MathDB
Problems
Contests
National and Regional Contests
Argentina Contests
Argentina National Olympiad
2006 Argentina National Olympiad
2006 Argentina National Olympiad
Part of
Argentina National Olympiad
Subcontests
(6)
6
1
Hide problems
a_1+a_2+... +a_n=a_1a_2 ... a_n=n
We will say that a natural number
n
n
n
is adequate if there exist
n
n
n
integers
a
1
,
a
2
,
…
,
a
n
a_1,a_2,\ldots ,a_n
a
1
,
a
2
,
…
,
a
n
(which are not necessarily positive and can be repeated) such that
a
1
+
a
2
+
⋯
+
a
n
=
a
1
a
2
⋯
a
n
=
n
.
a_1+a_2+\cdots +a_n=a_1a_2 \cdots a_n=n.
a
1
+
a
2
+
⋯
+
a
n
=
a
1
a
2
⋯
a
n
=
n
.
Determine all adequate numbers.
5
1
Hide problems
4000 gold coins among 40 pirates
The captain distributed
4000
4000
4000
gold coins among
40
40
40
pirates. A group of
5
5
5
pirates is called poor if those
5
5
5
pirates received, together,
500
500
500
coins or less. The captain made the distribution so that there were the minimum possible number of poor groups of
5
5
5
pirates. Determine how many poor
5
5
5
pirate groups there are. Clarification: Two groups of
5
5
5
pirates are considered different if there is at least one pirate in one of them who is not in the other.
4
1
Hide problems
1010 numbers in each rearrangement of the 2006 integer numbers
Find the greatest number
M
M
M
with the following property: in each rearrangement of the
2006
2006
2006
integer numbers
1
,
2
,
.
.
.
2006
1,2,...2006
1
,
2
,
...2006
there are
1010
1010
1010
numbers located consecutively in that rearrangement whose sum is greater than or equal to
M
M
M
.
3
1
Hide problems
succession of positive integers of $2006$ terms, 2 player game
Pablo and Nacho write together a succession of positive integers of
2006
2006
2006
terms, according to the following rules: Pablo begins, who in his first turn writes
1
1
1
, and from then on, each one in his turn writes an integer positive that is greater than or equal to the last number that the opponent wrote and less than or equal to triple the last number that the opponent wrote. When the two of them have written the
2006
2006
2006
numbers, the sum
S
S
S
of the first
2005
2005
2005
numbers written (all except the last one) and the sum
T
T
T
of the
2006
2006
2006
numbers written. If
S
S
S
and
T
T
T
are co-cousins, Nacho wins. Otherwise, Pablo wins. Determine which of the two players has a winning strategy, describe the strategy and demonstrate that it is a winning one.
1
1
Hide problems
na-a is positive integer, period of 10 digits
Let
A
A
A
be the set of positive real numbers less than
1
1
1
that have a periodic decimal expansion with a period of ten different digits. Find a positive integer
n
n
n
greater than
1
1
1
and less than
1
0
10
10^{10}
1
0
10
such that
n
a
−
a
na-a
na
−
a
is a positive integer for all
a
a
a
. of set
A
A
A
.
2
1
Hide problems
AB/BC wanted, MD_|_BD, angle bisector, midpoint related
In triangle
A
B
C
,
M
ABC, M
A
BC
,
M
is the midpoint of
A
B
AB
A
B
and
D
D
D
the foot of the bisector of angle
∠
A
B
C
\angle ABC
∠
A
BC
. If
M
D
MD
M
D
and
B
D
BD
B
D
are known to be perpendicular, calculate
A
B
B
C
\frac{AB}{BC}
BC
A
B
.