MathDB
Problems
Contests
International Contests
Cono Sur Olympiad
2017 Cono Sur Olympiad
2017 Cono Sur Olympiad
Part of
Cono Sur Olympiad
Subcontests
(6)
6
1
Hide problems
Sequence with nested square roots
The infinite sequence
a
1
,
a
2
,
a
3
,
…
a_1,a_2,a_3,\ldots
a
1
,
a
2
,
a
3
,
…
of positive integers is defined as follows:
a
1
=
1
a_1=1
a
1
=
1
, and for each
n
≥
2
n \ge 2
n
≥
2
,
a
n
a_n
a
n
is the smallest positive integer, distinct from
a
1
,
a
2
,
…
,
a
n
−
1
a_1,a_2, \ldots , a_{n-1}
a
1
,
a
2
,
…
,
a
n
−
1
such that:
a
n
+
a
n
−
1
+
…
+
a
2
+
a
1
\sqrt{a_n+\sqrt{a_{n-1}+\ldots+\sqrt{a_2+\sqrt{a_1}}}}
a
n
+
a
n
−
1
+
…
+
a
2
+
a
1
is an integer. Prove that all positive integers appear on the sequence
a
1
,
a
2
,
a
3
,
…
a_1,a_2,a_3,\ldots
a
1
,
a
2
,
a
3
,
…
5
1
Hide problems
Three sequences
Let
a
a
a
,
b
b
b
and
c
c
c
positive integers. Three sequences are defined as follows:[*]
a
1
=
a
a_1=a
a
1
=
a
,
b
1
=
b
b_1=b
b
1
=
b
,
c
1
=
c
c_1=c
c
1
=
c
[/*] [*]
a
n
+
1
=
⌊
a
n
b
n
⌋
a_{n+1}=\lfloor{\sqrt{a_nb_n}}\rfloor
a
n
+
1
=
⌊
a
n
b
n
⌋
,
b
n
+
1
=
⌊
b
n
c
n
⌋
\:b_{n+1}=\lfloor{\sqrt{b_nc_n}}\rfloor
b
n
+
1
=
⌊
b
n
c
n
⌋
,
c
n
+
1
=
⌊
c
n
a
n
⌋
\:c_{n+1}=\lfloor{\sqrt{c_na_n}}\rfloor
c
n
+
1
=
⌊
c
n
a
n
⌋
for
n
≥
1
n \ge 1
n
≥
1
[/*] [list = a] [*]Prove that for any
a
a
a
,
b
b
b
,
c
c
c
, there exists a positive integer
N
N
N
such that
a
N
=
b
N
=
c
N
a_N=b_N=c_N
a
N
=
b
N
=
c
N
.[/*] [*]Find the smallest
N
N
N
such that
a
N
=
b
N
=
c
N
a_N=b_N=c_N
a
N
=
b
N
=
c
N
for some choice of
a
a
a
,
b
b
b
,
c
c
c
such that
a
≥
2
a \ge 2
a
≥
2
y
b
+
c
=
2
a
−
1
b+c=2a-1
b
+
c
=
2
a
−
1
.[/*]
4
1
Hide problems
Circumcenter midpoint of segment
Let
A
B
C
ABC
A
BC
an acute triangle with circumcenter
O
O
O
. Points
X
X
X
and
Y
Y
Y
are chosen such that:[*]
∠
X
A
B
=
∠
Y
C
B
=
9
0
∘
\angle XAB = \angle YCB = 90^\circ
∠
X
A
B
=
∠
Y
CB
=
9
0
∘
[/*] [*]
∠
A
B
C
=
∠
B
X
A
=
∠
B
Y
C
\angle ABC = \angle BXA = \angle BYC
∠
A
BC
=
∠
BX
A
=
∠
B
Y
C
[/*] [*]
X
X
X
and
C
C
C
are in different half-planes with respect to
A
B
AB
A
B
[/*] [*]
Y
Y
Y
and
A
A
A
are in different half-planes with respect to
B
C
BC
BC
[/*]Prove that
O
O
O
is the midpoint of
X
Y
XY
X
Y
.
3
1
Hide problems
Tiling with Tetrominos
Let
n
n
n
be a positive integer. In how many ways can a
4
×
4
n
4 \times 4n
4
×
4
n
grid be tiled with the following tetromino? [asy] size(4cm); draw((1,0)--(3,0)--(3,1)--(0,1)--(0,0)--(1,0)--(1,2)--(2,2)--(2,0)); [/asy]
2
1
Hide problems
Areas inside convex polygon
Let
A
(
X
Y
Z
)
A(XYZ)
A
(
X
Y
Z
)
be the area of the triangle
X
Y
Z
XYZ
X
Y
Z
. A non-regular convex polygon
P
1
P
2
…
P
n
P_1 P_2 \ldots P_n
P
1
P
2
…
P
n
is called guayaco if exists a point
O
O
O
in its interior such that
A
(
P
1
O
P
2
)
=
A
(
P
2
O
P
3
)
=
⋯
=
A
(
P
n
O
P
1
)
.
A(P_1OP_2) = A(P_2OP_3) = \cdots = A(P_nOP_1).
A
(
P
1
O
P
2
)
=
A
(
P
2
O
P
3
)
=
⋯
=
A
(
P
n
O
P
1
)
.
Show that, for every integer
n
≥
3
n \ge 3
n
≥
3
, a guayaco polygon of
n
n
n
sides exists.
1
1
Hide problems
Sum of digits of $n^2$
A positive integer
n
n
n
is called guayaquilean if the sum of the digits of
n
n
n
is equal to the sum of the digits of
n
2
n^2
n
2
. Find all the possible values that the sum of the digits of a guayaquilean number can take.