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Problems
Contests
National and Regional Contests
France Contests
French Mathematical Olympiad
2001 French Mathematical Olympiad
2001 French Mathematical Olympiad
Part of
French Mathematical Olympiad
Subcontests
(1)
1
Hide problems
ab+bc+ca=0, u^2+3v^2 integers
A trio is any triple
(
a
,
b
,
c
)
(a,b,c)
(
a
,
b
,
c
)
of nonzero real numbers satisfying
a
b
+
b
c
+
c
a
=
0
ab+bc+ca=0
ab
+
b
c
+
c
a
=
0
. A triple is said to be reduced if
a
+
b
+
c
=
1
a+b+c=1
a
+
b
+
c
=
1
.[hide=Part 1] We denote by
C
C
C
the set of points
(
x
,
y
,
z
)
(x,y,z)
(
x
,
y
,
z
)
in coordinate space for which
(
x
,
y
,
z
)
(x,y,z)
(
x
,
y
,
z
)
is a trio, and by
Γ
\Gamma
Γ
the set of those for which
(
x
,
y
,
z
)
(x,y,z)
(
x
,
y
,
z
)
is a reduced trio. Let
O
O
O
be the origin and
P
P
P
be the plane given by
x
+
y
+
z
=
1
x+y+z=1
x
+
y
+
z
=
1
.(a) Does there exist a trio
(
a
,
b
,
c
)
(a,b,c)
(
a
,
b
,
c
)
such that
a
+
b
+
c
=
0
a+b+c=0
a
+
b
+
c
=
0
? (b) Prove that
C
C
C
is a union of lines passing through
O
O
O
, with
O
O
O
excluded. (c) Prove that
Γ
\Gamma
Γ
is the intersection of a plane and a sphere with center
O
O
O
. Describe
Γ
\Gamma
Γ
geometrically. (d) Describe
C
C
C
geometrically and sketch it. (e) Let
L
L
L
be a fixed point in
Γ
\Gamma
Γ
. If
L
’
L’
L
’
and
L
′
′
L''
L
′′
are arbitrary points on
Γ
\Gamma
Γ
, prove that the volume
V
V
V
of the tetrahedron
O
L
L
’
L
′
′
OLL’L''
O
LL
’
L
′′
is maximal when the lines
O
L
,
O
L
’
,
O
L
′
′
OL,OL’,OL''
O
L
,
O
L
’
,
O
L
′′
are orthogonal, and express the coordinates of
L
’
L’
L
’
and
L
′
′
L''
L
′′
in terms of those of
L
L
L
. (f) Prove that the product
a
b
c
abc
ab
c
attains its maximum and minimum values on
Γ
\Gamma
Γ
, and find the points at which those are attained.[hide=Part 2] A trio
(
a
,
b
,
c
)
(a,b,c)
(
a
,
b
,
c
)
is called rational if
a
,
b
,
c
a,b,c
a
,
b
,
c
are rational, and integer if
a
,
b
,
c
a,b,c
a
,
b
,
c
are integers. We say that an integer trio
(
a
,
b
,
c
)
(a,b,c)
(
a
,
b
,
c
)
is primitive if the greatest common divisor of
a
,
b
,
c
a,b,c
a
,
b
,
c
is
1
1
1
.(a) Describe the set
H
1
H_1
H
1
of points
(
x
,
y
,
1
)
(x,y,1)
(
x
,
y
,
1
)
such that
(
a
,
b
,
1
)
(a,b,1)
(
a
,
b
,
1
)
is a trio. Show that the point
Ω
1
(
−
1
,
−
1
,
1
)
\Omega_1(-1,-1,1)
Ω
1
(
−
1
,
−
1
,
1
)
is the center of symmetry of
H
1
H_1
H
1
. Find all points of
H
1
H_1
H
1
with integer coordinates. (b) For each nonzero integer
h
h
h
, denote by
Z
h
Z_h
Z
h
the set of integer trios
(
a
,
b
,
c
)
(a,b,c)
(
a
,
b
,
c
)
with
c
=
h
c=h
c
=
h
. Determine
Z
h
Z_h
Z
h
for
h
=
1
h=1
h
=
1
and
h
=
2
h=2
h
=
2
(c) Prove that
Z
h
Z_h
Z
h
is a finite set and find the number
N
(
h
)
N(h)
N
(
h
)
of its elements in terms of the number of divisors of
h
2
h^2
h
2
in
Z
\mathbb Z
Z
. Prove that
4
4
4
divides
N
(
h
)
−
2
N(h)-2
N
(
h
)
−
2
. (d) For every positive integer
h
h
h
, denote by
N
’
(
h
)
N’(h)
N
’
(
h
)
the number of integer trios
(
a
,
b
,
c
)
(a,b,c)
(
a
,
b
,
c
)
such that at least one of
a
,
b
,
c
a,b,c
a
,
b
,
c
is equal to
h
h
h
. Express
N
’
(
h
)
N’(h)
N
’
(
h
)
in terms of
N
(
h
)
N(h)
N
(
h
)
depending on the parity of
h
h
h
. (e) Prove that every integer trio
(
a
,
b
,
c
)
(a,b,c)
(
a
,
b
,
c
)
can be assigned a triple of integers
(
r
,
s
,
t
)
(r,s,t)
(
r
,
s
,
t
)
such that
r
r
r
and
s
s
s
are coprime,
s
s
s
is nonnegative, and
a
=
r
(
r
+
s
)
t
,
b
=
s
(
r
+
s
)
t
,
c
=
−
r
s
t
.
a=r(r+s)t,\qquad b=s(r+s)t,\qquad c=-rst.
a
=
r
(
r
+
s
)
t
,
b
=
s
(
r
+
s
)
t
,
c
=
−
rs
t
.
State and verify the converse. For which trios
(
a
,
b
,
c
)
(a,b,c)
(
a
,
b
,
c
)
is the triple
(
r
,
s
,
t
)
(r,s,t)
(
r
,
s
,
t
)
not unique? (f) Determine all triples
(
r
,
s
,
t
)
(r,s,t)
(
r
,
s
,
t
)
that are assigned to some primitive trios. Deduce that if
(
a
,
b
,
c
)
(a,b,c)
(
a
,
b
,
c
)
is a primitive trio, then
∣
a
b
c
∣
|abc|
∣
ab
c
∣
,
∣
a
+
b
∣
|a+b|
∣
a
+
b
∣
,
∣
b
+
c
∣
|b+c|
∣
b
+
c
∣
and
∣
c
+
a
∣
|c+a|
∣
c
+
a
∣
are perfect squares. (g) For each positive integer
h
h
h
, denote by
P
(
h
)
P(h)
P
(
h
)
the number of primitive trios
(
a
,
b
,
c
)
(a,b,c)
(
a
,
b
,
c
)
with
c
=
h
c=h
c
=
h
. Prove that
P
(
h
)
P(h)
P
(
h
)
is a power of
2
2
2
. For which
h
h
h
is
P
(
h
)
=
N
(
h
)
P(h)=N(h)
P
(
h
)
=
N
(
h
)
? Give a sequence of integers
(
h
n
)
(h_n)
(
h
n
)
for which the sequence
P
(
h
n
)
N
(
h
n
)
\frac{P(h_n)}{N(h_n)}
N
(
h
n
)
P
(
h
n
)
converges to zero. (h) Let
(
a
,
b
,
1
)
(a,b,1)
(
a
,
b
,
1
)
be a trio. Show that there exist sequence
(
x
n
)
,
(
y
n
)
(x_n),(y_n)
(
x
n
)
,
(
y
n
)
, and
(
z
n
)
(z_n)
(
z
n
)
converging respectively to
a
,
b
a,b
a
,
b
, and
c
c
c
such that
(
x
n
,
y
n
,
1
)
(x_n,y_n,1)
(
x
n
,
y
n
,
1
)
is a rational trio for all
n
n
n
. (i) Let
(
a
,
b
,
1
)
(a,b,1)
(
a
,
b
,
1
)
be a reduced trio. Show that there exist sequence
(
x
n
)
,
(
y
n
)
(x_n),(y_n)
(
x
n
)
,
(
y
n
)
, and
(
z
n
)
(z_n)
(
z
n
)
converging respectively to
a
,
b
a,b
a
,
b
, and
c
c
c
such that
(
x
n
,
y
n
,
z
n
)
(x_n,y_n,z_n)
(
x
n
,
y
n
,
z
n
)
is a rational reduced trio for all
n
n
n
.[hide=Part 3] Denote
j
=
e
2
i
π
/
3
=
−
1
2
+
i
3
2
j=e^{2i\pi/3}=-\frac12+i\frac{\sqrt3}2
j
=
e
2
iπ
/3
=
−
2
1
+
i
2
3
. For each trio
T
=
(
a
,
b
,
c
)
T=(a,b,c)
T
=
(
a
,
b
,
c
)
we define
T
=
(
a
,
c
,
b
)
\mathcal T=(a,c,b)
T
=
(
a
,
c
,
b
)
,
S
(
T
)
=
a
+
b
+
c
S(T)=a+b+c
S
(
T
)
=
a
+
b
+
c
and
z
(
T
)
=
a
+
b
j
+
c
j
2
z(T)=a+bj+cj^2
z
(
T
)
=
a
+
bj
+
c
j
2
.(a) Express the modulus of
z
(
T
)
z(T)
z
(
T
)
as a function of
S
(
T
)
S(T)
S
(
T
)
. Can we have
z
(
T
)
=
0
z(T)=0
z
(
T
)
=
0
? Compute the sine and the cosine of the argument
θ
\theta
θ
of
z
(
T
)
z(T)
z
(
T
)
in terms of
a
,
b
,
c
a,b,c
a
,
b
,
c
. (b) Let
z
0
z_0
z
0
be a given nonzero complex number. Find all trios
T
=
(
a
,
b
,
c
)
T=(a,b,c)
T
=
(
a
,
b
,
c
)
such that
z
(
T
)
=
z
0
z(T)=z_0
z
(
T
)
=
z
0
. (c) Given trios
T
1
T_1
T
1
and
T
2
T_2
T
2
, prove that there is a unique trio, to be denoted as
T
1
∗
T
2
T_1*T_2
T
1
∗
T
2
, satisfying
S
(
T
1
∗
T
2
)
=
S
(
T
1
)
S
(
T
2
)
S(T_1*T_2)=S(T_1)S(T_2)
S
(
T
1
∗
T
2
)
=
S
(
T
1
)
S
(
T
2
)
and
z
(
T
1
∗
T
2
)
=
z
(
T
1
)
z
(
T
2
)
z(T_1*T_2)=z(T_1)z(T_2)
z
(
T
1
∗
T
2
)
=
z
(
T
1
)
z
(
T
2
)
. Compute
T
1
∗
T
2
T_1*T_2
T
1
∗
T
2
in terms of
T
1
T_1
T
1
and
T
2
T_2
T
2
. What can be said about the argument of
z
(
T
1
∗
T
2
)
z(T_1*T_2)
z
(
T
1
∗
T
2
)
? What can be said about
z
(
T
1
∗
T
1
)
z(T_1*\mathcal T_1)
z
(
T
1
∗
T
1
)
? (d) If
T
1
T_1
T
1
and
T
2
T_2
T
2
are reduced trios, is
T
1
∗
T
2
T_1*T_2
T
1
∗
T
2
? The same question if the word ”reduced” is replaced by ”integer” and by ”primitive”. (e) Compare the trios
T
1
∗
T
2
T_1*T_2
T
1
∗
T
2
and
T
2
∗
T
1
T_2*T_1
T
2
∗
T
1
;
T
1
∗
(
T
2
∗
T
3
)
T_1*(T_2*T_3)
T
1
∗
(
T
2
∗
T
3
)
and
(
T
1
∗
T
2
)
∗
T
3
(T_1*T_2)*T_3
(
T
1
∗
T
2
)
∗
T
3
,
T
1
T_1
T
1
and
T
1
∗
(
1
,
0
,
0
)
T_1*(1,0,0)
T
1
∗
(
1
,
0
,
0
)
. (f) Given trios
T
1
T_1
T
1
and
T
2
T_2
T
2
, solve the equation
T
1
∗
T
=
T
2
T_1*T=T_2
T
1
∗
T
=
T
2
in
T
T
T
. (g) Given a trio
T
T
T
, define the sequence of trios
(
T
n
)
(T_n)
(
T
n
)
by
T
0
=
(
1
,
0
,
0
)
T_0=(1,0,0)
T
0
=
(
1
,
0
,
0
)
and
T
n
+
1
=
T
∗
T
n
T_{n+1}=T*T_n
T
n
+
1
=
T
∗
T
n
. Calculate
S
(
T
n
)
S(T_n)
S
(
T
n
)
. Given an integer
p
p
p
, find all
T
T
T
for which
T
p
=
T
0
T_p=T_0
T
p
=
T
0
.[hide=Part 4] Denote by
A
A
A
the set of integers
m
m
m
that are of the form
u
2
+
3
v
2
u^2+3v^2
u
2
+
3
v
2
, where
u
,
v
u,v
u
,
v
are integers. Denote by
A
’
A’
A
’
the set of nonzero complex numbers
z
=
u
+
i
v
3
z=u+iv\sqrt3
z
=
u
+
i
v
3
, where
u
,
v
u,v
u
,
v
are integers (note that
∣
z
∣
2
=
u
2
+
3
v
2
|z|^2=u^2+3v^2
∣
z
∣
2
=
u
2
+
3
v
2
). Denote by
B
B
B
the set of nonzero integers
n
n
n
of the form
r
2
+
r
s
+
s
2
r^2+rs+s^2
r
2
+
rs
+
s
2
, where
r
,
s
r,s
r
,
s
are integers.(a) Prove that a product of two elements of
A
’
A’
A
’
belongs to
A
’
A’
A
’
, and that a product of two elements of
A
A
A
belongs to
A
A
A
. (b) Show that if
p
∈
A
p\in A
p
∈
A
is a prime number, then
p
=
3
p=3
p
=
3
or
3
∣
p
−
1
3\mid p-1
3
∣
p
−
1
. (c) Prove that
A
=
B
A=B
A
=
B
(you may note that
r
2
+
r
s
+
s
2
=
(
r
+
s
)
2
−
(
r
+
s
)
s
+
s
2
r^2+rs+s^2=(r+s)^2-(r+s)s+s^2
r
2
+
rs
+
s
2
=
(
r
+
s
)
2
−
(
r
+
s
)
s
+
s
2
). (d) Prove that every even element of
A
A
A
is divisible by
4
4
4
and that its quarter belongs to
A
A
A
; then prove that each element of
A
A
A
is the product of a power of
4
4
4
and an odd element of
A
A
A
. (e) i. Suppose that there is an odd integer
m
=
u
2
+
3
v
2
m=u^2+3v^2
m
=
u
2
+
3
v
2
, where
u
,
v
u,v
u
,
v
are coprime integers, and there exists a prime divisor
p
p
p
of
m
m
m
not belonging to
A
A
A
. Prove that there exists the smallest positive integer
n
0
n_0
n
0
such that
n
0
p
n_0p
n
0
p
is in
A
A
A
, and that
n
0
n_0
n
0
is odd. ii. Verify the existence of integers
u
’
,
v
’
u’,v’
u
’
,
v
’
less than
p
2
\frac p2
2
p
in absolute value such that
u
−
u
’
u-u’
u
−
u
’
and
v
−
v
’
v-v’
v
−
v
’
are divisible by
p
p
p
. Prove that
p
p
p
divides the nonzero number
u
’
2
+
3
v
’
2
u’^2+3v’^2
u
’
2
+
3
v
’
2
and hence that
n
0
<
p
n_0<p
n
0
<
p
. iii. Verify the existence of coprime nonzero integers
u
0
,
v
0
u_0,v_0
u
0
,
v
0
such that
n
0
p
=
u
0
2
+
3
v
0
2
n_0p=u_0^2+3v_0^2
n
0
p
=
u
0
2
+
3
v
0
2
. iv. Verify the existence of integers
u
1
,
v
1
u_1,v_1
u
1
,
v
1
less than
n
0
2
\frac{n_0}2
2
n
0
in absolute value such that
u
1
−
u
0
u_1-u_0
u
1
−
u
0
and
v
1
−
v
0
v_1-v_0
v
1
−
v
0
are divisible by
n
n
n
. Prove that
n
0
n_0
n
0
divides the nonzero integer
u
1
2
+
3
v
1
2
u_1^2+3v_1^2
u
1
2
+
3
v
1
2
which we’ll denote by
n
0
n
1
n_0n_1
n
0
n
1
. v. Deduce that such an integer
m
m
m
cannot exist (you may consider the number
n
0
2
n
1
p
n_0^2n_1p
n
0
2
n
1
p
). (f) Prove that every element of
A
A
A
can be written in the form
m
=
C
2
p
1
⋯
p
k
m=C^2p_1\cdots p_k
m
=
C
2
p
1
⋯
p
k
, where
C
C
C
is a positive integer and
p
i
p_i
p
i
distinct prime factors of
A
A
A
. (g) i. Let
p
p
p
be a prime number with
3
∣
p
−
1
3\mid p-1
3
∣
p
−
1
, and
K
K
K
the set of triples
(
x
,
y
,
z
)
(x,y,z)
(
x
,
y
,
z
)
of integers with
0
<
x
,
y
,
z
<
p
0<x,y,z<p
0
<
x
,
y
,
z
<
p
such that
p
∣
x
y
z
−
1
p\mid xyz-1
p
∣
x
yz
−
1
. Prove that
k
k
k
has exactly
(
p
−
1
)
2
(p-1)^2
(
p
−
1
)
2
elements and that the number of those with
x
,
y
,
z
x,y,z
x
,
y
,
z
not equal is divisible by
3
3
3
. (h) Let
D
D
D
be the set of integers
d
d
d
for which there is an integer trio
(
a
,
b
,
c
)
(a,b,c)
(
a
,
b
,
c
)
satisfying
a
+
b
+
c
=
d
a + b + c = d
a
+
b
+
c
=
d
and
a
b
c
≠
0
abc\ne 0
ab
c
=
0
. Prove, using question (e) of part
2
2
2
, that every element of
D
D
D
has a prime divisor in
A
A
A
. Conversely, what can be said about the nonzero integers having a prime divisor in
A
A
A
? (i) Find the elements of
D
D
D
between
2001
2001
2001
and
2010
2010
2010
inclusive.