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International Contests
Cono Sur Olympiad
2020 Cono Sur Olympiad
2020 Cono Sur Olympiad
Part of
Cono Sur Olympiad
Subcontests
(6)
6
1
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2020 Cono Sur Math Olympiad P6
A
4
4
4
x
4
4
4
square board is called
b
r
a
s
u
c
a
brasuca
b
r
a
s
u
c
a
if it follows all the conditions: • each box contains one of the numbers
0
,
1
,
2
,
3
,
4
0, 1, 2, 3, 4
0
,
1
,
2
,
3
,
4
or
5
5
5
; • the sum of the numbers in each line is
5
5
5
; • the sum of the numbers in each column is
5
5
5
; • the sum of the numbers on each diagonal of four squares is
5
5
5
; • the number written in the upper left box of the board is less than or equal to the other numbers the board; • when dividing the board into four
2
2
2
×
2
2
2
squares, in each of them the sum of the four numbers is
5
5
5
.How many
"
b
r
a
s
u
c
a
s
"
"brasucas"
"
b
r
a
s
u
c
a
s
"
boards are there?
5
1
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Tokens and products in the board
There is a pile with
15
15
15
coins on a table. At each step, Pedro choses one of the piles in the table with
a
>
1
a>1
a
>
1
coins and divides it in two piles with
b
≥
1
b\geq1
b
≥
1
and
c
≥
1
c\geq1
c
≥
1
coins and writes in the board the product
a
b
c
abc
ab
c
. He continues until there are
15
15
15
piles with
1
1
1
coin each. Determine all possible values that the final sum of the numbers in the board can have.
4
1
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Reflections of A lie on BC
Let
A
B
C
ABC
A
BC
be an acute scalene triangle.
D
D
D
and
E
E
E
are variable points in the half-lines
A
B
AB
A
B
and
A
C
AC
A
C
(with origin at
A
A
A
) such that the symmetric of
A
A
A
over
D
E
DE
D
E
lies on
B
C
BC
BC
. Let
P
P
P
be the intersection of the circles with diameter
A
D
AD
A
D
and
A
E
AE
A
E
. Find the locus of
P
P
P
when varying the line segment
D
E
DE
D
E
.
3
1
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Cono Sur 2020 Problem 3
Let
A
B
C
ABC
A
BC
be an acute triangle such that
A
C
<
B
C
AC<BC
A
C
<
BC
and
ω
\omega
ω
its circumcircle.
M
M
M
is the midpoint of
B
C
BC
BC
. Points
F
F
F
and
E
E
E
are chosen in
A
B
AB
A
B
and
B
C
BC
BC
, respectively, such that
A
C
=
C
F
AC=CF
A
C
=
CF
and
E
B
=
E
F
EB=EF
EB
=
EF
. The line
A
M
AM
A
M
intersects
ω
\omega
ω
in
D
≠
A
D\neq A
D
=
A
. The line
D
E
DE
D
E
intersects the line
F
M
FM
FM
in
G
G
G
. Prove that
G
G
G
lies on
ω
\omega
ω
.
2
1
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b^2-4ac is not a perfect square
Given
2021
2021
2021
distinct positive integers non divisible by
2
1010
2^{1010}
2
1010
, show that it's always possible to choose
3
3
3
of them
a
a
a
,
b
b
b
and
c
c
c
, such that
∣
b
2
−
4
a
c
∣
|b^2-4ac|
∣
b
2
−
4
a
c
∣
is not a perfect square.
1
1
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Trinomials game
Ari and Beri play a game using a deck of
2020
2020
2020
cards with exactly one card with each number from
1
1
1
to
2020
2020
2020
. Ari gets a card with a number
a
a
a
and removes it from the deck. Beri sees the card, chooses another card from the deck with a number
b
b
b
and removes it from the deck. Then Beri writes on the board exactly one of the trinomials
x
2
−
a
x
+
b
x^2-ax+b
x
2
−
a
x
+
b
or
x
2
−
b
x
+
a
x^2-bx+a
x
2
−
b
x
+
a
from his choice. This process continues until no cards are left on the deck. If at the end of the game every trinomial written on the board has integer solutions, Beri wins. Otherwise, Ari wins. Prove that Beri can always win, no matter how Ari plays.