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International Contests
Cono Sur Olympiad
2019 Cono Sur Olympiad
2019 Cono Sur Olympiad
Part of
Cono Sur Olympiad
Subcontests
(6)
6
1
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2019 Cono Sur Mathematical Olympiad, P6
Let
A
B
C
ABC
A
BC
be an acute-angled triangle with
A
B
<
A
C
AB< AC
A
B
<
A
C
, and let
H
H
H
be its orthocenter. The circumference with diameter
A
H
AH
A
H
meets the circumscribed circumference of
A
B
C
ABC
A
BC
at
P
≠
A
P\neq A
P
=
A
. The tangent to the circumscribed circumference of
A
B
C
ABC
A
BC
through
P
P
P
intersects line
B
C
BC
BC
at
Q
Q
Q
. Show that
Q
P
=
Q
H
QP=QH
QP
=
Q
H
.
5
1
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2019 Cono Sur Mathematical Olympiad, P5
Let
n
≥
3
n\geq 3
n
≥
3
a positive integer. In each cell of a
n
×
n
n\times n
n
×
n
chessboard one must write
1
1
1
or
2
2
2
in such a way the sum of all written numbers in each
2
×
3
2\times 3
2
×
3
and
3
×
2
3\times 2
3
×
2
sub-chessboard is even. How many different ways can the chessboard be completed?
4
1
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2019 Cono Sur Mathematical Olympiad, P4
Find all positive prime numbers
p
,
q
,
r
,
s
p,q,r,s
p
,
q
,
r
,
s
so that
p
2
+
2019
=
26
(
q
2
+
r
2
+
s
2
)
p^2+2019=26(q^2+r^2+s^2)
p
2
+
2019
=
26
(
q
2
+
r
2
+
s
2
)
.
3
1
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2019 Cono Sur Mathematical Olympiad, P3
Let
n
≥
3
n\geq 3
n
≥
3
an integer. Determine whether there exist permutations
(
a
1
,
a
2
,
…
,
a
n
)
(a_1,a_2, \ldots, a_n)
(
a
1
,
a
2
,
…
,
a
n
)
of the numbers
(
1
,
2
,
…
,
n
)
(1,2,\ldots, n)
(
1
,
2
,
…
,
n
)
and
(
b
1
,
b
2
,
…
,
b
n
)
(b_1, b_2, \ldots, b_n)
(
b
1
,
b
2
,
…
,
b
n
)
of the numbers
(
n
+
1
,
n
+
2
,
…
,
2
n
)
(n+1,n+2,\ldots, 2n)
(
n
+
1
,
n
+
2
,
…
,
2
n
)
so that
(
a
1
b
1
,
a
2
b
2
,
…
a
n
b
n
)
(a_1b_1, a_2b_2, \ldots a_nb_n)
(
a
1
b
1
,
a
2
b
2
,
…
a
n
b
n
)
is a strictly increasing arithmetic progression.
2
1
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2019 Cono Sur Mathematical Olympiad, P2
We say that a positive integer
M
M
M
with
2
n
2n
2
n
digits is hypersquared if the following three conditions are met:[*]
M
M
M
is a perfect square. [*]The number formed by the first
n
n
n
digits of
M
M
M
is a perfect square. [*]The number formed by the last
n
n
n
digits of
M
M
M
is a perfect square and has exactly
n
n
n
digits (its first digit is not zero).Find a hypersquared number with
2000
2000
2000
digits.
1
1
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2019 Cono Sur Mathematical Olympiad, P1
Martin has two boxes
A
A
A
and
B
B
B
. In the box
A
A
A
there are
100
100
100
red balls numbered from
1
1
1
to
100
100
100
, each one with one of these numbers. In the box
B
B
B
there are
100
100
100
blue balls numbered from
101
101
101
to
200
200
200
, each one with one of these numbers. Martin chooses two positive integers
a
a
a
and
b
b
b
, both less than or equal to
100
100
100
, and then he takes out
a
a
a
balls from box
A
A
A
and
b
b
b
balls from box
B
B
B
, without replacement. Martin's goal is to have two red balls and one blue ball among all balls taken such that the sum of the numbers of two red balls equals the number of the blue ball.\\ What is the least possible value of
a
+
b
a+b
a
+
b
so that Martin achieves his goal for sure? For such a minimum value of
a
+
b
a+b
a
+
b
, give an example of
a
a
a
and
b
b
b
satisfying the goal and explain why every
a
a
a
and
b
b
b
with smaller sum cannot accomplish the aim.