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Contests
National and Regional Contests
France Contests
France Team Selection Test
2012 France Team Selection Test
2012 France Team Selection Test
Part of
France Team Selection Test
Subcontests
(3)
3
1
Hide problems
(Very) well-known equation by now
Let
p
p
p
be a prime number. Find all positive integers
a
,
b
,
c
≥
1
a,b,c\ge 1
a
,
b
,
c
≥
1
such that:
a
p
+
b
p
=
p
c
.
a^p+b^p=p^c.
a
p
+
b
p
=
p
c
.
2
2
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AP, BC and OH are concurrent iff AH=HN
Let
A
B
C
ABC
A
BC
be an acute-angled triangle with
A
B
≠
A
C
AB\not= AC
A
B
=
A
C
. Let
Γ
\Gamma
Γ
be the circumcircle,
H
H
H
the orthocentre and
O
O
O
the centre of
Γ
\Gamma
Γ
.
M
M
M
is the midpoint of
B
C
BC
BC
. The line
A
M
AM
A
M
meets
Γ
\Gamma
Γ
again at
N
N
N
and the circle with diameter
A
M
AM
A
M
crosses
Γ
\Gamma
Γ
again at
P
P
P
. Prove that the lines
A
P
,
B
C
,
O
H
AP,BC,OH
A
P
,
BC
,
O
H
are concurrent if and only if
A
H
=
H
N
AH=HN
A
H
=
H
N
.
The roots are the coefficients
Determine all non-constant polynomials
X
n
+
a
n
−
1
X
n
−
1
+
⋯
+
a
1
X
+
a
0
X^n+a_{n-1}X^{n-1}+\cdots +a_1X+a_0
X
n
+
a
n
−
1
X
n
−
1
+
⋯
+
a
1
X
+
a
0
with integer coefficients for which the roots are exactly the numbers
a
0
,
a
1
,
…
,
a
n
−
1
a_0,a_1,\ldots ,a_{n-1}
a
0
,
a
1
,
…
,
a
n
−
1
(with multiplicity).
1
2
Hide problems
A group of k people where some (n+1)-th knows all n
Let
n
n
n
and
k
k
k
be two positive integers. Consider a group of
k
k
k
people such that, for each group of
n
n
n
people, there is a
(
n
+
1
)
(n+1)
(
n
+
1
)
-th person that knows them all (if
A
A
A
knows
B
B
B
then
B
B
B
knows
A
A
A
). 1) If
k
=
2
n
+
1
k=2n+1
k
=
2
n
+
1
, prove that there exists a person who knows all others. 2) If
k
=
2
n
+
2
k=2n+2
k
=
2
n
+
2
, give an example of such a group in which no-one knows all others.
k-tastrophic functions
Let
k
>
1
k>1
k
>
1
be an integer. A function
f
:
N
∗
→
N
∗
f:\mathbb{N^*}\to\mathbb{N^*}
f
:
N
∗
→
N
∗
is called
k
k
k
-tastrophic when for every integer
n
>
0
n>0
n
>
0
, we have
f
k
(
n
)
=
n
k
f_k(n)=n^k
f
k
(
n
)
=
n
k
where
f
k
f_k
f
k
is the
k
k
k
-th iteration of
f
f
f
:
f
k
(
n
)
=
f
∘
f
∘
⋯
∘
f
⏟
k
times
(
n
)
f_k(n)=\underbrace{f\circ f\circ\cdots \circ f}_{k\text{ times}}(n)
f
k
(
n
)
=
k
times
f
∘
f
∘
⋯
∘
f
(
n
)
For which
k
k
k
does there exist a
k
k
k
-tastrophic function?