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National and Regional Contests
Argentina Contests
Argentina Iberoamerican TST
2009 Argentina Iberoamerican TST
2009 Argentina Iberoamerican TST
Part of
Argentina Iberoamerican TST
Subcontests
(3)
1
2
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Operation on a 31-gon
In the vertexes of a regular
31
31
31
-gon there are written the numbers from
1
1
1
to
31
31
31
, ordered increasingly, clockwise oriented. We are allowed to perform an operation which consists in taking any three vertexes, namely the ones who have written
a
a
a
,
b
b
b
, and
c
c
c
and change them into
c
c
c
, a\minus{}\frac{1}{10} and b\plus{}\frac{1}{10} respectively (
a
a
a
becomes
c
c
c
,
b
b
b
becomes a\minus{}\frac{1}{10} and
c
c
c
turns into b\plus{}\frac{1}{10} Prove that after applying several operations we can reach the state in which the numbers in the vertexes are the numbers from
1
1
1
to
31
31
31
, ordered increasingly,anti-clockwise oriented.
Find integers x and y
Find all positive integers
(
x
,
y
)
(x,y)
(
x
,
y
)
such that \frac{y^2x}{x\plus{}y} is a prime number
3
1
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Least value difference between person with max. nb. friends
Within a group of
2009
2009
2009
people, every two people has exactly one common friend. Find the least value of the difference between the person with maximum number of friends and the person with minimum number of friends.
2
2
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Prove the lead wins the game
There are m \plus{} 1 horizontal lines and
m
m
m
vertical lines on the plane so that m(m \plus{} 1) intersections are made. A mark is placed at one of the
m
m
m
points of the lowest horizontal line. 2 players play the game of the following rules on this lines and points. 1. Each player moves a mark from a point to a point along the lines in turns. 2. The segment is erased after a mark moved along it. 3. When a player cannot make a move, then he loses. Prove that the lead always wins the game. PS I haven't found a student who solved it. There can be no one.
Sequence and product of digits
Let
a
a
a
and
k
k
k
be positive integers. Let
a
i
a_i
a
i
be the sequence defined by a_1 \equal{} a and a_{n \plus{} 1} \equal{} a_n \plus{} k\pi(a_n) where
π
(
x
)
\pi(x)
π
(
x
)
is the product of the digits of
x
x
x
(written in base ten) Prove that we can choose
a
a
a
and
k
k
k
such that the infinite sequence
a
i
a_i
a
i
contains exactly
100
100
100
distinct terms