MathDB
Problems
Contests
International Contests
Cono Sur Olympiad
2024 Cono Sur Olympiad
2024 Cono Sur Olympiad
Part of
Cono Sur Olympiad
Subcontests
(6)
4
1
Hide problems
NT hand job
Let
N
N
N
be a positive integer with
2
k
2k
2
k
digits. Its chunks are defined by the two numbers formed by the digits from
1
1
1
to
k
k
k
and
k
+
1
k+1
k
+
1
to
2
k
2k
2
k
(e.g. the chunks of 142856 are 142 and 856). We define the
N
N
N
-reverse as the number formed by switching its chunks (e.g. the reverse of 142856 is 856142 and for 1401 it is 114). We call a number cearense is it satisfies the following conditions:[*] Has an even number of digits [*] Its chunks are relatively prime [*]Divides its reverseFind the two smallest cearense integer.
6
1
Hide problems
Cono Sur 2024 /P6
On a board of
8
×
8
8 \times 8
8
×
8
exists
64
64
64
kings, all initially placed in different squares. Alnardo and Bernaldo play alternately, with Arnaldo starting. On each move, one of the two players chooses a king and can move it one square to the right, one square up, or one square up to the right. In the event that a king is moved to an occupied square, both kings are removed from the game. The player who can remove two of the last kings or leave one last king in the upper right corner wins the game. Which of the two players can ensure victory?
5
1
Hide problems
Magical permututations!
A permutation of
{
1
,
2
⋯
,
n
}
\{1, 2 \cdots, n \}
{
1
,
2
⋯
,
n
}
is magic if each element
k
k
k
of it has at least
⌊
k
2
⌋
\left\lfloor \frac{k}{2} \right\rfloor
⌊
2
k
⌋
numbers less to it at the left. For each
n
n
n
find the number of magical permutations.
2
1
Hide problems
An ugly geo in Cono Sur
Let
A
B
C
ABC
A
BC
be a triangle. Let
A
1
A_1
A
1
and
A
2
A_2
A
2
be points on side
B
C
,
B
1
BC, B_1
BC
,
B
1
and
B
2
B_2
B
2
be points on side
C
A
CA
C
A
and
C
1
C_1
C
1
and
C
2
C_2
C
2
be points on side
A
B
AB
A
B
such that
A
1
A
2
B
1
B
2
C
1
C
2
A_1A_2B_1B_2C_1C_2
A
1
A
2
B
1
B
2
C
1
C
2
is a convex hexagon and that
B
,
A
1
,
A
2
B,A_1,A_2
B
,
A
1
,
A
2
and
C
C
C
are located in that order on side
B
C
BC
BC
. We say that triangles
A
B
2
C
1
,
B
A
1
C
2
AB_2C_1, BA_1C_2
A
B
2
C
1
,
B
A
1
C
2
and
C
A
2
B
1
CA_2B_1
C
A
2
B
1
are glueable if there exists a triangle
P
Q
R
PQR
PQR
and there exist
X
,
Y
X,Y
X
,
Y
and
Z
Z
Z
on sides
Q
R
,
R
P
QR, RP
QR
,
RP
and
P
Q
PQ
PQ
respectively, such that triangle
A
B
2
C
1
AB_2C_1
A
B
2
C
1
is congruent in that order to triangle
P
Y
Z
PYZ
P
Y
Z
, triangle
B
A
1
C
2
BA_1C_2
B
A
1
C
2
is congruent in that order to triangle
Q
X
Z
QXZ
QXZ
and triangle
C
A
2
B
1
CA_2B_1
C
A
2
B
1
is congruent in that order to triangle
R
X
Y
RXY
RX
Y
. Prove that triangles
A
B
2
C
1
,
B
A
1
C
2
AB_2C_1, BA_1C_2
A
B
2
C
1
,
B
A
1
C
2
and
C
A
2
B
1
CA_2B_1
C
A
2
B
1
are glueable if and only if the centroids of triangles
A
1
B
1
C
1
A_1B_1C_1
A
1
B
1
C
1
and
A
2
B
2
C
2
A_2B_2C_2
A
2
B
2
C
2
coincide.
3
1
Hide problems
Cono Sur 2024/P3
Find all positive integers
n
n
n
such that
3
n
−
2
n
−
1
3^n - 2^n - 1
3
n
−
2
n
−
1
is a perfect square.
1
1
Hide problems
Cono Sur 24/P1
Prove that there are infinitely many quadruplets of positive integers
(
a
,
b
,
c
,
d
)
(a,b,c,d)
(
a
,
b
,
c
,
d
)
, such that\\
a
b
+
1
ab+1
ab
+
1
,
b
c
+
16
bc+16
b
c
+
16
,
c
d
+
4
cd+4
c
d
+
4
,
a
d
+
9
ad+9
a
d
+
9
\\ are perfect squares