taken by the function
f
f
f
.
Problem 2
1
Hide problems
prove periodicity of sequence
Let
(
u
n
)
(u_n)
(
u
n
)
be a sequence of real numbers which satisfies
u
n
+
2
=
∣
u
n
+
1
∣
−
u
n
for all
n
∈
N
.
u_{n+2}=|u_{n+1}|-u_n\qquad\text{for all }n\in\mathbb N.
u
n
+
2
=
∣
u
n
+
1
∣
−
u
n
for all
n
∈
N
.
Prove that there exists a positive integer
p
p
p
such that
u
n
=
u
n
+
p
u_n=u_{n+p}
u
n
=
u
n
+
p
holds for all
n
∈
N
n\in\mathbb N
n
∈
N
.
Problem 1
1
Hide problems
tetrahedron, minimize expression gives side lengths
A tetrahedron
A
B
C
D
ABCD
A
BC
D
satisfies the following conditions: the edges
A
B
,
A
C
AB,AC
A
B
,
A
C
and
A
D
AD
A
D
are pairwise orthogonal,
A
B
=
3
AB=3
A
B
=
3
and
C
D
=
2
CD=\sqrt2
C
D
=
2
. Find the minimum possible value of
B
C
6
+
B
D
6
−
A
C
6
−
A
D
6
.
BC^6+BD^6-AC^6-AD^6.
B
C
6
+
B
D
6
−
A
C
6
−
A
D
6
.