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Grade 12 - Exp.
/ Annals on matrices
15 exercises (including 13 corrected)
a
Matrices and sequences
(1 exercice)
Transition matrix
(3 exercices)
Transition matrix type: X=AX+B
(1 exercice)
Matrices and arithmetic
(1 exercice)
O
E
F
G
E
F
G
O
E
F
G
1.
Matric
es
and
se
quenc
es
E.5954
Soft
w
are
can
b
e
used
to
transform
a
rectangular
el-
emen
t
in
a
photograph.
Th
us,
the
initial
rectan-
gle
O
E
F
G
is
transformed
in
to
a
rectangle
O
E
F
G
,
called
the
O
E
F
G
image.
The
purp
ose
of
this
exer-
cise
is
to
study
the
rectan-
gle
ob-
held
after
sev
eral
successiv
e
transformations.
P
art
A
The
plane
is
referred
to
an
orthonormal
frame
of
reference
O
;
−
→
i
;
−
→
j
.
The
p
oin
ts
E
,
F
and
G
ha
v
e
resp
ectiv
e
co
ordinates
(
2
;
2
)
,
(
−
1
;
5
)
et
(
−
3
;
3
)
.
The
soft
w
are
transformation
asso-
ciates
to
an
y
p
oin
t
M
(
x
;
y
)
of
the
plane
the
p
oin
t
M
(
x
;
y
)
,
image
of
the
p
oin
t
M
suc
h
that
:
x
=
5
4
x
+
3
4
y
y
=
3
4
x
+
5
4
y
1
a
Calculate
the
co
ordinates
E
,
F
and
G
,
images
of
the
p
oin
ts
E
,
F
and
G
b
y
this
transformation.
b
Compare
the
lengths
O
E
and
O
E
on
the
one
hand,
O
G
and
O
G
on
the
other.
Giv
e
the
square
matrix
of
order
2
,
denoted
A
,
suc
h
that
:
x
y
=
A
·
x
y
P
art
B
In
this
part,
w
e
study
the
co
ordinates
of
successiv
e
images
of
v
ertex
F
of
rectangle
O
E
F
G
when
the
soft
w
are
transforma-
tion
is
applied
sev
eral
times.
1
Consider
the
follo
wing
algorithm
designed
to
displa
y
the
co
ordinates
of
these
successiv
e
images.
An
error
has
b
een
made.
Mo
dify
this
algorithm
so
that
it
displa
ys
these
co
ordi-
nates
:
Input
En
ter
a
non-zero
natural
n
um
b
er
N
In
tialization
Assign
x
the
v
alue
−
1
Assign
to
y
the
v
alue
5
Pro
cessing
POUR
i
RANGING
FR
OM
1
TO
N
Assign
to
a
the
v
alue
5
4
x
+
3
4
y
Assign
to
b
the
v
alue
3
4
x
+
5
4
y
Assign
to
x
the
v
alue
a
Assign
to
y
the
v
alue
b
FIN
F
OR
Output
Displa
y
x
,
displa
y
y
2
The
follo
wing
table
is
obtained
:
i
1
2
3
4
5
10
15
x
2.5
7.25
15.625
31.8125
63.9063
2047.9971
65535.9999
y
5.5
8.75
16.375
32.1875
64.0938
2048.0029
65536.0001
Conjecture
the
b
eha
vior
of
the
sequence
of
successiv
e
im-
ages
of
the
p
oin
t
F
.
P
art
C
In
this
part,
w
e
study
the
co
ordinates
of
the
successiv
e
im-
ages
of
the
v
ertex
E
of
the
rectangle
O
E
F
G
.
W
e
define
the
sequence
of
p
oin
ts
E
n
(
x
n
;
y
n
)
of
the
plane
b
y
E
0
=
E
and
the
recurrence
relation
:
x
n
+1
y
n
+1
=
A
·
x
n
y
n
où
(
x
n
+
;
y
n
+1
)
denote
the
co
ordinates
of
the
p
oin
t
E
n
+1
.
Th
us,
x
0
=
2
and
y
0
=
2
.
1
W
e
admit
that,
for
an
y
in
teger
n
1
,
the
matrix
A
n
can
b
e
written
as
:
A
n
=
¸
n
˛
n
˛
n
¸
n
Demonstrate
b
y
recurrence
that,
for
an
y
natural
n
um
b
er
n
1
,
w
e
ha
v
e
:
¸
n
=
2
n
−
1
+
1
2
n
+1
;
˛
=
2
n
−
1
−
1
2
n
+1
2
a
Sho
w
that,
for
an
y
natural
n
um
b
er
n
,
the
p
oin
t
E
n
lies
on
the
line
of
equation
y
=
x
.
W
e
can
use
that,
for
an
y
natural
n
um
b
er
n
,
the
co
or-
dinates
(
x
n
;
y
n
)
of
the
p
oin
t
E
n
v
erify
:
x
n
y
n
=
A
n
·
2
2
b
Sho
w
that
the
length
O
E
n
tends
to
+
∞
when
n
tends
to
+
∞
.
2.
T
r
ansition
matrix
E.5439
W
e
study
the
ev
olution
o
v
er
time
of
the
n
um
b
er
of
y
oung
and
adults
in
a
p
opulation
of
animals.
F
or
an
y
natural
n
um
b
er
n
,
note
j
n
the
n
um
b
er
of
y
oung
ani-
mals
after
n
y
ears
of
observ
ation
and
a
n
the
n
um
b
er
of
adult
animals
after
n
y
ears
of
observ
ation.
A
t
the
start
of
the
first
y
ear
of
the
study
,
there
are
200
y
oung
animals
and
500
adult
h
ttps:/
/c
hingmath.fr
chapExoCorrec/5954
sacados/5954
Asie Juin 2013
O
E
F
G
E
F
G
O
E
F
G
chapExoCorrec/5439
sacados/5439
animals.
Th
us
:
j
0
=
200
and
a
0
=
500
.
W
e
admit
that
for
an
y
natural
n
um
b
er
n
,
w
e
ha
v
e
:
j
n
+1
=
0.125
·
j
n
+
0.525
·
a
n
a
n
+1
=
0.625
·
j
n
+
0.625
·
a
n
The
follo
wing
matrices
are
in
tro
duced
:
A
=
0.125
0.525
0.625
0.625
;
U
n
=
j
n
a
n
1
a
Sho
w
that
for
natural
n
um
b
er
n
:
U
n
+1
=
A
×
U
n
.
b
Calculate
the
n
um
b
er
of
y
oung
and
adult
animals
af-
ter
one
y
ear
of
observ
ation
and
then
after
t
w
o
y
ears
of
observ
ation
(r
esults
r
ounde
d
to
the
ne
ar
est
unit
by
default)
c
F
or
an
y
non-zero
natural
n
um
b
er
n
,
express
U
n
as
a
function
of
A
n
and
U
0
.
2
The
follo
wing
matrices
are
in
tro
duced
:
Q
=
7
3
−
5
5
;
D
=
−
0.25
0
0
1
a
W
e
admit
that
the
matrix
Q
is
in
v
ertible
and
that
:
Q
−
1
=
0.1
−
0.06
0.1
0.14
Sho
w
that
:
Q
×
D
×
Q
−
1
02
=
A
b
Sho
w,
b
y
recurrence
on
n
,
that
for
an
y
non-zero
natu-
ral
n
um
b
er
n
:
A
n
=
Q
×
D
n
×
Q
−
1
c
F
or
an
y
non-zero
natural
n
um
b
er
n
,
determine
D
n
as
a
function
of
n
.
3
W
e
admit
that
for
an
y
non-zero
natural
n
um
b
er
n
,
A
n
=
0.3
+
0.7
×
(
−
0.25)
n
0.42
−
0.42
×
(
−
0.25)
n
0.5
−
0.5
×
(
−
0.25)
n
0.7
+
0.3
×
(
−
0.25)
n
a
Deduce
the
expressions
for
j
n
and
a
n
as
a
function
of
n
.
Determine
the
limits
of
these
t
w
o
sequences.
b
What
can
b
e
concluded
for
the
p
opulation
of
animals
studied?
E.5955
One
bird
sp
ecies
liv
es
on
only
t
w
o
islands
A
and
B
of
an
arc
hip
elago.
A
t
the
b
eginning
of
the
y
ear
2013
,
20
million
birds
of
this
sp
ecies
are
presen
t
on
the
island
and
10
million
on
the
island
B
.
Observ
ations
o
v
er
sev
eral
y
ears
ha
v
e
enabled
aornithologists
to
estimate
that,
taking
in
to
accoun
t
births,
deaths,
and
mi-
grations
b
et
w
een
the
t
w
o
islands,
the
follo
wing
prop
osals
are
found
at
the
b
eginning
of
eac
h
y
ear
:
On
the
island
A
:
80
%
of
the
n
um
b
er
of
birds
presen
t
on
the
island
A
at
the
b
eginning
of
the
previous
y
ear
and
30
%
of
the
n
um
b
er
of
birds
presen
t
on
the
island
B
at
the
start
of
the
previous
y
ear
;
on
the
island
B
:
20
%
of
the
n
um
b
er
of
birds
presen
t
on
the
island
A
at
the
b
eginning
of
the
previous
y
ear
and
70
%
of
the
n
um
b
er
of
birds
presen
t
on
the
island
B
at
the
start
of
the
previous
y
ear.
F
or
an
y
natural
n
um
b
er
n
,
w
e
note
a
n
(r
esp
e
ctively
b
n
)
the
n
um
b
er
of
birds
(in
mil
lions)
presen
t
on
the
island
A
(r
esp
e
c-
tively
B
)
at
the
b
eginning
of
the
y
ear
(2013+
n
)
.
P
art
A
-
Algorithms
and
conjectures
Giv
en
b
elo
w
is
a
function
f
,
deriv
ed
from
an
algorithm,
tak-
ing
as
argumen
t
an
in
teger
n
greater
than
or
equal
to
2013
represen
ting
the
y
ear
of
study
and
returning
the
n
um
b
er
of
birds
living
on
eac
h
of
the
t
w
o
islands
for
that
y
ear.
Function
f(n)
a
←
20
b
←
10
i
←
2013
As
long
as
i<
n
c
←
(0.8a+
0.3b)
b
←
(0.2a+
0.7b)
a
←
c
End
As
long
as
Renvoyer
(
a
;
b
)
1
The
function
co
de
f
con
tains
omissions
in
pro
cessing.
Iden
tify
these
omissions
and
correct
them.
2
Belo
w
is
giv
en
a
table
represen
ting
the
v
alues
successiv
ely
tak
en
b
y
the
v
ariabbles
of
the
function
f
during
its
step-
b
y-step
execution
when
called
with
the
v
alue
2020
.
n
a
b
2013
20
10
2014
19
11
2015
18.5
11.5
2016
18.25
11.75
2017
18.125
11.875
2018
18.0425
11.9375
2019
18.03125
11.96875
2020
18.015625
11.984375
In
view
of
these
results,
mak
e
conjectures
concerning
the
direction
of
v
ariation
and
con
v
ergence
of
the
sequences
a
n
and
b
n
.
P
art
B
-
Mathematical
study
h
ttps:/
/c
hingmath.fr
chapExoCorrec/5955
sacados/5955
Centres etrangers Juin 2013
I
M
S
1
2
1
3
1
3
1
2
1
1
3
Note
U
n
the
column
matrix
a
n
b
n
1
Sho
w
that,
for
an
y
natural
n
um
b
er
n
:
U
n
+1
=
M
·
U
n
où
M
is
a
square
matrix
of
order
2
to
b
e
determined.
W
e
then
admit
that
U
n
=
M
n
·
U
0
for
an
y
natural
n
um
b
er
n
1
.
2
Using
reasoning
b
y
recurrence,
justify
that,
for
an
y
nat-
ural
n
um
b
er
n
1
:
M
n
=
0.6
+
0.4
×
0.5
n
0.6
−
0.6
×
0.5
n
0.4
−
0.4
×
0.5
n
0.4
+
0.6
×
0.5
n
W
e
will
only
detail
the
calculation
for
the
first
of
the
co
efficien
ts
of
the
M
n
matrix.
3
Express
a
n
as
a
function
of
n
,
for
an
y
natural
n
um
b
er
n
1
.
4
With
this
mo
del,
can
w
e
sa
y
that
after
a
large
n
um
b
er
of
y
ears,
the
n
um
b
er
of
birds
on
the
island
A
will
stabilize?
If
so,
sp
ecify
to
w
ards
what
v
alue.
E.5957
P
arts
A
and
B
can
b
e
pro
cessed
indep
enden
tly
of
eac
h
other
In
a
remote
imaginary
village,
a
new
con
tagious
but
non-
lethal
disease
has
app
eared.
Scien
tists
so
on
disco
v
ered
that
an
individual
could
b
e
in
one
of
three
states
:
S
:
ˇ
the
individual
is
health
y
,
i.e.
not
ill
and
not
infecté
ı,
I
:
ˇ
the
individual
is
a
health
y
carrier,
i.e.
not
ill
but
infecté
ı,
M
:
ˇ
the
individual
is
ill
and
infecté
ı.
P
art
A
Scien
tists
estimate
that
only
one
individual
causes
the
disease
out
of
the
100
p
eople
in
the
p
opulation
and
that,
from
one
w
eek
to
the
next,
an
individual
c
hanges
state
according
to
the
follo
wing
pro
cess
:
among
health
y
individuals,
the
prop
ortion
of
those
who
b
ecome
health
y
carriers
is
equal
to
1
3
and
the
prop
ortion
of
those
who
b
ecome
ill
is
equal
to
1
3
.
among
health
y
carrier
individuals,
the
prop
ortion
of
those
who
b
ecome
ill
is
equal
to
1
2
.
The
situation
can
b
e
represen
ted
b
y
a
probabilistic
graph
as
sho
wn
opp
osite.
W
e
note
P
n
=
s
n
i
n
m
n
the
ro
w
matrix
giving
the
probabilistic
state
after
n
w
eeks
où
s
n
,
i
n
and
m
n
denote,
resp
ectiv
ely
,
the
probabilit
y
of
the
individual
b
eing
health
y
,
a
health
y
carrier
or
sic
k
at
n
-w
eek.
Then
w
e
ha
v
e
P
0
=
0.99
0
0.01
and
for
an
y
natural
n
um
b
er
n
,
s
n
+1
=
1
3
s
n
i
n
+1
=
1
3
s
n
+
1
2
i
n
m
n
+1
=
1
3
s
n
+
1
2
i
n
+
m
n
1
W
rite
the
matrix
A
called
tr
ansition
matrix
,
suc
h
that
for
an
y
natural
n
um
b
er
n
:
P
n
+1
=
P
n
·
A
2
Demonstrate
b
y
recurrence
that
for
an
y
non-zero
natural
n
um
b
er
n
:
P
n
=
P
0
·
A
n
3
Determine
the
probabilistic
state
P
4
after
four
w
eeks.
V
alues
ma
y
b
e
rounded
to
10
−
2
.
What
is
the
probabilit
y
that
an
individual
will
b
e
health
y
after
four
w
eeks?
P
art
B
The
disease
do
esn’t
actually
ev
olv
e
according
to
the
previous
mo
del,
since
after
4
w
eeks
of
researc
h,
scien
tists
disco
v
er
a
v
accine
that
halts
the
endemic
and
immediately
treats
the
en
tire
p
opulation.
The
w
eekly
ev
olution
of
the
disease
after
v
accination
is
giv
en
b
y
the
transition
matrix
:
B
=
5
12
1
4
1
3
5
12
1
4
1
3
1
6
1
2
1
3
Let’s
note
Q
n
the
ro
w
matrix
giving
the
probabilistic
state
after
n
w
eeks
follo
wing
the
implemen
tation
of
these
new
v
ac-
cination
measures.
Th
us
:
Q
n
=
S
n
I
n
M
n
où
S
n
,
I
n
and
M
n
refer
resp
ectiv
ely
to
the
probabilit
y
of
the
individual
b
eing
health
y
,
a
health
y
carrier
and
ill
on
the
n
-th
w
eek
after
v
accination.
F
or
an
y
natural
n
um
b
er
n
,
w
e
then
ha
v
e
:
Q
n
+1
=
Q
n
·
B
F
rom
part
A
,
Q
0
=
P
4
.
F
or
the
rest,
w
e
tak
e
:
Q
0
=
0.01
0.10
0.89
où
co
efficien
ts
ha
v
e
b
een
rounded
to
10
−
2
.
1
Express
S
n
+1
,
I
n
+1
and
M
n
+1
in
terms
of
S
n
,
I
n
and
M
n
.
2
Determine
the
real
constan
t
k
suc
h
that
B
2
=
k
·
J
où
J
is
the
square
matrix
of
order
3
whose
co
efficien
ts
are
all
equal
to
1
.
W
e
deduce
that
for
an
y
in
teger
n
greater
than
or
equal
to
2
:
B
n
=
B
2
3
a
Sho
w
that
for
an
y
in
teger
n
greater
than
or
equal
to
2
:
Q
n
=
1
3
1
3
1
3
b
In
terpret
this
result
in
terms
of
disease
ev
olution.
Can
w
e
hop
e
to
eradicate
the
disease
with
the
v
accine?
3.
T
r
ansition
matrix
typ
e
:
X=AX+B
h
ttps:/
/c
hingmath.fr
chapExoCorrec/5957
sacados/5957
Metropole Septembre 2013
I
M
S
1
2
1
3
1
3
1
2
1
1
3
E.5953
A
telephone
op
erator
A
wishes
to
forecast
the
ev
olution
of
its
subscrib
er
n
um
b
ers
in
a
large
cit
y
compared
to
its
main
comp
etitor
B
from
2013
.
In
2013
,
op
erators
A
and
B
eac
h
ha
v
e
300
thousand
sub-
scrib
ers.
F
or
an
y
natural
n
um
b
er
n
,
let
a
n
b
e
the
n
um
b
er
of
subscrib
ers,
in
thousands,
of
the
op
erator
A
in
the
n
-th
y
ear
after
2013
,
and
b
n
the
n
um
b
er
of
subscrib
ers,
in
thousands,
of
op
erator
B
the
n
-th
y
ear
after
2013
.
Th
us
:
a
0
=
300
and
b
0
=
300
.
Observ
ations
made
in
previous
y
ears
lead
us
to
mo
del
the
situation
b
y
the
follo
wing
relationship
:
a
n
+1
=
0.7
a
n
+
0.2
b
n
+
60
b
n
+1
=
0.1
a
n
+
0.6
b
n
+
70
,
for
an
y
in
teger
n
∈
N
.
Consider
the
matrices
:
M
=
0.7
0.2
0.1
0.6
;
P
=
60
70
.
F
or
an
y
natural
n
um
b
er
n
,
let
U
n
=
a
n
b
n
1
a
Determine
U
1
.
b
V
erify
that,
for
an
y
natural
n
um
b
er
n
:
U
n
+1
=
M
×
U
n
+
P
.
2
W
e
note
I
the
matrix
1
0
0
1
a
Calculate
:
(
I
−
M
)
×
4
2
1
3
.
b
Deduce
that
the
matrix
I
−
M
is
in
v
ertible
and
sp
ecify
its
in
v
erse.
c
Determine
the
matrix
suc
h
that
:
U
=
M
×
U
+
P
3
F
or
an
y
natural
n
um
b
er,
w
e
p
ose
:
V
n
=
U
n
−
U
.
a
Justify
that,
for
an
y
natural
n
um
b
er
n
:
V
n
+1
=
M
×
V
n
.
b
Deduce
that,
for
an
y
natural
n
um
b
er
n
:
V
n
=
M
n
×
V
0
4
W
e
admit
that,
for
an
y
natural
n
um
b
er
n
:
V
n
=
−
100
3
×
0.8
n
−
140
3
×
0.5
n
−
50
3
×
0.8
n
+
140
3
×
0.5
n
a
F
or
an
y
natural
n
um
b
er
n
,
express
U
n
as
a
function
of
n
and
deduce
the
limit
of
the
sequence
a
n
.
b
Estimate
the
n
um
b
er
of
subscrib
ers
to
op
erator
A
in
the
long
term.
4.
Matric
es
and
arithmetic
E.5956
P
art
A
Consider
the
function
f
,
tak
en
from
an
algorithm,
taking
as
argumen
t
a
natural
in
teger
A
and
reviewing
at
the
end
of
execution
the
v
alue
of
the
v
ariable
X
:
Function
f(A)
X
←
A
As
long
as
X
greater
than
or
equal
to
26
X
←
X
−
26
End
As
long
as
Return
X
1
What
v
alue
is
returned
b
y
calling
the
function
f
when
the
v
alue
supplied
as
an
argumen
t
is
the
n
um
b
er
3
?
2
What
is
the
v
alue
returned
b
y
calling
the
function
f
when
the
v
alue
supplied
as
an
argumen
t
is
the
n
um
b
er
55
?
3
F
or
an
y
in
teger
en
tered,
what
is
the
result
returned
b
y
this
function?
P
art
B
W
e
w
an
t
to
co
de
a
blo
c
k
of
t
w
o
letters
according
to
the
fol-
lo
wing
pro
cedure
(detaile
d
in
four
steps)
:
Step
1
:
eac
h
letter
of
the
blo
c
k
is
replaced
b
y
an
in
teger
using
the
table
b
elo
w
:
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
W
e
obtain
a
column
matrix
x
1
y
2
où
x
1
corresp
onds
to
the
first
letter
of
the
w
ord
and
x
2
corresp
onds
to
the
second
letter
of
the
w
ord.
Step
2
:
x
1
y
1
is
transformed
in
to
y
1
y
2
suc
h
that
:
y
1
y
2
=
3
1
5
2
·
x
1
x
2
The
matrix
C
=
3
1
5
2
is
called
the
co
ding
matrix.
Step
3
:
y
1
y
2
is
transformed
in
to
z
1
z
2
suc
h
that
:
z
1
≡
y
1
(
mo
d.
26)
a
v
ec
0
z
1
25
z
2
≡
y
2
(
mo
d.
26)
a
v
ec
0
z
2
25
Step
4
:
z
1
z
2
is
transformed
in
to
a
t
w
o-letter
blo
c
k
using
the
mapping
table
giv
en
in
step
1
Exemple
:
RE
↦−
→
17
4
↦−
→
55
93
↦−
→
3
15
↦−
→
DP
Justify
the
c
hange
from
17
4
to
55
93
to
3
15
1
Let
x
1
,
x
2
,
x
1
,
x
2
four
in
tegers
b
et
w
een
0
and
25
suc
h
that
x
1
x
2
and
x
1
x
2
are
transformed
during
the
co
ding
h
ttps:/
/c
hingmath.fr
chapExoCorrec/5953
sacados/5953
Polynesie Juin 2013
chapExoCorrec/5956
sacados/5956
Antilles-Guyane Septembre 2013
pro
cess
in
to
z
1
z
2
.
a
Mon
trer
que
3
x
1
+
x
2
≡
3
x
1
+
x
2
(
mo
d.
26)
5
x
1
+
2
x
2
≡
5
x
1
+
2
x
2
(
mo
d.
26)
b
Deduce
x
1
≡
x
1
(
mo
d.
26)
and
x
2
≡
x
2
(
mo
d.
26)
,
then
x
1
=
x
1
and
x
2
=
x
2
.
2
W
e
w
an
t
to
find
a
deco
ding
metho
d
for
the
D
P
blo
c
k
a
V
erify
that
the
matrix
C
=
2
−
1
−
5
3
is
the
in
v
erse
matrix
of
C
.
b
Calculate
y
1
y
2
suc
h
that
:
y
1
y
2
=
2
−
1
−
5
3
3
15
c
Calculate
x
1
x
2
suc
h
that
:
x
1
≡
y
1
(
mo
d.
26)
a
v
ec
0
x
1
25
x
2
≡
y
2
(
mo
d.
26)
a
v
ec
0
x
2
25
d
What
general
deco
ding
pro
cedure
can
b
e
conjectured?
3
In
this
question,
w
e
will
generalize
this
deco
ding
pro
ce-
dure.
Consider
a
blo
c
k
of
t
w
o
letters
and
call
z
1
and
z
2
the
t
w
o
in
tegers
b
et
w
een
0
and
25
asso
ciated
with
these
letters
at
step
3
.
W
e
try
to
find
t
w
o
in
tegers
x
1
and
x
2
b
et
w
een
0
and
25
that
giv
e
the
column
matrix
z
1
z
2
b
y
steps
2
and
3
of
the
co
ding
pro
cess.
Let
y
1
and
y
2
b
e
suc
h
that
:
y
1
y
2
=
C
·
z
1
z
2
où
C
=
2
−
1
−
5
3
Let
x
1
and
x
2
b
e
in
tegers
suc
h
that
:
x
1
≡
y
1
(
mo
d.
26)
a
v
ec
0
x
1
25
x
2
≡
y
2
(
mo
d.
26)
a
v
ec
0
x
2
25
Sho
w
that
:
3
x
1
+
x
2
≡
z
1
(
mo
d.
26)
5
x
1
+
2
x
2
≡
z
2
(
mo
d.
26)
Conclude.
4
Deco
de
QC
.
5.
Unclassifie
d
financial
ye
ars
E.6067
Ev
ery
y
oung
paren
t
uses
just
one
brand
of
bab
y
fo
o
d
ev
ery
mon
th.
Three
brands
X
,
Y
and
Z
share
the
mark
et.
Let
n
b
e
a
natural
n
um
b
er.
Let
:
X
n
the
ev
en
t
ˇ
the
brand
X
is
used
in
the
mon
th
n
ı
;
Y
n
the
ev
en
t
ˇ
the
mark
Y
is
used
the
mon
th
n
ı
;
Z
n
the
ev
en
t
ˇ
the
mark
Z
is
used
the
mon
th
n
ı
;
The
probabilities
of
the
ev
en
ts
X
n
,
Y
n
,
Z
n
are
noted
resp
ec-
tiv
ely
x
n
,
y
n
,
z
n
.
Eac
h
brand’s
adv
ertising
campaign
c
hanges
the
distribution
:
A
buy
er
of
the
brand
X
the
mon
th
n
has
the
follo
wing
mon
th
:
50
%
c
hance
of
remaining
lo
y
al
to
this
brand.
40
%
c
hance
of
buying
the
brand
Y
.
10
%
c
hance
of
buying
brand
Z
.
A
buy
er
of
brand
Y
the
mon
th
n
has
the
follo
wing
mon
th
:
30
%
c
hance
of
remaining
lo
y
al
to
this
brand
;
50
%
c
hance
of
buying
the
brand
X
;
20
%
c
hance
of
buying
brand
Z
.
A
buy
er
of
brand
Z
the
mon
th
n
has
the
follo
wing
mon
th
:
70
%
c
hance
of
remaining
lo
y
al
to
this
brand
;
10
%
c
hance
of
buying
the
brand
X
;
20
%
c
hance
of
buying
brand
Y
.
1
a
Express
x
n
+1
as
a
function
of
x
n
,
y
n
and
z
n
.
W
e
assume
that
:
y
n
+1
=
0.4
x
n
+0.3
y
n
+0.2
z
n
;
z
n
+1
=
0.1
x
n
+0.2
y
n
+0.7
z
n
b
Express
z
n
as
a
function
of
x
n
and
y
n
.
Deduce
the
expression
of
x
n
+1
and
y
n
+1
as
a
function
of
x
n
and
y
n
.
2
W
e
define
the
sequence
U
n
b
y
U
n
=
x
n
y
n
for
an
y
nat-
ural
n
um
b
er
n
.
W
e
admit
that,
for
an
y
natural
n
um
b
er
n
:
U
n
+1
=
A
·
U
n
+
B
où
:
A
=
0.4
0.4
0.2
0.1
;
B
=
0.1
0.2
A
t
the
start
of
the
statistical
study
(month
of
January
2014
:
n
=
0
)
,
w
e
estimate
that
:
U
0
=
0.5
0.3
Consider
the
function
f
of
the
follo
wing
algorithm
:
Function
f(n)
i
←
0
A
←
0.4
0.4
0.2
0.1
B
←
0.1
0.2
U
←
0.5
0.3
As
long
as
i<
n
U
←
A
·
U+B
i
←
i+
1
End
of
As
long
as
Return
U
a
Giv
e
the
v
alues
returned
b
y
this
function
when
called
with
the
v
alues
n
=
1
then
for
n
=
3
.
h
ttps:/
/c
hingmath.fr
chapExoCorrec/6067
sacados/6067
b
What
is
the
probabilit
y
of
using
brand
X
in
the
mon
th
of
April?
In
the
remainder
of
the
exercise,
w
e
seek
to
determine
an
expression
for
U
n
as
a
function
of
n
.
W
e
denote
I
the
matrix
1
0
0
1
and
N
the
matrix
I
−
A
.
3
W
e
denote
b
y
C
a
t
w
o-ro
w
column
matrix.
a
Demonstrate
that
C
=
A
·
C
+
B
is
equiv
alen
t
to
N
·
C
=
B
.
b
W
e
admit
that
N
is
an
in
v
ertible
matrix
and
that
:
N
−
1
=
45
23
20
23
10
23
30
23
Deduce
that
:
C
=
17
46
7
23
4
Note
V
n
the
matrix
suc
h
that
V
n
=
U
n
−
C
for
an
y
natural
n
um
b
er
n
.
a
Sho
w
that,
for
an
y
natural
n
um
b
er
n
:
V
n
+1
=
A
·
V
n
b
W
e
admit
that
:
U
n
=
A
n
·
U
0
−
C
+
C
.
What
are
the
probabilities
of
using
the
brands
X
,
Y
and
Z
in
the
mon
th
of
Ma
y?
E.6253
P
art
A
:
preliminaries
1
a
Let
n
and
N
b
e
t
w
o
natural
n
um
b
ers
greater
than
or
equal
to
2
,
suc
h
that
:
n
2
≡
N
−
1
(
mo
d.
N
)
Sho
w
that
:
n
×
n
3
≡
1
(
mo
d.
N
)
b
Deduce
from
the
previous
question
an
in
teger
k
1
suc
h
that
:
5
·
k
1
≡
1
(
mo
d.
26)
W
e’ll
admit
that
the
unique
in
teger
k
suc
h
that
:
0
k
25
;
5
·
k
≡
1
(
mo
d.
26)
is
w
orth
21
.
2
W
e
giv
e
the
matrices
:
A
=
4
1
3
2
;
B
=
2
−
1
−
3
4
;
X
=
x
1
x
2
;
Y
=
y
1
y
2
a
Calculate
matrix
:
6
A
−
A
2
.
b
Deduce
that
A
is
in
v
ertible
and
that
its
in
v
erse
matrix,
denoted
A
−
1
,
can
b
e
written
as
:
A
−
1
=
¸
·
I
+
˛
·
A
où
¸
and
˛
are
t
w
o
real
n
um
b
ers
to
b
e
determined.
c
Chec
k
that
:
B
=
5
·
A
−
1
d
Sho
w
that
if
A
·
X
=
Y
then
5
·
X
=
B
·
Y
.
P
art
B
:
co
ding
pro
cedure
Co
de
the
w
ord
ˇ
ET
ı,
using
the
co
ding
pro
cedure
describ
ed
b
elo
w.
The
w
ord
to
b
e
co
ded
is
replaced
b
y
the
matrix
X
=
x
1
x
2
,
où
x
1
is
the
in
teger
represen
ting
the
first
letter
of
the
w
ord
and
x
2
the
in
teger
represen
ting
the
second
according
to
the
corresp
ondence
table
b
elo
w
:
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
matrix
X
is
transformed
in
to
the
matrix
y
1
y
2
suc
h
that
:
Y
=
A
·
X
.
matrix
Y
is
transformed
in
to
the
matrix
R
=
r
1
r
2
,
où
r
1
is
the
remainder
of
the
Euclidean
division
of
y
1
b
y
26
and
r
2
the
remainder
of
the
Euclidean
division
of
y
2
b
y
26
.
The
in
tegers
r
1
and
r
2
giv
e
the
letters
of
the
co
de
w
ord,
according
to
the
corresp
ondence
table
ab
o
v
e.
Example
:
ˇ
Or
ı
(c
o
de
wor
d)
X
14
20
Y
=
76
82
R
=
24
4
ˇ
YE
ı
(mot
c
o
dé)
P
art
C
:
deco
ding
pro
cedure
(we
ke
ep
the
same
notations
as
for
c
o
ding)
During
enco
ding,
the
matrix
X
w
as
transformed
in
to
the
ma-
trix
Y
=
y
1
y
2
suc
h
that
:
Y
=
A
·
X
h
ttps:/
/c
hingmath.fr
chapExoCorrec/6253
sacados/6253
1
Demonstrate
that
:
5
·
x
1
=
2
·
y
1
−
y
2
5
·
x
2
=
−
3
·
y
1
+
4
·
y
2
2
Using
question
1
b
from
part
A
,
establish
that
:
x
1
≡
16
·
y
1
+
5
·
y
2
(
mo
d.
26)
x
2
≡
15
·
y
1
+
6
·
y
2
(
mo
d.
26)
3
Deco
de
the
w
ord
ˇ
QP
ı.
E.6938
P
art
A
Consider
matrices
M
of
the
form
M
=
a
b
5
3
où
a
and
b
are
in
tegers.
The
n
um
b
er
3
a
−
5
b
is
called
the
determinan
t
of
M
.
It
is
noted
det
(
M
)
.
Th
us
:
det
(
M
)
=
3
a
−
5
b
1
In
this
question,
w
e
assume
that
det
(
M
)
=
0
and
p
ose
:
N
=
1
det
(
M
)
·
3
−
b
−
5
a
.
Justify
that
N
is
the
in
v
erse
of
M
.
2
Consider
the
equation
(
E
)
:
det
(
M
)
=
3
W
e
wish
to
determine
all
pairs
of
in
tegers
(
a
;
b
)
solutions
of
the
equation
(
E
)
.
a
V
erify
that
the
couple
(
6
;
3
)
is
a
solution
of
(
E
)
.
b
Sho
w
that
the
pair
of
in
tegers
(
a
;
b
)
is
a
solution
of
(
E
)
if,
and
only
if,
3
·
(
a
−
6)
=
5
·
(
b
−
3)
Deduce
the
set
of
solutions
of
the
equation
(
E
)
.
P
art
B
1
W
e
p
ose
:
Q
=
6
3
5
3
Using
the
A
part,
determine
the
in
v
erse
matrix
of
Q
.
2
Co
ding
with
matrix
Q
T
o
co
de
a
t
w
o-letter
w
ord
using
the
matrix
Q
=
6
3
5
3
,
the
follo
wing
pro
cedure
is
used
:
Step
1
:
The
w
ord
is
asso
ciated
with
the
matrix
X
=
x
1
x
2
où
x
1
is
the
in
teger
corresp
onding
to
the
first
letter
of
the
w
ord
and
x
2
the
in
teger
corresp
ond-
ing
to
the
second
letter
of
the
w
ord
according
to
the
corresp
ondence
table
b
elo
wb
elo
w
:
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
Step
2
:
The
matrix
X
is
transformed
in
to
the
matrix
Y
=
y
1
y
2
suc
h
that
Y
=
Q
·
X
.
Step
3
:
matrix
Y
is
transformed
in
to
the
matrix
R
=
r
1
r
2
suc
h
that
r
1
is
the
remainder
of
the
Eu-
clidean
division
of
y
1
b
y
26
and
r
2
is
the
remainder
of
the
Euclidean
division
of
y
2
b
y
26
.
Step
4
:
T
o
the
matrix
R
=
r
1
r
2
,
w
e
asso
ciate
a
t
w
o-
letter
w
ord
according
to
the
corresp
ondence
table
in
step
1.
Example
:
J
E
↦→
X
=
9
4
↦→
Y
=
66
57
↦→
R
=
14
5
↦→
O
F
Le
mot
J
E
est
co
dé
en
le
mot
O
E
.
Co
de
the
w
ord
D
O
.
3
Deco
ding
pro
cedure
W
e
retain
the
same
notations
as
for
enco
ding.
During
enco
ding,
the
matrix
X
w
as
transformed
in
to
the
matrix
Y
suc
h
that
Y
=
Q
·
X
.
a
Demonstrate
that
3
X
=
3
·
Q
−
1
·
Y
then
that
:
3
·
x
1
≡
3
·
r
1
−
3
·
r
2
(
mo
d.
26)
3
·
x
2
≡
−
5
·
r
1
+
6
·
r
2
(
mo
d.
26)
b
Noting
that
9
×
3
≡
1
(
mo
d.
26)
,
sho
w
that
:
x
1
≡
r
1
−
r
2
(
mo
d.
26)
x
2
≡
7
·
r
1
+
2
·
r
2
(
mo
d.
26)
c
Deco
de
the
w
ord
S
G
.
E.6940
There
are
t
w
o
urns
U
and
V
eac
h
con
taining
t
w
o
balls.
Initially
,
urn
U
con
tains
t
w
o
white
balls
and
urn
V
con
tains
t
w
o
blac
k
balls.
Successiv
e
dra
ws
are
made
from
these
urns
in
the
follo
wing
w
a
y
:
eac
h
dra
w
consists
of
randomly
taking,
sim
ultaneously
,
one
ball
from
eac
h
urn
and
putting
it
in
the
other
urn.
F
or
an
y
non-zero
natural
n
um
b
er
n
,
note
X
n
the
random
v
ari-
able
equal
to
the
n
um
b
er
of
white
balls
con
tained
in
the
U
urn
at
the
end
of
the
n
-th
dra
w.
1
a
T
ranslate
the
probabilit
y
in
to
a
sen
tence
:
P
(
X
n
=1)
X
n
+1
=
1
then
determine
the
follo
wing
conditional
probabilities
:
P
(
X
n
=0)
X
n
+1
=
1
;
P
(
X
n
=1)
X
n
+1
=
1
;
P
(
X
n
=2)
X
n
+1
=
1
b
Exprimer
P
X
n
+1
=
1
en
fonction
de
P
X
n
=
0
,
P
X
n
=
1
et
P
X
n
=
2
2
F
or
an
y
non-zero
natural
n
um
b
er
n
,
let
R
n
denote
the
ro
w
matrix
c
hallenged
b
y
:
R
n
=
P
X
n
=
0
P
X
n
=
1
P
X
n
=
2
and
consider
M
the
matrix
:
0
1
0
1
4
1
2
1
4
0
1
0
Let
R
0
b
e
the
ro
w
matrix
0
0
1
.
It
will
b
e
admitted
hereafter
that,
for
an
y
n
∈
N
:
R
n
+1
=
R
n
·
M
Determine
R
1
and
justify
that,
for
an
y
n
∈
N
:
R
n
=
R
0
·
M
n
.
3
W
e
admit
that
M
=
P
·
D
·
P
−
1
with
:
P
=
1
6
·
2
3
1
−
1
0
1
2
−
3
1
;
D
=
−
1
2
0
0
0
0
0
0
0
1
P
−
1
=
1
−
2
1
1
0
−
1
1
4
1
Establish
that,
for
an
y
n
∈
N
:
M
n
=
P
·
D
n
·
P
−
1
.
h
ttps:/
/c
hingmath.fr
chapExoCorrec/6938
sacados/6938
chapExoCorrec/6940
sacados/6940
W
e’ll
admit
that,
for
an
y
natural
n
um
b
er
:
D
n
=
−
1
2
n
0
0
0
0
0
0
0
1
4
a
Calculate
D
n
·
P
−
1
as
a
function
of
n
.
b
Kno
wing
that
R
0
·
P
=
1
3
−
1
2
1
6
,
determine
the
co
efficien
ts
of
R
n
as
a
function
of
n
.
5
Determine
:
lim
n
↦→
+
∞
P
X
n
=
0
,
lim
n
↦→
+
∞
P
X
n
=
1
et
lim
n
↦→
+
∞
P
X
n
=
2
.
In
terpret
these
results.
E.6942
The
aim
of
this
exercise
is
to
study
,
on
an
example,
an
encryption
metho
d
published
in
1929
b
y
mathematician
and
cryptologist
Lester
Hill.
This
cipher
re-
lies
on
the
data
of
a
matrix
A
,
kno
wn
only
to
the
sender
and
the
receiv
er.
Throughout
the
exercise,
w
e
will
note
A
the
matrix
defined
b
y
:
A
=
5
2
7
7
P
art
A
-
Hill
encryption
Here
are
the
differen
t
encryption
steps
for
a
w
ord
with
an
ev
en
n
um
b
er
of
letters
:
Step
1
:
W
e
divide
the
w
ord
in
to
blo
c
ks
of
t
w
o
consecutiv
e
letters
and
then,
for
eac
h
blo
c
k,
p
erform
eac
h
of
the
follo
wing
steps.
Step
2
:
The
t
w
o
letters
in
the
blo
c
k
are
asso
ciated
with
the
t
w
o
in
tegers
x
1
and
x
2
b
oth
b
et
w
een
0
and
25
,
whic
h
corre-
sp
ond
to
the
t
w
o
letters
in
the
same
order,
in
the
follo
w-
ing
table
:
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
Step
3
:
W
e
transform
the
matrix
X
=
x
1
x
2
in
to
the
matrix
Y
=
y
1
y
2
v
érifian
t
Y
=
A
·
X
.
Step
4
:
W
e
transform
the
matrix
Y
=
y
1
y
2
in
to
the
matrix
R
=
r
1
r
2
,
où
r
1
is
the
remainder
of
the
Euclidean
division
of
y
1
b
y
26
and
r
2
that
of
the
Euclidean
division
of
y
2
b
y
26
.
Step
5
:
The
in
tegers
r
1
and
r
2
are
asso
ciated
with
the
t
w
o
corre-
sp
onding
letters
from
the
table
in
step
2
.
The
encrypted
blo
c
k
is
the
blo
c
k
obtained
b
y
juxtap
osing
these
t
w
o
letters.
Question
:
use
the
encryption
metho
d
sho
wn
to
encrypt
the
w
ord
ˇ
HILL
ı.
P
art
B
-
Some
mathematical
to
ols
needed
for
decryp-
tion
1
Let
a
b
e
a
prime
relativ
e
in
teger
with
26
.
Pro
v
e
that
there
exists
a
relativ
e
in
teger
u
suc
h
that
:
u
×
a
≡
1
(
mo
d.
26)
.
2
Consider
the
function
f
of
an
algorithm
taking
as
argu-
men
t
a
natural
n
um
b
er
a
prime
with
26
.
Function
f(a)
u
←
0
r
←
0
As
long
as
r
=
1
u
←
u+1
r
←
remainder
of
Euclidean
division
of
u
×
a
by
26
End
of
As
long
as
Return
u
W
e
call
the
function
f
with
parameter
v
alue
a=21
.
a
Repro
duce
on
the
cop
y
and
complete
the
follo
wing
ta-
ble,
with
the
differen
t
v
alues
tak
en
b
y
the
v
ariables
u
and
v
when
calling
the
function
f
.
u
0
1
2
·
·
·
r
0
21
·
·
·
·
·
·
b
Infer
that
:
5
×
21
≡
1
(
mo
d.
26)
.
3
Recall
that
A
is
the
matrix
A
=
5
2
7
7
and
note
I
the
matrix
:
I
=
1
0
0
1
a
Calculate
the
matrix
:
12
·
A
−
A
2
.
b
Deduce
the
matrix
B
suc
h
that
:
B
·
A
=
21
·
I
c
Demonstrate
that
if
A
·
X
=
Y
then
21
·
X
=
B
·
Y
.
P
art
C
-
Decryption
W
e
w
an
t
to
decrypt
the
w
ord
V
LU
P
.
W
e
note
X
=
x
1
x
2
the
matrix
asso
ciated,
according
to
the
corresp
ondence
table,
to
a
t
w
o-letter
blo
c
k
b
efore
encryption,
and
Y
=
y
1
y
2
the
matrix
defined
b
y
the
equalit
y
:
Y
=
A
·
X
=
5
2
7
7
·
X
If
r
1
and
r
2
are
the
resp
ectiv
e
remainders
of
y
1
and
y
2
in
Eu-
clidean
division
b
y
26
,
the
t
w
o-letter
blo
c
k
after
encryption
is
asso
ciated
with
the
matrix
R
=
r
1
r
2
.
1
Sho
w
that
:
21
·
x
1
=
7
·
y
1
−
2
·
y
2
21
·
x
2
=
−
7
·
y
1
+
5
·
y
2
2
Using
question
B
2
,
establish
that
:
x
1
≡
9
·
r
1
+
16
·
r
2
(
mo
d.
26)
x
2
≡
17
·
r
1
+
25
·
r
2
(
mo
d.
26)
3
Decipher
the
w
ord
VLUP
,
asso
ciated
with
the
matrices
21
11
and
20
15
h
ttps:/
/c
hingmath.fr
chapExoCorrec/6942
sacados/6942
E.6948
A
smok
er
decides
to
stop
smoking.
W
e
c
ho
ose
to
use
the
follo
wing
mo
del
:
if
he
do
esn’t
smok
e
on
a
giv
en
da
y
,
he
do
esn’t
smok
e
the
next
da
y
with
a
probabilit
y
of
0.9
;
if
he
smok
es
on
a
giv
en
da
y
,
he
smok
es
the
next
da
y
with
a
probabilit
y
of
0.6
.
W
e
call
p
n
the
probabilit
y
of
not
smoking
on
the
n
-th
da
y
after
his
decision
to
stop
smoking
and
q
n
,
the
probabilit
y
of
smoking
on
the
n
-th
da
y
after
his
decision
to
stop
smoking.
W
e
assume
that
p
0
=
0
and
q
0
=
1
.
1
Calculate
p
1
and
q
1
.
2
A
spreadsheet
is
used
to
automate
the
calculation
of
the
successiv
e
terms
of
the
sequences
p
n
and
q
n
.
A
screenshot
of
this
spreadsheet
is
pro
vided
b
elo
w
:
A
B
C
D
1
n
p
n
q
n
2
0
0
1
1
3
1
4
2
5
3
In
the
column
A
are
the
v
alues
of
the
natural
n
um
b
er
n
.
What
form
ulas
can
b
e
written
in
the
cells
B3
and
C3
so
that
b
y
cop
ying
them
do
wn,
w
e
obtain
resp
ectiv
ely
in
the
columns
B
and
C
the
successiv
e
terms
of
the
sequences
p
n
and
q
n
?
3
W
e
define
the
matrices
M
and,
for
an
y
natural
n
um
b
er
n
,
X
n
b
y
:
M
=
0.9
0.4
0.1
0.6
et
X
n
=
p
n
q
n
W
e
admit
that
X
n
+1
=
M
·
X
n
and
that,
for
an
y
natural
n
um
b
er
n
,
X
n
=
M
n
·
X
0
W
e
define
the
matrices
A
and
B
b
y
:
A
=
0.8
0.8
0.2
0.2
et
B
=
0.2
−
0.8
−
0.2
0.8
a
Demonstrate
that
:
M
=
A
+0.5
·
B
b
Chec
k
that
A
2
=
A
and
that
:
A
·
B
=
B
·
A
=
0
0
0
0
.
In
the
follo
wing,
w
e
admit
that,
for
an
y
strictly
p
ositiv
e
natural
n
um
b
er
n
:
A
n
=
A
;
B
n
=
B
c
Demonstrate
that,
for
an
y
natural
n
um
b
er
:
M
n
=
A
+
0.5
n
·
B
d
Deduce,
that
for
an
y
natural
n
um
b
er
n
:
p
n
=
0.8
−
0.8
×
0.5
n
e
In
the
long
term,
can
w
e
sa
y
with
certain
t
y
that
the
smok
er
will
stop
smoking?
E.6949
W
e
giv
e
the
matrices
:
M
=
1
1
1
1
−
1
1
4
2
1
;
I
=
1
0
0
0
1
0
0
0
1
P
art
A
1
Determine
the
matrix
M
2
.
W
e
giv
e
:
M
3
=
20
10
11
12
2
9
42
20
21
2
Chec
k
that
:
M
3
=
M
2
+8
·
M
+6
·
I
3
Deduce
that
M
is
in
v
ertible
and
that
:
M
−
1
=
1
6
·
M
2
−
M
−
8
·
I
.
P
art
B
:
Study
of
a
particular
case
W
e
seek
to
determine
three
in
tegers
a
,
b
and
c
suc
h
that
the
parab
ola
of
equation
y
=
a
·
x
2
+
b
·
x
+
c
passes
through
the
p
oin
ts
:
A
(
1
;
1
)
;
B
(
−
1
;
−
1
)
;
C
(
2
;
5
)
1
Demonstrate
that
the
problem
to
lo
ok
for
three
in
tegers
a
,
b
and
c
suc
h
that
:
M
·
a
b
c
=
1
−
1
5
2
Calculate
the
n
um
b
ers
a
,
b
and
c
and
c
hec
k
that
these
n
um
b
ers
are
in
tegers.
P
art
C
:
Bac
k
to
the
general
case
The
n
um
b
ers
a
,
b
,
c
,
p
,
q
,
r
are
in
tegers.
In
a
frame
of
reference
O
;
−
→
i
;
−
→
j
,
w
e
consider
the
p
oin
ts
A
(
1
;
p
)
,
B
(
−
1
;
q
)
et
C
(
2
;
r
)
.
W
e
are
lo
oking
for
the
v
alues
of
p
,
q
and
r
so
that
there
exists
a
parab
ola
with
equation
:
y
=
a
·
x
2
+
b
·
x
+
c
passing
through
A
,
B
and
C
.
1
Démon
trer
que
si
a
b
c
=
M
−
1
·
p
q
r
a
v
ec
a
,
b
and
c
in
te-
gers,
then
:
−
3
p
+
q
+
2
r
≡
0
(
mo
d.
6)
3
p
−
3
q
≡
0
(
mo
d.
6)
6
p
+
2
q
−
2
r
≡
0
(
mo
d.
6)
2
Deduce
that
:
q
−
r
≡
0
(
mo
d.
3)
p
−
q
≡
0
(
mo
d.
2)
3
Recipro
cally
,
w
e
admit
that
if
:
q
−
r
≡
0
(
mo
d.
3)
p
−
q
≡
0
(
mo
d.
2)
A
,
B
,
C
are
not
alignés
then
there
are
three
in
tegers
a
,
b
and
c
suc
h
that
the
parab
ola
of
equations
:
y
=
a
·
x
2
+
b
·
x
+
c
passes
through
the
p
oin
ts
A
,
B
and
C
.
a
Sho
w
that
the
p
oin
ts
A
,
B
and
C
are
aligned
if,
and
only
if
:
2
·
r
+
q
−
3
·
p
=
0
.
b
W
e
c
ho
ose
p
=
7
.
Determine
in
tegers
q
,
r
,
a
,
b
and
c
suc
h
that
the
parab
ola
of
equation
y
=
a
·
x
2
+
b
·
x
+
c
passes
through
the
p
oin
ts
A
,
B
and
C
.
E.8146
ASie
June
2018
E.8150
New
Caledonia
No
v
em
b
er
2018
5
p
oin
ts
h
ttps:/
/c
hingmath.fr
chapExoCorrec/6948
sacados/6948
Liban Mai 2015
chapExoCorrec/6949
sacados/6949
sacados/8146
Asie Juin 2018 Asie
sacados/8150
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