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Contests
National and Regional Contests
Argentina Contests
Argentina Team Selection Test
2006 Argentina Team Selection Test
2006 Argentina Team Selection Test
Part of
Argentina Team Selection Test
Subcontests
(5)
3
1
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Prove that chords are equal
In a circumference with center
O
O
O
we draw two equal chord AB\equal{}CD and if AB \cap CD \equal{}L then
A
L
>
B
L
AL>BL
A
L
>
B
L
and
D
L
>
C
L
DL>CL
D
L
>
C
L
We consider
M
∈
A
L
M \in AL
M
∈
A
L
and
N
∈
D
L
N \in DL
N
∈
D
L
such that \widehat {ALC} \equal{}2 \widehat {MON} Prove that the chord determined by extending
M
N
MN
MN
has the same as length as both
A
B
AB
A
B
and
C
D
CD
C
D
4
1
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Easy equation in integers
Find all integers solutions for xy\plus{}yz\plus{}zx\minus{}xyz\equal{}2
6
1
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Find the number of sequences
Let
n
n
n
be a natural number, and we consider the sequence
a
1
,
a
2
…
,
a
2
n
a_1, a_2 \ldots , a_{2n}
a
1
,
a
2
…
,
a
2
n
where a_i \in (\minus{}1,0,1) If we make the sum of consecutive members of the sequence, starting from one with an odd index and finishing in one with and even index, the result is
≤
2
\le 2
≤
2
and \ge \minus{}2 How many sequence are there satisfying this conditions?
1
1
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Sets include the sum or the difference of every two elements
Let
A
A
A
be a set of natural numbers in which if
a
a
a
,
b
b
b
belong to
A
A
A
(
a
>
b
a>b
a
>
b
) then either a\plus{}b or a\minus{}b belong to
A
A
A
( both cases may be posible at the same time). Decide wheter there is or not a set
A
A
A
consisting on exactly
100
100
100
elements which has four elements
x
x
x
,
y
y
y
,
z
z
z
,
w
w
w
( not necesarilly distinct) that satisfy x\minus{}y\equal{}512 and z\minus{}w\equal{}460 Daniel
2
1
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given 365 cards, in which distinct numbers are written
Given 365 cards, in which distinct numbers are written. We may ask for any three cards, the order of numbers written in them. Is it always possible to find out the order of all 365 cards by 2000 such questions?