MathDB
Problems
Contests
National and Regional Contests
Bolivia Contests
Bolivian Ibero TST
2021 Bolivia Ibero TST
2021 Bolivia Ibero TST
Part of
Bolivian Ibero TST
Subcontests
(4)
4
1
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Why no normal geometry :(
On a isosceles triangle
△
A
B
C
\triangle ABC
△
A
BC
with
A
B
=
B
C
AB=BC
A
B
=
BC
let
K
,
M
K,M
K
,
M
be the midpoints of
A
B
,
A
C
AB,AC
A
B
,
A
C
respectivily. Let
(
C
K
B
)
(CKB)
(
C
K
B
)
intersect
B
M
BM
BM
at
N
≠
M
N \ne M
N
=
M
, the line through
N
N
N
parallel to
A
C
AC
A
C
intersects
(
A
B
C
)
(ABC)
(
A
BC
)
at
A
1
,
C
1
A_1,C_1
A
1
,
C
1
. Show that
△
A
1
B
C
1
\triangle A_1BC_1
△
A
1
B
C
1
is equilateral.
3
1
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a,b,c cubed give different residues modulo p
Let
p
=
a
b
+
b
c
+
a
c
p=ab+bc+ac
p
=
ab
+
b
c
+
a
c
be a prime number where
a
,
b
,
c
a,b,c
a
,
b
,
c
are different two by two, show that
a
3
,
b
3
,
c
3
a^3,b^3,c^3
a
3
,
b
3
,
c
3
gives different residues modulo
p
p
p
2
1
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How to make weird functions be like:
Let
f
:
Z
+
→
Z
f: \mathbb Z^+ \to \mathbb Z
f
:
Z
+
→
Z
be a function such that a)
f
(
p
)
=
1
f(p)=1
f
(
p
)
=
1
for every prime
p
p
p
. b)
f
(
x
y
)
=
x
f
(
y
)
+
y
f
(
x
)
f(xy)=xf(y)+yf(x)
f
(
x
y
)
=
x
f
(
y
)
+
y
f
(
x
)
for every pair of positive integers
x
,
y
x,y
x
,
y
Find the least number
n
≥
2021
n \ge 2021
n
≥
2021
such that
f
(
n
)
=
n
f(n)=n
f
(
n
)
=
n
1
1
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n times n grid coloring
Let
n
n
n
be a posititve integer. On a
n
×
n
n \times n
n
×
n
grid there are
n
2
n^2
n
2
unit squares and on these we color the sides with blue such that every unit square has exactly one side with blue. a) Find the maximun number of blue unit sides we can have on the
n
×
n
n \times n
n
×
n
grid. b) Find the minimun number of blue unit sides we can have on the
n
×
n
n \times n
n
×
n
grid.