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Problems
Contests
National and Regional Contests
Argentina Contests
Argentina Team Selection Test
2010 Argentina Team Selection Test
2010 Argentina Team Selection Test
Part of
Argentina Team Selection Test
Subcontests
(6)
4
1
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Game in rhombus - Argentina TST 2010
Two players,
A
A
A
and
B
B
B
, play a game on a board which is a rhombus of side
n
n
n
and angles of
6
0
∘
60^{\circ}
6
0
∘
and
12
0
∘
120^{\circ}
12
0
∘
, divided into
2
n
2
2n^2
2
n
2
equilateral triangles, as shown in the diagram for
n
=
4
n=4
n
=
4
.
A
A
A
uses a red token and
B
B
B
uses a blue token, which are initially placed in cells containing opposite corners of the board (the
6
0
∘
60^{\circ}
6
0
∘
ones). In turns, players move their token to a neighboring cell (sharing a side with the previous one). To win the game, a player must either place his token on the cell containing the other player's token, or get to the opposite corner to the one where he started. If
A
A
A
starts the game, determine which player has a winning strategy.
6
1
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Divide {1,2,...,2010} in subsets - Argentina TST 2010
Suppose
a
1
,
a
2
,
.
.
.
,
a
r
a_1, a_2, ..., a_r
a
1
,
a
2
,
...
,
a
r
are integers with
a
i
≥
2
a_i \geq 2
a
i
≥
2
for all
i
i
i
such that
a
1
+
a
2
+
.
.
.
+
a
r
=
2010
a_1 + a_2 + ... + a_r = 2010
a
1
+
a
2
+
...
+
a
r
=
2010
. Prove that the set
{
1
,
2
,
3
,
.
.
.
,
2010
}
\{1,2,3,...,2010\}
{
1
,
2
,
3
,
...
,
2010
}
can be partitioned in
r
r
r
subsets
A
1
,
A
2
,
.
.
.
,
A
r
A_1, A_2, ..., A_r
A
1
,
A
2
,
...
,
A
r
each with
a
1
,
a
2
,
.
.
.
,
a
r
a_1, a_2, ..., a_r
a
1
,
a
2
,
...
,
a
r
elements respectively, such that the sum of the numbers on each subset is divisible by
2011
2011
2011
. Decide whether this property still holds if we replace
2010
2010
2010
by
2011
2011
2011
and
2011
2011
2011
by
2012
2012
2012
(that is, if the set to be partitioned is
{
1
,
2
,
3
,
.
.
.
,
2011
}
\{1,2,3,...,2011\}
{
1
,
2
,
3
,
...
,
2011
}
).
1
1
Hide problems
Football tournament - Argentina TST 2010
In a football tournament there are
8
8
8
teams, each of which plays exacly one match against every other team. If a team
A
A
A
defeats team
B
B
B
, then
A
A
A
is awarded
3
3
3
points and
B
B
B
gets
0
0
0
points. If they end up in a tie, they receive
1
1
1
point each. It turned out that in this tournament, whenever a match ended up in a tie, the two teams involved did not finish with the same final score. Find the maximum number of ties that could have happened in such a tournament.
5
1
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Sequence with primes - Argentina TST 2010
Let
p
p
p
and
q
q
q
be prime numbers. The sequence
(
x
n
)
(x_n)
(
x
n
)
is defined by
x
1
=
1
x_1 = 1
x
1
=
1
,
x
2
=
p
x_2 = p
x
2
=
p
and
x
n
+
1
=
p
x
n
−
q
x
n
−
1
x_{n+1} = px_n - qx_{n-1}
x
n
+
1
=
p
x
n
−
q
x
n
−
1
for all
n
≥
2
n \geq 2
n
≥
2
. Given that there is some
k
k
k
such that
x
3
k
=
−
3
x_{3k} = -3
x
3
k
=
−
3
, find
p
p
p
and
q
q
q
.
2
1
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Concyclic points - Argentina TST 2010
Let
A
B
C
ABC
A
BC
be a triangle with
A
B
=
A
C
AB = AC
A
B
=
A
C
. The incircle touches
B
C
BC
BC
,
A
C
AC
A
C
and
A
B
AB
A
B
at
D
D
D
,
E
E
E
and
F
F
F
respectively. Let
P
P
P
be a point on the arc \overarc{EF} that does not contain
D
D
D
. Let
Q
Q
Q
be the second point of intersection of
B
P
BP
BP
and the incircle of
A
B
C
ABC
A
BC
. The lines
E
P
EP
EP
and
E
Q
EQ
EQ
meet the line
B
C
BC
BC
at
M
M
M
and
N
N
N
, respectively. Prove that the four points
P
,
F
,
B
,
M
P, F, B, M
P
,
F
,
B
,
M
lie on a circle and
E
M
E
N
=
B
F
B
P
\frac{EM}{EN} = \frac{BF}{BP}
EN
EM
=
BP
BF
.
3
1
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Functional equation - Argentina TST 2010
Find all functions
f
:
R
→
R
f: \mathbb R \rightarrow \mathbb R
f
:
R
→
R
such that
f
(
x
+
x
y
+
f
(
y
)
)
=
(
f
(
x
)
+
1
2
)
(
f
(
y
)
+
1
2
)
f(x+xy+f(y)) = \left(f(x)+\frac{1}{2}\right) \left(f(y)+\frac{1}{2}\right)
f
(
x
+
x
y
+
f
(
y
))
=
(
f
(
x
)
+
2
1
)
(
f
(
y
)
+
2
1
)
holds for all real numbers
x
,
y
x,y
x
,
y
.