MathDB
Problems
Contests
National and Regional Contests
Peru Contests
Peru Cono Sur TST
2021 Peru Cono Sur TST.
2021 Peru Cono Sur TST.
Part of
Peru Cono Sur TST
Subcontests
(6)
P6
1
Hide problems
Prove that there are no such 2021 numbers
Prove that there are no positive integers
a
1
,
a
2
,
…
,
a
2021
a_1, a_2, \ldots , a_{2021}
a
1
,
a
2
,
…
,
a
2021
(not necessarily distinct) such that for
k
=
1
,
2
,
3
,
…
,
2021
k = 1, 2, 3, \ldots , 2021
k
=
1
,
2
,
3
,
…
,
2021
the number of elements in the set
A
k
=
{
j
∈
N
:
1
≤
j
≤
2021
and
a
j
∣
k
}
A_k = \{ j \in \mathbb{N} : 1 \le j \le 2021 \text{ and } a_j|k \}
A
k
=
{
j
∈
N
:
1
≤
j
≤
2021
and
a
j
∣
k
}
be exactly
a
k
a_k
a
k
.
P5
1
Hide problems
2n concentric circles
Let
n
≥
2
n\ge 2
n
≥
2
be an integer. They are given
n
+
1
n + 1
n
+
1
red points in the plane. Prove that there exist
2
n
2n
2
n
circles
C
1
,
C
2
,
…
,
C
n
,
D
1
,
D
2
,
…
,
D
n
C_1 , C_2 , \ldots , C_n , D_1 , D_2 , \ldots , D_n
C
1
,
C
2
,
…
,
C
n
,
D
1
,
D
2
,
…
,
D
n
such that:
∙
\bullet
∙
C
1
,
C
2
,
…
,
C
n
C_1 , C_2 ,\ldots , C_n
C
1
,
C
2
,
…
,
C
n
are concentric.
∙
\bullet
∙
D
1
,
D
2
,
…
,
D
n
D_1 , D_2 ,\ldots , D_n
D
1
,
D
2
,
…
,
D
n
are concentric.
∙
\bullet
∙
For
k
=
1
,
2
,
3
,
…
,
n
k = 1, 2, 3,\ldots , n
k
=
1
,
2
,
3
,
…
,
n
the circles
C
k
C_k
C
k
and
D
k
D_k
D
k
are disjoint.
∙
\bullet
∙
For
k
=
1
,
2
,
3
,
…
,
n
k = 1, 2, 3,\ldots , n
k
=
1
,
2
,
3
,
…
,
n
it is true that
C
k
C_k
C
k
contains exactly
k
k
k
red dots in its interior and
D
k
D_k
D
k
contains exactly
n
+
1
−
k
n + 1 - k
n
+
1
−
k
red dots in its interior.
P4
1
Hide problems
Very interesting subset problem
Let
n
≥
5
n\ge 5
n
≥
5
be an integer. Consider
2
n
−
1
2n-1
2
n
−
1
subsets
A
1
,
A
2
,
A
3
,
…
,
A
2
n
−
1
A_1, A_2, A_3, \ldots , A_{2n-1}
A
1
,
A
2
,
A
3
,
…
,
A
2
n
−
1
of the set
{
1
,
2
,
3
,
…
,
n
}
\{ 1, 2, 3,\ldots , n \}
{
1
,
2
,
3
,
…
,
n
}
, these subsets have the property that each of them has
2
2
2
elements (that is that is, for
1
≤
i
≤
2
n
−
1
1 \le i \le 2n-1
1
≤
i
≤
2
n
−
1
it is true that
A
i
A_i
A
i
has
2
2
2
elements). Show that it is always possible to select
n
n
n
of these subsets in such a way that the union of these
n
n
n
subsets has at most
2
3
n
+
1
\frac{2}{3}n + 1
3
2
n
+
1
elements in total.
P2
1
Hide problems
Fascinating numbers
For each positive integer
k
k
k
we denote by
S
(
k
)
S(k)
S
(
k
)
the sum of its digits, for example
S
(
132
)
=
6
S(132)=6
S
(
132
)
=
6
and
S
(
1000
)
=
1
S(1000)=1
S
(
1000
)
=
1
. A positive integer
n
n
n
is said to be
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
f
a
s
c
i
n
a
t
i
n
g
<
/
s
p
a
n
>
<span class='latex-bold'>fascinating</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
f
a
sc
ina
t
in
g
<
/
s
p
an
>
if it holds that
n
=
k
S
(
k
)
n = \frac{k}{S(k)}
n
=
S
(
k
)
k
for some positive integer
k
k
k
. For example, the number
11
11
11
is
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
f
a
s
c
i
n
a
t
i
n
g
<
/
s
p
a
n
>
<span class='latex-bold'>fascinating</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
f
a
sc
ina
t
in
g
<
/
s
p
an
>
since
11
=
198
S
(
198
)
(
11 = \frac{198}{S(198)} (
11
=
S
(
198
)
198
(
since
198
S
(
198
)
=
198
1
+
9
+
8
=
198
18
=
11
)
\frac{198}{S(198)}=\frac{198}{1+9+8}=\frac{198}{18} = 11)
S
(
198
)
198
=
1
+
9
+
8
198
=
18
198
=
11
)
. Prove that there exists a positive integer less than
2021
2021
2021
and that it is not
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
f
a
s
c
i
n
a
t
i
n
g
<
/
s
p
a
n
>
<span class='latex-bold'>fascinating</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
f
a
sc
ina
t
in
g
<
/
s
p
an
>
.
P1
1
Hide problems
Values that take the maximum integer of m^2+\sqrt{2}n
Find the set of all possible values of the expression
⌊
m
2
+
2
n
⌋
\lfloor m^2+\sqrt{2} n \rfloor
⌊
m
2
+
2
n
⌋
, where
m
m
m
and
n
n
n
are positive integers. Note: The symbol
⌊
x
⌋
\lfloor x\rfloor
⌊
x
⌋
denotes the largest integer less than or equal to
x
x
x
.
P3
1
Hide problems
BFG is a line
Let
A
B
C
ABC
A
BC
be a triangle and
D
D
D
is a point in
B
C
BC
BC
. The line
D
A
DA
D
A
cuts the circumcircle of
A
B
C
ABC
A
BC
in the point
E
E
E
. Let
M
M
M
and
N
N
N
be the midpoints of
A
B
AB
A
B
and
C
D
CD
C
D
, respectively. Let
F
=
M
N
∩
A
D
F=MN\cap AD
F
=
MN
∩
A
D
and
G
≠
F
G\neq F
G
=
F
is the point of intersection of the circumcircles of
△
D
N
F
\triangle DNF
△
D
NF
and
△
E
C
F
\triangle ECF
△
ECF
. Prove that
B
,
F
B,F
B
,
F
and
G
G
G
are collinears.