MathDB
Problems
Contests
National and Regional Contests
France Contests
French Mathematical Olympiad
1989 French Mathematical Olympiad
1989 French Mathematical Olympiad
Part of
French Mathematical Olympiad
Subcontests
(5)
Problem 5
1
Hide problems
sum a_k and sum(a_k)^(1-1/k)
Let
a
1
,
a
2
,
…
,
a
n
a_1,a_2,\ldots,a_n
a
1
,
a
2
,
…
,
a
n
be positive real numbers. Denote
s
=
∑
k
=
1
n
a
k
and
s
′
=
∑
k
=
1
n
a
k
1
−
1
k
.
s=\sum_{k=1}^na_k\text{ and }s'=\sum_{k=1}^na_k^{1-\frac1k}.
s
=
k
=
1
∑
n
a
k
and
s
′
=
k
=
1
∑
n
a
k
1
−
k
1
.
(a) Let
λ
>
1
\lambda>1
λ
>
1
be a real number. Show that
s
′
<
λ
s
+
λ
λ
−
1
s'<\lambda s+\frac\lambda{\lambda-1}
s
′
<
λ
s
+
λ
−
1
λ
. (b) Deduce that
s
′
<
s
+
1
\sqrt{s'}<\sqrt s+1
s
′
<
s
+
1
.
Problem 4
1
Hide problems
recurrence operation, exponential
For natural numbers
x
1
,
…
,
x
k
x_1,\ldots,x_k
x
1
,
…
,
x
k
, let
[
x
k
,
…
,
x
1
]
[x_k,\ldots,x_1]
[
x
k
,
…
,
x
1
]
be defined recurrently as follows:
[
x
2
,
x
1
]
=
x
2
x
1
[x_2,x_1]=x_2^{x_1}
[
x
2
,
x
1
]
=
x
2
x
1
and, for
k
≥
3
k\ge3
k
≥
3
,
[
x
k
,
x
k
−
1
,
…
,
x
1
]
=
x
k
[
x
k
−
1
,
…
,
x
1
]
[x_k,x_{k-1},\ldots,x_1]=x_k^{[x_{k-1},\ldots,x_1]}
[
x
k
,
x
k
−
1
,
…
,
x
1
]
=
x
k
[
x
k
−
1
,
…
,
x
1
]
.(a) Let
3
≤
a
1
≤
a
2
≤
…
≤
a
n
3\le a_1\le a_2\le\ldots\le a_n
3
≤
a
1
≤
a
2
≤
…
≤
a
n
be integers. For a permutation
σ
\sigma
σ
of the set
{
1
,
2
,
…
,
n
}
\{1,2,\ldots,n\}
{
1
,
2
,
…
,
n
}
, we set
P
(
σ
)
=
[
a
σ
(
n
)
,
a
σ
(
n
−
1
)
,
…
,
a
σ
(
1
)
]
P(\sigma)=[a_{\sigma(n)},a_{\sigma(n-1)},\ldots,a_{\sigma(1)}]
P
(
σ
)
=
[
a
σ
(
n
)
,
a
σ
(
n
−
1
)
,
…
,
a
σ
(
1
)
]
. Find the permutations
σ
\sigma
σ
for which
P
(
σ
)
P(\sigma)
P
(
σ
)
is minimal or maximal. (b) Find all integers
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
, greater than or equal to
2
2
2
, for which
[
178
,
9
]
≤
[
a
,
b
,
c
,
d
]
≤
[
198
,
9
]
[178,9]\le[a,b,c,d]\le[198,9]
[
178
,
9
]
≤
[
a
,
b
,
c
,
d
]
≤
[
198
,
9
]
.
Problem 3
1
Hide problems
tetrahedron inequality, with parameter
Find the greatest real
k
k
k
such that, for every tetrahedron
A
B
C
D
ABCD
A
BC
D
of volume
V
V
V
, the product of areas of faces
A
B
C
,
A
B
D
ABC,ABD
A
BC
,
A
B
D
and
A
C
D
ACD
A
C
D
is at least
k
V
2
kV^2
k
V
2
.
Problem 2
1
Hide problems
complex numbers form trapezoid (a), equality of locuses (b)
(a) Let
z
1
,
z
2
z_1,z_2
z
1
,
z
2
be complex numbers such that
z
1
z
2
=
1
z_1z_2=1
z
1
z
2
=
1
and
∣
z
1
−
z
2
∣
=
2
|z_1-z_2|=2
∣
z
1
−
z
2
∣
=
2
. Let
A
,
B
,
M
1
,
M
2
A,B,M_1,M_2
A
,
B
,
M
1
,
M
2
denote the points in complex plane corresponding to
−
1
,
1
,
z
1
,
z
2
-1,1,z_1,z_2
−
1
,
1
,
z
1
,
z
2
, respectively. Show that
A
M
1
B
M
2
AM_1BM_2
A
M
1
B
M
2
is a trapezoid and compute the lengths of its non-parallel sides. Specify the particular cases. (b) Let
C
1
\mathcal C_1
C
1
and
C
2
\mathcal C_2
C
2
be circles in the plane with centers
O
1
O_1
O
1
and
O
2
O_2
O
2
, respectively, and with radius
d
2
d\sqrt2
d
2
, where
2
d
=
O
1
O
2
2d=O_1O_2
2
d
=
O
1
O
2
. Let
P
P
P
and
Q
Q
Q
be two variable points on
C
1
\mathcal C_1
C
1
and
C
2
\mathcal C_2
C
2
respectively, both on
O
1
O
2
O_1O_2
O
1
O
2
on on different sides of
O
1
O
2
O_1O_2
O
1
O
2
, such that
P
Q
=
2
d
PQ=2d
PQ
=
2
d
. Prove that the locus of midpoints
I
I
I
of segments
P
Q
PQ
PQ
is the same as the locus of points
M
M
M
with
M
O
1
⋅
M
O
2
=
m
MO_1\cdot MO_2=m
M
O
1
⋅
M
O
2
=
m
for some
m
m
m
.
Problem 1
1
Hide problems
central symmetries of plane figures
Given a figure
B
B
B
in the plane, consider the figures
A
A
A
, containing
B
\mathcal B
B
, with the property
(
P
)
(P)
(
P
)
: a composition of an odd number of central symmetries with centers in
A
A
A
is also a central symmetry with center in
A
A
A
. The task of this problem is to determine the smallest such figure, denoted by
A
\mathcal A
A
, that is contained in every figure
A
A
A
.(a) Determine the figure
A
\mathcal A
A
if
B
B
B
consists of:
(
1
)
(1)
(
1
)
two distinct points
I
,
J
I,J
I
,
J
;
(
2
)
(2)
(
2
)
three non-collinear points
I
,
J
,
K
I,J,K
I
,
J
,
K
. (b) Determine
A
\mathcal A
A
if
B
B
B
is a circle (with nonzero radius). (c) Give some examples of figures
B
B
B
whose associated figures
A
\mathcal A
A
are mutually distinct and distinct from the above ones.