MathDB
Problems
Contests
International Contests
Cono Sur Olympiad
2022 Cono Sur
2022 Cono Sur
Part of
Cono Sur Olympiad
Subcontests
(6)
4
1
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Ana versus Beto coloring a grid
Ana and Beto play on a grid of
2022
×
2022
2022 \times 2022
2022
×
2022
. Ana colors the sides of some squares on the board red, so that no square has two red sides that share a vertex. Next, Bob must color a blue path that connects two of the four corners of the board, following the sides of the squares and not using any red segments. If Beto succeeds, he is the winner, otherwise Ana wins. Who has a winning strategy?
1
1
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Two digit problems on the same day
A positive integer is happy if:1. All its digits are different and not
0
0
0
, 2. One of its digits is equal to the sum of the other digits.For example, 253 is a happy number. How many happy numbers are there?
6
1
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Colored sums divide colored sums
On a blackboard the numbers
1
,
2
,
3
,
…
,
170
1,2,3,\dots,170
1
,
2
,
3
,
…
,
170
are written. You want to color each of these numbers with
k
k
k
colors
C
1
,
C
2
,
…
,
C
k
C_1,C_2, \dots, C_k
C
1
,
C
2
,
…
,
C
k
, such that the following condition is satisfied: for each
i
i
i
with
1
≤
i
<
k
1 \leq i < k
1
≤
i
<
k
, the sum of all numbers with color
C
i
C_i
C
i
divide the sum of all numbers with color
C
i
+
1
C_{i+1}
C
i
+
1
. Determine the largest possible value of
k
k
k
for which it is possible to do that coloring.
3
1
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Blocks in powers
Prove that for every positive integer
n
n
n
there exists a positive integer
k
k
k
, such that each of the numbers
k
,
k
2
,
…
,
k
n
k, k^2, \dots, k^n
k
,
k
2
,
…
,
k
n
have at least one block of
2022
2022
2022
in their decimal representation.For example, the numbers 4202213 and 544202212022 have at least one block of
2022
2022
2022
in their decimal representation.
5
1
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Differences of adjacent divisors are also divisors
An integer
n
>
1
n>1
n
>
1
, whose positive divisors are
1
=
d
1
<
d
2
<
⋯
<
d
k
=
n
1=d_1<d_2< \cdots <d_k=n
1
=
d
1
<
d
2
<
⋯
<
d
k
=
n
, is called
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
s
o
u
t
h
e
r
n
<
/
s
p
a
n
>
<span class='latex-italic'>southern</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
so
u
t
h
er
n
<
/
s
p
an
>
if all the numbers
d
2
−
d
1
,
d
3
−
d
2
,
⋯
,
d
k
−
d
k
−
1
d_2-d_1, d_3- d_2 , \cdots, d_k-d_{k-1}
d
2
−
d
1
,
d
3
−
d
2
,
⋯
,
d
k
−
d
k
−
1
are divisors of
n
n
n
.a) Find a positive integer that is
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
n
o
t
s
o
u
t
h
e
r
n
<
/
s
p
a
n
>
<span class='latex-italic'>not southern</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
n
o
t
so
u
t
h
er
n
<
/
s
p
an
>
and has exactly
2022
2022
2022
positive divisors that are
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
s
o
u
t
h
e
r
n
<
/
s
p
a
n
>
<span class='latex-italic'>southern</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
so
u
t
h
er
n
<
/
s
p
an
>
.b) Show that there are infinitely many positive integers that are
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
n
o
t
s
o
u
t
h
e
r
n
<
/
s
p
a
n
>
<span class='latex-italic'>not southern</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
n
o
t
so
u
t
h
er
n
<
/
s
p
an
>
and have exactly
2022
2022
2022
positive divisors that are
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
s
o
u
t
h
e
r
n
<
/
s
p
a
n
>
<span class='latex-italic'>southern</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
so
u
t
h
er
n
<
/
s
p
an
>
.
2
1
Hide problems
Equal segments in an incircle configuration
Given is a triangle
A
B
C
ABC
A
BC
with incircle
ω
\omega
ω
, tangent to
B
C
,
C
A
,
A
B
BC, CA, AB
BC
,
C
A
,
A
B
at
D
,
E
,
F
D, E, F
D
,
E
,
F
. The perpendicular from
B
B
B
to
B
C
BC
BC
meets
E
F
EF
EF
at
M
M
M
, and the perpendicular from
C
C
C
to
B
C
BC
BC
meets
E
F
EF
EF
at
N
N
N
. Let
D
M
DM
D
M
and
D
N
DN
D
N
meet
ω
\omega
ω
at
P
P
P
and
Q
Q
Q
. Prove that
D
P
=
D
Q
DP=DQ
D
P
=
D
Q
.