MathDB
Problems
Contests
National and Regional Contests
France Contests
France Team Selection Test
2002 France Team Selection Test
2002 France Team Selection Test
Part of
France Team Selection Test
Subcontests
(3)
3
2
Hide problems
a_{2i} is a permutation of {1,2,3...,n} iff a_1-a_2n=n
Let
n
n
n
be a positive integer and let
(
a
1
,
a
2
,
…
,
a
2
n
)
(a_1,a_2,\ldots ,a_{2n})
(
a
1
,
a
2
,
…
,
a
2
n
)
be a permutation of
1
,
2
,
…
,
2
n
1,2,\ldots ,2n
1
,
2
,
…
,
2
n
such that the numbers
∣
a
i
+
1
−
a
i
∣
|a_{i+1}-a_i|
∣
a
i
+
1
−
a
i
∣
are pairwise distinct for
i
=
1
,
…
,
2
n
−
1
i=1,\ldots ,2n-1
i
=
1
,
…
,
2
n
−
1
. Prove that
{
a
2
,
a
4
,
…
,
a
2
n
}
=
{
1
,
2
,
…
,
n
}
\{a_2,a_4,\ldots ,a_{2n}\}=\{1,2,\ldots ,n\}
{
a
2
,
a
4
,
…
,
a
2
n
}
=
{
1
,
2
,
…
,
n
}
if and only if
a
1
−
a
2
n
=
n
a_1-a_{2n}=n
a
1
−
a
2
n
=
n
.
All sums a_i + a_j are distinct
Let
p
≥
3
p\ge 3
p
≥
3
be a prime number. Show that there exist
p
p
p
positive integers
a
1
,
a
2
,
…
,
a
p
a_1,a_2,\ldots ,a_p
a
1
,
a
2
,
…
,
a
p
not exceeding
2
p
2
2p^2
2
p
2
such that the
p
(
p
−
1
)
2
\frac{p(p-1)}{2}
2
p
(
p
−
1
)
sums
a
i
+
a
j
(
i
<
j
)
a_i+a_j\ (i<j)
a
i
+
a
j
(
i
<
j
)
are all distinct.
1
2
Hide problems
MA_1 is tangent to (A_1B_1C)
In an acute-angled triangle
A
B
C
ABC
A
BC
,
A
1
A_1
A
1
and
B
1
B_1
B
1
are the feet of the altitudes from
A
A
A
and
B
B
B
respectively, and
M
M
M
is the midpoint of
A
B
AB
A
B
. a) Prove that
M
A
1
MA_1
M
A
1
is tangent to the circumcircle of triangle
A
1
B
1
C
A_1B_1C
A
1
B
1
C
. b) Prove that the circumcircles of triangles
A
1
B
1
C
,
B
M
A
1
A_1B_1C,BMA_1
A
1
B
1
C
,
BM
A
1
, and
A
M
B
1
AMB_1
A
M
B
1
have a common point.
Three Colleges
There are three colleges in a town. Each college has
n
n
n
students. Any student of any college knows
n
+
1
n+1
n
+
1
students of the other two colleges. Prove that it is possible to choose a student from each of the three colleges so that all three students would know each other.
2
2
Hide problems
Elements in S such that difference of divisors is 22
Consider the set
S
S
S
of integers
k
k
k
which are products of four distinct primes. Such an integer
k
=
p
1
p
2
p
3
p
4
k=p_1p_2p_3p_4
k
=
p
1
p
2
p
3
p
4
has
16
16
16
positive divisors
1
=
d
1
<
d
2
<
…
<
d
15
<
d
16
=
k
1=d_1<d_2<\ldots <d_{15}<d_{16}=k
1
=
d
1
<
d
2
<
…
<
d
15
<
d
16
=
k
. Find all elements of
S
S
S
less than
2002
2002
2002
such that
d
9
−
d
8
=
22
d_9-d_8=22
d
9
−
d
8
=
22
.
condition for angle AIO < 90
Let
A
B
C
ABC
A
BC
be a non-equilateral triangle. Denote by
I
I
I
the incenter and by
O
O
O
the circumcenter of the triangle
A
B
C
ABC
A
BC
. Prove that
∠
A
I
O
≤
π
2
\angle AIO\leq\frac{\pi}{2}
∠
A
I
O
≤
2
π
holds if and only if 2\cdot BC\leq AB\plus{}AC.