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Problems
Contests
International Contests
Cono Sur Olympiad
2016 Cono Sur Olympiad
2016 Cono Sur Olympiad
Part of
Cono Sur Olympiad
Subcontests
(6)
6
1
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Friendly numbers
We say that three different integers are friendly if one of them divides the product of the other two. Let
n
n
n
be a positive integer.a) Show that, between
n
2
n^2
n
2
and
n
2
+
n
n^2+n
n
2
+
n
, exclusive, does not exist any triplet of friendly numbers.b) Determine if for each
n
n
n
exists a triplet of friendly numbers between
n
2
n^2
n
2
and
n
2
+
n
+
3
n
n^2+n+3\sqrt{n}
n
2
+
n
+
3
n
, exclusive.
5
1
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Incenter and excenter
Let
A
B
C
ABC
A
BC
be a triangle inscribed on a circle with center
O
O
O
. Let
D
D
D
and
E
E
E
be points on the sides
A
B
AB
A
B
and
B
C
BC
BC
,respectively, such that
A
D
=
D
E
=
E
C
AD = DE = EC
A
D
=
D
E
=
EC
. Let
X
X
X
be the intersection of the angle bisectors of
∠
A
D
E
\angle ADE
∠
A
D
E
and
∠
D
E
C
\angle DEC
∠
D
EC
. If
X
≠
O
X \neq O
X
=
O
, show that, the lines
O
X
OX
OX
and
D
E
DE
D
E
are perpendicular.
4
1
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Sum of digits equation
Let
S
(
n
)
S(n)
S
(
n
)
be the sum of the digits of the positive integer
n
n
n
. Find all
n
n
n
such that
S
(
n
)
(
S
(
n
)
−
1
)
=
n
−
1
S(n)(S(n)-1)=n-1
S
(
n
)
(
S
(
n
)
−
1
)
=
n
−
1
.
3
1
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Winning Strategy
There are
2016
2016
2016
positions marked around a circle, with a token on one of them. A legitimate move is to move the token either 1 position or 4 positions from its location, clockwise. The restriction is that the token can not occupy the same position more than once. Players
A
A
A
and
B
B
B
take turns making moves. Player
A
A
A
has the first move. The first player who cannot make a legitimate move loses. Determine which of the two players has a winning strategy.
2
1
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Diophantic equations
For every
k
=
1
,
2
,
…
k= 1,2, \ldots
k
=
1
,
2
,
…
let
s
k
s_k
s
k
be the number of pairs
(
x
,
y
)
(x,y)
(
x
,
y
)
satisfying the equation
k
x
+
(
k
+
1
)
y
=
1001
−
k
kx + (k+1)y = 1001 - k
k
x
+
(
k
+
1
)
y
=
1001
−
k
with
x
x
x
,
y
y
y
non-negative integers. Find
s
1
+
s
2
+
⋯
+
s
200
s_1 + s_2 + \cdots + s_{200}
s
1
+
s
2
+
⋯
+
s
200
.
1
1
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Special Numbers
Let
a
b
c
d
‾
\overline{abcd}
ab
c
d
be one of the 9999 numbers
0001
,
0002
,
0003
,
…
,
9998
,
9999
0001, 0002, 0003, \ldots, 9998, 9999
0001
,
0002
,
0003
,
…
,
9998
,
9999
. Let
a
b
c
d
‾
\overline{abcd}
ab
c
d
be an special number if
a
b
−
c
d
ab-cd
ab
−
c
d
and
a
b
+
c
d
ab+cd
ab
+
c
d
are perfect squares,
a
b
−
c
d
ab-cd
ab
−
c
d
divides
a
b
+
c
d
ab+cd
ab
+
c
d
and also
a
b
+
c
d
ab+cd
ab
+
c
d
divides
a
b
c
d
abcd
ab
c
d
. For example 2016 is special. Find all the
a
b
c
d
‾
\overline{abcd}
ab
c
d
special numbers. Note: If
a
b
c
d
‾
=
0206
\overline{abcd}=0206
ab
c
d
=
0206
, then
a
b
=
02
ab=02
ab
=
02
and
c
d
=
06
cd=06
c
d
=
06
.