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Problems
Contests
National and Regional Contests
France Contests
French Mathematical Olympiad
1994 French Mathematical Olympiad
1994 French Mathematical Olympiad
Part of
French Mathematical Olympiad
Subcontests
(4)
Problem 5
1
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f(m^2+n^2)=f(m)^2+f(n)^2 over N
Assume
f
:
N
0
→
N
0
f:\mathbb N_0\to\mathbb N_0
f
:
N
0
→
N
0
is a function such that
f
(
1
)
>
0
f(1)>0
f
(
1
)
>
0
and, for any nonnegative integers
m
m
m
and
n
n
n
,
f
(
m
2
+
n
2
)
=
f
(
m
)
2
+
f
(
n
)
2
.
f\left(m^2+n^2\right)=f(m)^2+f(n)^2.
f
(
m
2
+
n
2
)
=
f
(
m
)
2
+
f
(
n
)
2
.
(a) Calculate
f
(
k
)
f(k)
f
(
k
)
for
0
≤
k
≤
12
0\le k\le12
0
≤
k
≤
12
. (b) Calculate
f
(
n
)
f(n)
f
(
n
)
for any natural number
n
n
n
.
Problem 3
1
Hide problems
f(2n)=2f(n)+1,f(2n+1)=2f(n)
Let us define a function
f
:
N
→
N
0
f:\mathbb N\to\mathbb N_0
f
:
N
→
N
0
by
f
(
1
)
=
0
f(1)=0
f
(
1
)
=
0
and, for all
n
∈
N
n\in\mathbb N
n
∈
N
,
f
(
2
n
)
=
2
f
(
n
)
+
1
,
f
(
2
n
+
1
)
=
2
f
(
n
)
.
f(2n)=2f(n)+1,\qquad f(2n+1)=2f(n).
f
(
2
n
)
=
2
f
(
n
)
+
1
,
f
(
2
n
+
1
)
=
2
f
(
n
)
.
Given a positive integer
p
p
p
, define a sequence
(
u
n
)
(u_n)
(
u
n
)
by
u
0
=
p
u_0=p
u
0
=
p
and
u
k
+
1
=
f
(
u
k
)
u_{k+1}=f(u_k)
u
k
+
1
=
f
(
u
k
)
whenever
u
k
≠
0
u_k\ne0
u
k
=
0
.(a) Prove that, for each
p
∈
N
p\in\mathbb N
p
∈
N
, there is a unique integer
v
(
p
)
v(p)
v
(
p
)
such that
u
v
(
p
)
=
0
u_{v(p)}=0
u
v
(
p
)
=
0
. (b) Compute
v
(
1994
)
v(1994)
v
(
1994
)
. What is the smallest integer
p
>
0
p>0
p
>
0
for which
v
(
p
)
=
v
(
1994
)
v(p)=v(1994)
v
(
p
)
=
v
(
1994
)
. (c) Given an integer
N
N
N
, determine the smallest integer
p
p
p
such that
v
(
p
)
=
N
v(p)=N
v
(
p
)
=
N
.
Problem 2
1
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maximize cylinder volume
Let be given a semi-sphere
Σ
\Sigma
Σ
whose base-circle lies on plane
p
p
p
. A variable plane
Q
Q
Q
, parallel to a fixed plane non-perpendicular to
P
P
P
, cuts
Σ
\Sigma
Σ
at a circle
C
C
C
. We denote by
C
′
C'
C
′
the orthogonal projection of
C
C
C
onto
P
P
P
. Find the position of
Q
Q
Q
for which the cylinder with bases
C
C
C
and
C
′
C'
C
′
has the maximum volume.
Problem 1
1
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# of n for which 50^n<7^p<50^(p+1) forms a sequence
For each positive integer
n
n
n
, let
I
n
I_n
I
n
denote the number of integers
p
p
p
for which
5
0
n
<
7
p
<
5
0
n
+
1
50^n<7^p<50^{n+1}
5
0
n
<
7
p
<
5
0
n
+
1
.(a) Prove that, for each
n
n
n
,
I
n
I_n
I
n
is either
2
2
2
or
3
3
3
. (b) Prove that
I
n
=
3
I_n=3
I
n
=
3
for infinitely many
n
∈
N
n\in\mathbb N
n
∈
N
, and find at least one such
n
n
n
.