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Problems
Contests
National and Regional Contests
Honduras Contests
Honduras Virtual Math Competition
CVM 2020
CVM 2020
Part of
Honduras Virtual Math Competition
Subcontests
(10)
2
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2020 CVM - RE - P6
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P
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<span class='latex-bold'>Problem 6+.</span>
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P
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6
+
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Find all functions
f
:
Z
→
Z
f: \mathbb{Z} \rightarrow \mathbb{Z}
f
:
Z
→
Z
such that for all integers
x
,
y
x,y
x
,
y
holds
f
(
x
−
f
(
x
y
)
)
=
f
(
x
)
f
(
1
−
y
)
f(x-f(xy)) = f(x)f(1-y)
f
(
x
−
f
(
x
y
))
=
f
(
x
)
f
(
1
−
y
)
Proposed by Manuel Aguilera, Valle
CVM - RE - P6
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P
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<span class='latex-bold'>Problem 5+.</span>
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P
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+
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There is a
2020
−
2020-
2020
−
regular agon its vertices are numbered with
2020
2020
2020
consecutive natural numbers. Cobyi wants to travel moving from vertex to vertex (forming a single line) with the following conditions:
(
a
)
(a)
(
a
)
Start at the vertex with the smallest number.
(
b
)
(b)
(
b
)
When making the firts movement, the other movements dependent on the sum of the indices through which Cobyi has alrready passed.
(
c
)
(c)
(
c
)
if the sum of the indices exceeds the largest number listed in the vertices, the vertices change index with the smallest number equal to
1
1
1
, consequently Cobyi goes to number
1
1
1
and the process repeats.Determine the smallest and longest distance Cobyi can travelProposed by David Cruz, Francisco Morazan
Problem 3+
1
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CVM 2020 - RE - P3
Consider
(
△
n
=
A
n
B
n
C
n
)
n
≥
1
\left(\triangle_n=A_nB_nC_n\right)_{n\ge 1}
(
△
n
=
A
n
B
n
C
n
)
n
≥
1
. We define points
A
n
′
,
B
n
′
,
C
n
′
A_n',B_n',C_n'
A
n
′
,
B
n
′
,
C
n
′
in sides
C
n
B
n
,
A
n
C
n
,
B
n
A
n
C_nB_n,A_nC_n,B_nA_n
C
n
B
n
,
A
n
C
n
,
B
n
A
n
such that
(
n
+
1
)
B
n
A
n
′
=
C
n
A
n
′
,
(
n
+
1
)
C
n
B
n
′
=
A
n
B
n
′
,
(
n
+
1
)
A
n
C
n
′
=
B
n
C
n
′
(n+1)B_nA_n'=C_nA_n',~(n+1)C_nB_n'=A_nB_n',~(n+1)A_nC_n'=B_nC_n'
(
n
+
1
)
B
n
A
n
′
=
C
n
A
n
′
,
(
n
+
1
)
C
n
B
n
′
=
A
n
B
n
′
,
(
n
+
1
)
A
n
C
n
′
=
B
n
C
n
′
△
n
+
1
\triangle_{n+1}
△
n
+
1
is defined by the intersections of
A
n
A
n
′
,
B
n
B
n
′
,
C
n
C
n
′
A_nA_n',B_nB_n',C_nC_n'
A
n
A
n
′
,
B
n
B
n
′
,
C
n
C
n
′
. If
S
n
S_n
S
n
denotes the area of
△
n
\triangle_n
△
n
. Find
S
1
S
2020
\frac{S_1}{S_{2020}}
S
2020
S
1
.Proposed by Alejandro Madrid, Valle
Problem 2+
1
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CVM 2020 - RE - P2
Find all the real solutions to
n
=
∑
i
=
1
n
x
i
=
∑
1
≤
i
<
j
≤
n
x
i
x
j
n=\sum_{i=1}^n x_i=\sum_{1\le i<j\le n} x_ix_j
n
=
i
=
1
∑
n
x
i
=
1
≤
i
<
j
≤
n
∑
x
i
x
j
Proposed by Carlos Dominguez, Valle
Problem 1+
1
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CVM 2020 - RE - P1
Given the number
a
1
a
2
⋯
a
n
‾
\overline{a_1a_2\cdots a_n}
a
1
a
2
⋯
a
n
such that
a
n
⋯
a
2
a
1
‾
∣
a
1
a
2
⋯
a
n
‾
\overline{a_n\cdots a_2a_1}\mid \overline{a_1a_2\cdots a_n}
a
n
⋯
a
2
a
1
∣
a
1
a
2
⋯
a
n
Then show
(
a
1
a
2
⋯
a
n
‾
)
(
a
n
⋯
a
2
a
1
‾
)
(\overline{a_1a_2\cdots a_n})(\overline{a_n\cdots a_2a_1})
(
a
1
a
2
⋯
a
n
)
(
a
n
⋯
a
2
a
1
)
is a perfect square. Proposed by Ezra Guerrero, Francisco Morazan
Problem 6
1
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CVM 2020 - Problem 6
Let
P
(
x
)
P(x)
P
(
x
)
be a monic cubic polynomial. The lines
y
=
0
y = 0
y
=
0
and
y
=
m
y = m
y
=
m
intersect
P
(
x
)
P(x)
P
(
x
)
at points
A
A
A
,
C
C
C
,
E
E
E
and
B
B
B
,
D
D
D
,
F
F
F
from left to right for a positive real number
m
m
m
. If
A
B
=
7
AB = \sqrt{7}
A
B
=
7
,
C
D
=
15
CD = \sqrt{15}
C
D
=
15
, and
E
F
=
10
EF = \sqrt{10}
EF
=
10
, what is the value of
m
m
m
?
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6.1.
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<span class='latex-bold'>6.1.</span>
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6.1.
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A monic polynomial is one that has a main coefficient equal to
1
1
1
. For example, the polynomial
P
(
x
)
=
x
3
+
5
x
2
−
3
x
+
7
P(x) = x^3 + 5x^2 - 3x + 7
P
(
x
)
=
x
3
+
5
x
2
−
3
x
+
7
is a monic polynomial Proposed by Lenin Vasquez, Copan
Problem 5
1
Hide problems
CVM 2020 - Problem 5
In a room with
9
9
9
students, there are
n
n
n
clubs with
4
4
4
participants in each club. For any pairs of clubs no more than
2
2
2
students belong to both clubs. Prove that
n
≤
18
n \le 18
n
≤
18
Proposed by Manuel Aguilera, Valle
Problem 4
1
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CVM 2020 - Problem 4
Consider an
A
B
C
D
ABCD
A
BC
D
parallelogram with
A
D
‾
\overline{AD}
A
D
=
=
=
B
D
‾
\overline{BD}
B
D
. Point E lies in segment
B
D
‾
\overline{BD}
B
D
in such a way that
A
E
‾
\overline{AE}
A
E
=
=
=
D
E
‾
\overline{DE}
D
E
. The extension of line
A
E
‾
\overline{AE}
A
E
cuts segment
B
C
‾
\overline{BC}
BC
and
F
F
F
. if line
D
F
‾
\overline{DF}
D
F
is the bisector of the
∠
C
E
D
\angle CED
∠
CE
D
. Find the value of the
∠
A
B
D
\angle ABD
∠
A
B
D
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4.1.
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<span class='latex-bold'>4.1.</span>
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4.1.
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Point
E
E
E
lies in segment
B
D
‾
\overline{BD}
B
D
means that exits a point
E
E
E
in the segment
B
D
‾
\overline{BD}
B
D
in other words lies refers to the same thing foundProposed by Alicia Smith, Francisco Morazan
Problem 3
1
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2020 CVM - Problem 3
In
△
A
B
C
\triangle ABC
△
A
BC
we consider the points
A
′
,
B
′
,
C
′
A',B',C'
A
′
,
B
′
,
C
′
in sides
B
C
,
A
C
,
A
B
BC,AC,AB
BC
,
A
C
,
A
B
such that
3
B
A
′
=
C
A
′
,
3
C
B
′
=
A
B
′
,
3
A
C
′
=
B
A
′
3BA'=CA',~3CB'=AB',~3AC'=BA'
3
B
A
′
=
C
A
′
,
3
C
B
′
=
A
B
′
,
3
A
C
′
=
B
A
′
△
D
E
F
\triangle DEF
△
D
EF
is defined by the intersections of
A
A
′
,
B
B
′
,
C
C
′
AA',BB',CC'
A
A
′
,
B
B
′
,
C
C
′
. If the are of
△
A
B
C
\triangle ABC
△
A
BC
is
2020
2020
2020
find the area of
△
D
E
F
\triangle DEF
△
D
EF
.Proposed by Alejandro Madrid, Valle
Problem 2
1
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2020 CVM - Problem 2
Find all
(
x
,
y
,
z
)
∈
R
3
(x,y,z)\in\mathbb R^3
(
x
,
y
,
z
)
∈
R
3
such that
x
+
y
+
z
=
x
y
+
y
z
+
z
x
=
3
x+y+z=xy+yz+zx=3
x
+
y
+
z
=
x
y
+
yz
+
z
x
=
3
Proposed by Ezra Guerrero, Francisco Morazan
Problem 1
1
Hide problems
2020 CVM - Problem 1
How many numbers
a
b
c
‾
\overline{abc}
ab
c
with
a
,
b
,
c
>
0
a,b,c>0
a
,
b
,
c
>
0
there exists such that
c
b
a
‾
∣
a
b
c
‾
\overline{cba}\mid \overline{abc}
c
ba
∣
ab
c
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<span class='latex-bold'>1.1.</span>
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1.1.
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The vertical line denotes that
c
b
a
‾
\overline{cba}
c
ba
divides
a
b
c
‾
.
\overline{abc}.
ab
c
.
Proposed by Roger Carranza, Choluteca