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Problems
Contests
National and Regional Contests
France Contests
French Mathematical Olympiad
1988 French Mathematical Olympiad
1988 French Mathematical Olympiad
Part of
French Mathematical Olympiad
Subcontests
(4)
Problem 4
1
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circle with 2n+1 points, product of distances
A circle
C
\mathcal C
C
and five distinct points
M
1
,
M
2
,
M
3
,
M
4
M_1,M_2,M_3,M_4
M
1
,
M
2
,
M
3
,
M
4
and
M
M
M
on
C
\mathcal C
C
are given in the plane. Prove that the product of the distances from
M
M
M
to lines
M
1
M
2
M_1M_2
M
1
M
2
and
M
3
M
4
M_3M_4
M
3
M
4
is equal to the product of the distances from
M
M
M
to the lines
M
1
M
3
M_1M_3
M
1
M
3
and
M
2
M
4
M_2M_4
M
2
M
4
. What can one deduce for
2
n
+
1
2n+1
2
n
+
1
distinct points
M
1
,
…
,
M
2
n
,
M
M_1,\ldots,M_{2n},M
M
1
,
…
,
M
2
n
,
M
on
C
\mathcal C
C
?
Problem 3
1
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tangents to sphere, minimum length
Consider two spheres
Σ
1
\Sigma_1
Σ
1
and
Σ
2
\Sigma_2
Σ
2
and a line
Δ
\Delta
Δ
not meeting them. Let
C
i
C_i
C
i
and
r
i
r_i
r
i
be the center and radius of
Σ
i
\Sigma_i
Σ
i
, and let
H
i
H_i
H
i
and
d
i
d_i
d
i
be the orthogonal projection of
C
i
C_i
C
i
onto
Δ
\Delta
Δ
and the distance of
C
i
C_i
C
i
from
Δ
(
i
=
1
,
2
)
\Delta~(i=1,2)
Δ
(
i
=
1
,
2
)
. For a point
M
M
M
on
Δ
\Delta
Δ
, let
δ
i
(
M
)
\delta_i(M)
δ
i
(
M
)
be the length of a tangent
M
T
i
MT_i
M
T
i
to
Σ
i
\Sigma_i
Σ
i
, where
T
i
∈
Σ
i
(
i
=
1
,
2
)
T_i\in\Sigma_i~(i=1,2)
T
i
∈
Σ
i
(
i
=
1
,
2
)
. Find
M
M
M
on
Δ
\Delta
Δ
for which
δ
1
(
M
)
+
δ
2
(
M
)
\delta_1(M)+\delta_2(M)
δ
1
(
M
)
+
δ
2
(
M
)
is minimal.
Problem 2
1
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on n^6+5n^5*sin(n)+1
For each
n
∈
N
n\in\mathbb N
n
∈
N
, determine the sign of
n
6
+
5
n
5
sin
n
+
1
n^6+5n^5\sin n+1
n
6
+
5
n
5
sin
n
+
1
. For which
n
∈
N
n\in\mathbb N
n
∈
N
does it hold that
n
2
+
5
n
cos
n
+
1
n
6
+
5
n
5
sin
n
+
1
≥
1
0
−
4
\frac{n^2+5n\cos n+1}{n^6+5n^5\sin n+1}\ge10^{-4}
n
6
+
5
n
5
s
i
n
n
+
1
n
2
+
5
n
c
o
s
n
+
1
≥
1
0
−
4
.
Problem 1
1
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matrix of n rows, conditions hold
Let us consider a matrix
T
T
T
with n rows denoted
1
,
…
,
n
1,\ldots,n
1
,
…
,
n
and
p
p
p
columns
1
,
…
,
p
1,\ldots,p
1
,
…
,
p
. Its entries
a
i
k
(
1
≤
i
≤
n
,
1
≤
k
≤
p
)
a_{ik}~(1\le i\le n,1\le k\le p)
a
ik
(
1
≤
i
≤
n
,
1
≤
k
≤
p
)
are integers such that
1
≤
a
i
k
≤
N
1\le a_{ik}\le N
1
≤
a
ik
≤
N
, where
N
N
N
is a given natural number. Let
E
i
E_i
E
i
be the set of numbers that appear on the
i
i
i
-th row. Answer question (a) or (b).(a) Assume
T
T
T
satisfies the following conditions:
(
1
)
(1)
(
1
)
E
i
E_i
E
i
has exactly
p
p
p
elements for each
i
i
i
, and
(
2
)
(2)
(
2
)
all
E
i
E_i
E
i
's are mutually distinct. Let
m
m
m
be the smallest value of
N
N
N
that permits a construction of such an
n
×
p
n\times p
n
×
p
table
T
T
T
. i. Compute
m
m
m
if
n
=
p
+
1
n=p+1
n
=
p
+
1
. ii. Compute
m
m
m
if
n
=
1
0
30
n=10^{30}
n
=
1
0
30
and
p
=
1998
p=1998
p
=
1998
. iii. Determine
lim
n
→
∞
m
p
n
\lim_{n\to\infty}\frac{m^p}n
lim
n
→
∞
n
m
p
, where
p
p
p
is fixed.(b) Assume
T
T
T
satisfies the following conditions instead:
(
1
)
(1)
(
1
)
p
=
n
p=n
p
=
n
,
(
2
)
(2)
(
2
)
whenever
i
,
k
i,k
i
,
k
are integers with
i
+
k
≤
n
i+k\le n
i
+
k
≤
n
, the number
a
i
k
a_{ik}
a
ik
is not in the set
E
i
+
k
E_{i+k}
E
i
+
k
. i. Prove that all
E
i
E_i
E
i
's are mutually distinct. ii. Prove that if
n
≥
2
q
n\ge2^q
n
≥
2
q
for some integer
q
>
0
q>0
q
>
0
, then
N
≥
q
+
1
N\ge q+1
N
≥
q
+
1
. iii. Let
n
=
2
r
−
1
n=2^r-1
n
=
2
r
−
1
for some integer
r
>
0
r>0
r
>
0
. Prove that
N
≥
r
N\ge r
N
≥
r
and show that there is such a table with
N
=
r
N=r
N
=
r
.