MathDB
Problems
Contests
National and Regional Contests
Peru Contests
Peru Cono Sur TST
2015 Peru Cono Sur TST
2015 Peru Cono Sur TST
Part of
Peru Cono Sur TST
Subcontests
(10)
P10
1
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Another card game with NT
Let
n
n
n
be a positive integer. There is a collection of cards that meets the following properties:
∙
\bullet
∙
Each card has a number written in the form
m
!
m!
m
!
, where
m
m
m
is a positive integer.
∙
\bullet
∙
For every positive integer
t
≤
n
!
t\le n!
t
≤
n
!
, it is possible to choose one or more cards from the collection in such a way
\text{ }
that the sum of the numbers of those cards is
t
t
t
. Determine, based on
n
n
n
, the smallest number of cards that this collection can have.
P9
1
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A lot of combinatorics... right?
Let
m
m
m
and
n
n
n
be positive integers. A child walks the Cartesian plane taking a few steps. The child begins its journey at the point
(
0
,
n
)
(0, n)
(
0
,
n
)
and ends at the point
(
m
,
0
)
(m, 0)
(
m
,
0
)
in such a way that:
∙
\bullet
∙
Each step has length
1
1
1
and is parallel to either the
X
X
X
or
Y
Y
Y
axis.
∙
\bullet
∙
For each point
(
x
,
y
)
(x, y)
(
x
,
y
)
of its path it is true that
x
≥
0
x\ge 0
x
≥
0
and
y
≥
0
y\ge 0
y
≥
0
. For each step of the child, the distance between the child and the axis to which said step is parallel is calculated. If the step causes the child to be further from the point
(
0
,
0
)
(0, 0)
(
0
,
0
)
than before, we consider that distance as positive, otherwise, we consider that distance as negative. Prove that at the end of the boy's journey, the sum of all the distances is
0
0
0
.
P7
1
Hide problems
What did the cosine of 18 have to do with it?
In the plan
6
6
6
points were located such that the distance between two damages of them is greater than or equal to
1
1
1
. Prove that it is possible to choose two of those points such that their distance is greater than or equal to
2
cos
18
2 \cos{18}
2
cos
18
Observation: It might help you to know that
cos
18
=
0.95105
…
\cos{18} = 0.95105\ldots
cos
18
=
0.95105
…
and
cos
24
=
0.91354
…
\cos{24} = 0.91354\ldots
cos
24
=
0.91354
…
P6
1
Hide problems
Get chessboard from simple operations
Let
n
n
n
be a positive integer. On a
2
n
×
2
n
2n\times 2n
2
n
×
2
n
board, the
2
n
2
2n^2
2
n
2
squares were painted white and the other
2
n
2
2n^2
2
n
2
squares were painted black. One operation is to choose a
2
×
2
2\times 2
2
×
2
subtable and mirror its
4
4
4
cells about the vertical or horizontal axis of symmetry of that subtable. For what values of
n
n
n
is it always possible to obtain a chess-like coloring from any initial coloring?
P5
1
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The smallest value of the sequence with a_1 too large
Find the smallest term of the sequence
a
1
,
a
2
,
a
3
,
…
a_1, a_2, a_3, \ldots
a
1
,
a
2
,
a
3
,
…
defined by
a
1
=
201
4
201
5
2016
a_1=2014^{2015^{2016}}
a
1
=
201
4
201
5
2016
and
a
n
+
1
=
{
a
n
2
if
a
n
is even
a
n
+
7
if
a
n
is odd
a_{n+1}= \begin{cases} \frac{a_n}{2} & \text{ if } a_n \text{ is even} \\ a_n + 7 & \text{ if } a_n \text{ is odd} \\ \end{cases}
a
n
+
1
=
{
2
a
n
a
n
+
7
if
a
n
is even
if
a
n
is odd
P4
1
Hide problems
Small town means n is small? XD
In a small city there are
n
n
n
bus routes, with
n
>
1
n > 1
n
>
1
, and each route has exactly
4
4
4
stops. If any two routes have exactly one common stop, and each pair of stops belongs to exactly one route, find all possible values of
n
n
n
.
P2
1
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Numbers from 1 to 15 with rare properties
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
and
d
d
d
be elements of the set
{
1
,
2
,
3
,
…
,
2014
,
2015
}
\{ 1, 2, 3,\ldots , 2014, 2015 \}
{
1
,
2
,
3
,
…
,
2014
,
2015
}
such that
a
<
b
<
c
<
d
a < b < c < d
a
<
b
<
c
<
d
,
a
+
b
a + b
a
+
b
is a divisor of
c
+
d
c + d
c
+
d
, and
a
+
c
a + c
a
+
c
is a divisor of
b
+
d
b + d
b
+
d
. Determine the largest value that
a
a
a
can take.
P1
1
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High score with dominoes
A
A
A
writes, at his choice,
8
8
8
ones and
8
8
8
twos on a
4
×
4
4\times 4
4
×
4
board. Then
B
B
B
covers the board with
8
8
8
dominoes and for each domino she finds the smaller of the two numbers that that domino covers. Finally,
A
A
A
adds these
8
8
8
numbers and the result is her score. What is the highest score
A
A
A
can secure, no matter how
B
B
B
plays? Clarification: A domino is a
1
×
2
1\times 2
1
×
2
or
2
×
1
2\times 1
2
×
1
rectangle that covers exactly two squares on the board.
P3
1
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Fixed point
Let
A
B
C
D
ABCD
A
BC
D
be a parallelogram, let
X
X
X
and
Y
Y
Y
in the segments
A
B
AB
A
B
and
C
D
CD
C
D
, respectively. The segments
A
Y
AY
A
Y
and
D
X
DX
D
X
intersects in
P
P
P
and the segments
B
Y
BY
B
Y
and
D
X
DX
D
X
intersects in
Q
Q
Q
, show that the line
P
Q
PQ
PQ
passes in a fixed point(independent of the positions of the points
X
X
X
and
Y
Y
Y
). I guess that the fixed point is the midpoint of
B
D
BD
B
D
.
P8
1
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Points and circles
Let
A
B
C
D
ABCD
A
BC
D
be a cyclic quadrilateral such that the lines
A
B
AB
A
B
and
C
D
CD
C
D
intersects in
K
K
K
, let
M
M
M
and
N
N
N
be the midpoints of
A
C
AC
A
C
and
C
K
CK
C
K
respectively. Find the possible value(s) of
∠
A
D
C
\angle ADC
∠
A
D
C
if the quadrilateral
M
B
N
D
MBND
MBN
D
is cyclic.