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National and Regional Contests
Argentina Contests
Argentina Team Selection Test
2008 Argentina Team Selection Test
2008 Argentina Team Selection Test
Part of
Argentina Team Selection Test
Subcontests
(6)
1
1
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Find the number of arrangments in 100-gon
In the vertexes of a regular
100
100
100
-gon we place the numbers from
1
1
1
to
100
100
100
, in some order, every number appearing exactly once. We say that an arrangment of the numbers is happy if for every simmetry axis of the polygon, the numbers which are from one side of the axis are greater that their respective simmetrics (we don't take into consideration the numbers which are on the axis) Find all happy arrangments (If two happy arrangments are the same under the rotation they are considered as only one)
5
1
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If ABPE is cyclic, DP parallel to AB
Let
A
B
C
ABC
A
BC
be a triangle,
D
D
D
,
E
E
E
and
F
F
F
the points of tangency of the incircle with sides
B
C
BC
BC
,
C
A
CA
C
A
,
A
B
AB
A
B
respectively. Let
P
P
P
be the second point of intersection of
C
F
CF
CF
and the incircle. If
A
B
P
E
ABPE
A
BPE
is a cyclic quadrilateral prove that
D
P
DP
D
P
is parellel to
A
B
AB
A
B
2
1
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Circle + ratios
Triangle
A
B
C
ABC
A
BC
is inscript in a circumference
Γ
\Gamma
Γ
. A chord MN\equal{}1 of
Γ
\Gamma
Γ
intersects the sides
A
B
AB
A
B
and
A
C
AC
A
C
at
X
X
X
and
Y
Y
Y
respectively, with
M
M
M
,
X
X
X
,
Y
Y
Y
,
N
N
N
in that order in
M
N
MN
MN
. Let
U
V
UV
U
V
be the diameter of
Γ
\Gamma
Γ
perpendicular to
M
N
MN
MN
with
U
U
U
and
A
A
A
in the same semiplane respect to
M
N
MN
MN
. Lines
A
V
AV
A
V
,
B
U
BU
B
U
and
C
U
CU
C
U
cut
M
N
MN
MN
in the ratios
3
2
\frac{3}{2}
2
3
,
4
5
\frac{4}{5}
5
4
and
7
6
\frac{7}{6}
6
7
respectively (starting counting from
M
M
M
). Find
X
Y
XY
X
Y
3
1
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function equation from R+->R+
-Find all the functions from \mathbb R^\plus{} \to \mathbb R^\plus{} that satisfy: x^2(f(x)\plus{}f(y))\equal{}(x\plus{}y)(f(yf(x)))
4
1
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n/d(n)=p
Let
d
(
n
)
d(n)
d
(
n
)
be the number of positive divisors of the natural number
n
n
n
. Find all
n
n
n
such that \frac {n} {d(n)}\equal{}p where
p
p
p
is a prime number Daniel
6
1
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Triangle Ineq
Show that in acute triangle ABC we have:
a
5
+
b
5
+
c
5
a
4
+
b
4
+
c
4
≥
3
R
\frac{a^5+b^5+c^5}{a^4+b^4+c^4} \ge \sqrt{3}R
a
4
+
b
4
+
c
4
a
5
+
b
5
+
c
5
≥
3
R