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Problems
Contests
National and Regional Contests
Argentina Contests
Argentina National Olympiad
2007 Argentina National Olympiad
2007 Argentina National Olympiad
Part of
Argentina National Olympiad
Subcontests
(6)
4
1
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k not integers sum among 45 sums of pairs of 2 real numbers among 10
10
10
10
real numbers are given
a
1
,
a
2
,
…
,
a
10
a_1,a_2,\ldots ,a_{10}
a
1
,
a
2
,
…
,
a
10
, and the
45
45
45
sums of two of these numbers are formed
a
i
+
a
j
a_i+a_j
a
i
+
a
j
, 1\leq i<j\leq 10 . It is known that not all these sums are integers. Determine the minimum value of
k
k
k
such that it is possible that among the
45
45
45
sums there are
k
k
k
that are not integers and
45
−
k
45-k
45
−
k
that are integers.
6
1
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2007 points on plane
Julián chooses
2007
2007
2007
points of the plane between which there are no
3
3
3
aligned, and draw with red all the segments that join two of those points. Next, Roberto draws several lines. Its objective is for each red segment to be cut inside by (at least) one of the lines. Determine the minor
ℓ
\ell
ℓ
lines such that, no matter how Julián chooses the
2007
2007
2007
points, with the properly chosen
ℓ
\ell
ℓ
lines, Roberto will achieve his objective with certainty.
5
1
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sum of its digits is divisible by 31
We will say that a positive integer is lucky [/i ]if the sum of its digits is divisible by
31
31
31
. What is the maximum possible difference between two consecutive lucky [/i ] numbers?
2
1
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4 colors in grid
The pieces in a game are squares of side
1
1
1
with their sides colored with
4
4
4
colors: blue, red, yellow and green, so that each piece has one side of each color. There are pieces in every possible color arrangement, and the game has a million pieces of each kind. With the pieces, rectangular puzzles are assembled, without gaps or overlaps, so that two pieces that share a side have that side of the same color. Determine if with this procedure you can make a rectangle of
99
×
2007
99\times 2007
99
×
2007
with one side of each color. And
100
×
2008
100\times 2008
100
×
2008
? And
99
×
2008
99\times 2008
99
×
2008
?
1
1
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p^2+q=37q^2+p
Find all the prime numbers
p
p
p
and
q
q
q
such that
p
2
+
q
=
37
q
2
+
p
p^2+q=37q^2+p
p
2
+
q
=
37
q
2
+
p
. Clarification:
1
1
1
is not a prime number.
3
1
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Find ratio
Let
A
B
C
D
ABCD
A
BC
D
be a parellogram with
A
B
>
A
D
AB>AD
A
B
>
A
D
. Suposse the ratio between diagonals
A
C
AC
A
C
and
B
D
BD
B
D
is \frac {AC} {BD}\equal{}3. Let
r
r
r
be the line symmetric to
A
D
AD
A
D
with respect to
A
C
AC
A
C
and
s
s
s
the line symmetric to
B
C
BC
BC
with respect to
B
D
BD
B
D
. If
r
r
r
and
s
s
s
intersect at
P
P
P
, find the ratio
P
A
P
B
\frac {PA} {PB}
PB
P
A
Daniel