MathDB
Problems
Contests
National and Regional Contests
Peru Contests
Peru Cono Sur TST
2019 Peru Cono Sur TST
2019 Peru Cono Sur TST
Part of
Peru Cono Sur TST
Subcontests
(4)
P3
1
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Partitions {1,2,3,...,n}
Let
A
A
A
be the number of ways in which the set
{
1
,
2
,
.
.
.
,
n
}
\{ 1, 2, . . . , n\}
{
1
,
2
,
...
,
n
}
can be partitioned into non-empty subsets. Let
B
B
B
be the number of ways in which the set
{
1
,
2
,
.
.
.
,
n
,
n
+
1
}
\{ 1, 2, . . . , n, n + 1 \}
{
1
,
2
,
...
,
n
,
n
+
1
}
can be partitioned into non-empty subsets such that consecutive numbers belong to distinct subsets. Partitions that differ only in the order of the subsets are considered equal. Prove that
A
=
B
A = B
A
=
B
.
P6
1
Hide problems
polynomials, absolute value, calculus
Two polynomials of the same degree
A
(
x
)
=
a
n
x
n
+
⋯
+
a
1
x
+
a
0
A(x)=a_nx^n+ \cdots + a_1x+a_0
A
(
x
)
=
a
n
x
n
+
⋯
+
a
1
x
+
a
0
and
B
(
x
)
=
b
n
x
n
+
⋯
+
b
1
x
+
b
0
B(x)=b_nx^n+\cdots+b_1x+b_0
B
(
x
)
=
b
n
x
n
+
⋯
+
b
1
x
+
b
0
are called friends is the coefficients
b
0
,
b
1
,
…
,
b
n
b_0,b_1, \ldots, b_n
b
0
,
b
1
,
…
,
b
n
are a permutation of the coefficients
a
0
,
a
1
,
…
,
a
n
a_0,a_1, \ldots, a_n
a
0
,
a
1
,
…
,
a
n
.
P
(
x
)
P(x)
P
(
x
)
and
Q
(
x
)
Q(x)
Q
(
x
)
be two friendly polynomials with integer coefficients. If
P
(
16
)
=
3
2020
P(16)=3^{2020}
P
(
16
)
=
3
2020
, the smallest possible value of
∣
Q
(
3
2020
)
∣
|Q(3^{2020})|
∣
Q
(
3
2020
)
∣
.
P1
1
Hide problems
one more number theory problem
Find all a positive integers
a
a
a
and
b
b
b
, such that
a
b
+
b
a
a
a
−
b
b
\frac{a^b+b^a}{a^a-b^b}
a
a
−
b
b
a
b
+
b
a
is an integer
P2
1
Hide problems
Peruvian geometry
Let
A
B
AB
A
B
be a diameter of a circle
Γ
\Gamma
Γ
with center
O
O
O
. Let
C
D
CD
C
D
be a chord where
C
D
CD
C
D
is perpendicular to
A
B
AB
A
B
, and
E
E
E
is the midpoint of
C
O
CO
CO
. The line
A
E
AE
A
E
cuts
Γ
\Gamma
Γ
in the point
F
F
F
, the segment
B
C
BC
BC
cuts
A
F
AF
A
F
and
D
F
DF
D
F
in
M
M
M
and
N
N
N
, respectively. The circumcircle of
D
M
N
DMN
D
MN
intersects
Γ
\Gamma
Γ
in the point
K
K
K
. Prove that
K
M
=
M
B
KM=MB
K
M
=
MB
.