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Problems
Contests
National and Regional Contests
Argentina Contests
Argentina Team Selection Test
2009 Argentina Team Selection Test
2009 Argentina Team Selection Test
Part of
Argentina Team Selection Test
Subcontests
(6)
6
1
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A game on combinatorial number theory
Let
n
≥
3
n \geq 3
n
≥
3
be an odd integer. We denote by [\minus{}n,n] the set of all integers greater or equal than \minus{}n and less or equal than
n
n
n
. Player
A
A
A
chooses an arbitrary positive integer
k
k
k
, then player
B
B
B
picks a subset of
k
k
k
(distinct) elements from [\minus{}n,n]. Let this subset be
S
S
S
. If all numbers in [\minus{}n,n] can be written as the sum of exactly
n
n
n
distinct elements of
S
S
S
, then player
A
A
A
wins the game. If not,
B
B
B
wins. Find the least value of
k
k
k
such that player
A
A
A
can always win the game.
5
1
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Indirect friends
There are several contestants at a math olympiad. We say that two contestants
A
A
A
and
B
B
B
are indirect friends if there are contestants
C
1
,
C
2
,
.
.
.
,
C
n
C_1, C_2, ..., C_n
C
1
,
C
2
,
...
,
C
n
such that
A
A
A
and
C
1
C_1
C
1
are friends,
C
1
C_1
C
1
and
C
2
C_2
C
2
are friends,
C
2
C_2
C
2
and
C
3
C_3
C
3
are friends, ...,
C
n
C_n
C
n
and
B
B
B
are friends. In particular, if
A
A
A
and
B
B
B
are friends themselves, they are indirect friends as well. Some of the contestants were friends before the olympiad. During the olympiad, some contestants make new friends, so that after the olympiad every contestant has at least one friend among the other contestants. We say that a contestant is special if, after the olympiad, he has exactly twice as indirect friends as he had before the olympiad. Prove that the number of special contestants is less or equal than
2
3
\frac{2}{3}
3
2
of the total number of contestants.
4
1
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Find all n
Find all positive integers
n
n
n
such that 20^n \minus{} 13^n \minus{} 7^n is divisible by
309
309
309
.
1
1
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No right-angled triangle
On a
50
×
50
50 \times 50
50
×
50
board, the centers of several unit squares are colored black. Find the maximum number of centers that can be colored black in such a way that no three black points form a right-angled triangle.
2
1
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Find the maximum value
Let
a
1
,
a
2
,
.
.
.
,
a
300
a_1, a_2, ..., a_{300}
a
1
,
a
2
,
...
,
a
300
be nonnegative real numbers, with \sum_{i\equal{}1}^{300} a_i \equal{} 1. Find the maximum possible value of
∑
i
≠
j
,
i
∣
j
a
i
a
j
\sum_{i \neq j, i|j} a_ia_j
∑
i
=
j
,
i
∣
j
a
i
a
j
.
3
1
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Prove AP/AQ = 3/2
Let
A
B
C
ABC
A
BC
be a triangle,
B
1
B_1
B
1
the midpoint of side
A
B
AB
A
B
and
C
1
C_1
C
1
the midpoint of side
A
C
AC
A
C
. Let
P
P
P
be the point of intersection (
≠
A
\neq A
=
A
) of the circumcircles of triangles
A
B
C
1
ABC_1
A
B
C
1
and
A
B
1
C
AB_1C
A
B
1
C
. Let
Q
Q
Q
be the point of intersection (
≠
A
\neq A
=
A
) of the line
A
P
AP
A
P
and the circumcircle of triangle
A
B
1
C
1
AB_1C_1
A
B
1
C
1
. Prove that \frac{AP}{AQ} \equal{} \frac{3}{2}.