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Contests
National and Regional Contests
Argentina Contests
Argentina Cono Sur TST
2013 Argentina Cono Sur TST
2013 Argentina Cono Sur TST
Part of
Argentina Cono Sur TST
Subcontests
(5)
6
1
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Argentina Cono Sur TST 2013. Problem 6
Let
m
≥
4
m\geq 4
m
≥
4
and
n
≥
4
n\geq 4
n
≥
4
. An integer is written on each cell of a
m
×
n
m \times n
m
×
n
board. If each cell has a number equal to the arithmetic mean of some pair of numbers written on its neighbouring cells, determine the maximum amount of distinct numbers that the board may have. Note: two neighbouring cells share a common side.
5
1
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Argentina Cono Sur TST 2013. Problem 5
Let
A
B
C
ABC
A
BC
be an equilateral triangle and
D
D
D
a point on side
A
C
AC
A
C
. Let
E
E
E
be a point on
B
C
BC
BC
such that
D
E
⊥
B
C
DE \perp BC
D
E
⊥
BC
,
F
F
F
on
A
B
AB
A
B
such that
E
F
⊥
A
B
EF \perp AB
EF
⊥
A
B
, and
G
G
G
on
A
C
AC
A
C
such that
F
G
⊥
A
C
FG \perp AC
FG
⊥
A
C
. Lines
F
G
FG
FG
and
D
E
DE
D
E
intersect in
P
P
P
. If
M
M
M
is the midpoint of
B
C
BC
BC
, show that
B
P
BP
BP
bisects
A
M
AM
A
M
.
4
1
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Show that a number is always a perfect square
Show that the number
N
=
44
…
4
⏟
n
88
…
8
⏟
n
−
1
33
…
3
⏟
n
−
1
2
\begin{matrix} \\ N= \end{matrix} \underbrace{44 \ldots 4}_{n} \underbrace{88 \ldots 8}_{n} - 1\underbrace{33 \ldots3 }_{n-1}2
N
=
n
44
…
4
n
88
…
8
−
1
n
−
1
33
…
3
2
is a perfect square for all positive integers
n
n
n
.
3
1
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1390 Ants near a line
1390
1390
1390
ants are placed near a line, such that the distance between their heads and the line is less than
1
cm
1\text{cm}
1
cm
and the distance between the heads of two ants is always larger than
2
cm
2\text{cm}
2
cm
. Show that there is at least one pair of ants such that the distance between their heads is at least
10
10
10
meters (consider the head of an ant as point).
1
1
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Truth-tellers and liars on a line
2000
2000
2000
people are standing on a line. Each one of them is either a liar, who will always lie, or a truth-teller, who will always tell the truth. Each one of them says: "there are more liars to my left than truth-tellers to my right". Determine, if possible, how many people from each class are on the line.