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Contests
National and Regional Contests
Bolivia Contests
Bolivian Cono Sur TST
2021 Bolivian Cono Sur TST
2021 Bolivian Cono Sur TST
Part of
Bolivian Cono Sur TST
Subcontests
(3)
3
1
Hide problems
A weird ratio of areas on a rectangle with a trick :)
Let
A
B
C
D
ABCD
A
BC
D
be a rectangle with sides
A
B
,
B
C
,
C
D
AB,BC,CD
A
B
,
BC
,
C
D
and
D
A
DA
D
A
. Let
K
,
L
K,L
K
,
L
be the midpoints of the sides
B
C
,
D
A
BC,DA
BC
,
D
A
respectivily. The perpendicular from
B
B
B
to
A
K
AK
A
K
hits
C
L
CL
C
L
at
M
M
M
. Find
[
A
B
K
M
]
[
A
B
C
L
]
\frac{[ABKM]}{[ABCL]}
[
A
BC
L
]
[
A
B
K
M
]
2
3
Hide problems
A nice and strage NT equation with involves LCM!?
Find all posible pairs of positive integers
x
,
y
x,y
x
,
y
such that
lcm
(
x
,
y
+
3001
)
=
lcm
(
y
,
x
+
3001
)
\text{lcm}(x,y+3001)=\text{lcm}(y,x+3001)
lcm
(
x
,
y
+
3001
)
=
lcm
(
y
,
x
+
3001
)
When the replace is sus (variants involved)
The numbers
1
,
2
,
.
.
.
,
100
1,2,...,100
1
,
2
,
...
,
100
are written in a board. We are allowed to choose any two numbers from the board
a
,
b
a,b
a
,
b
to delete them and replace on the board the number
a
+
b
−
1
a+b-1
a
+
b
−
1
. What are the possible numbers u can get after
99
99
99
consecutive operations of these?
3D combo makes u cry! (No algebra on both tests :'c)
Let
n
n
n
be a posititve integer and let
M
M
M
the set of all all integer cordinates
(
a
,
b
,
c
)
(a,b,c)
(
a
,
b
,
c
)
such that
0
≤
a
,
b
,
c
≤
n
0 \le a,b,c \le n
0
≤
a
,
b
,
c
≤
n
. A frog needs to go from the point
(
0
,
0
,
0
)
(0,0,0)
(
0
,
0
,
0
)
to the point
(
n
,
n
,
n
)
(n,n,n)
(
n
,
n
,
n
)
with the following rules:
⋅
\cdot
⋅
The frog can jump only in points of
M
M
M
⋅
\cdot
⋅
The frog can't jump more than
1
1
1
time over the same point.
⋅
\cdot
⋅
In each jump the frog can go from
(
x
,
y
,
z
)
(x,y,z)
(
x
,
y
,
z
)
to
(
x
+
1
,
y
,
z
)
(x+1,y,z)
(
x
+
1
,
y
,
z
)
,
(
x
,
y
+
1
,
z
)
(x,y+1,z)
(
x
,
y
+
1
,
z
)
,
(
x
,
y
,
z
+
1
)
(x,y,z+1)
(
x
,
y
,
z
+
1
)
or
(
x
,
y
,
z
−
1
)
(x,y,z-1)
(
x
,
y
,
z
−
1
)
In how many ways the Frog can make his target?
1
3
Hide problems
True coin, there is 1 fake coin AMONG US. Can u find the SUSSY impostor?
a) Among
9
9
9
apparently identical coins, one is false and lighter than the others. How can you discover the fake coin by making
2
2
2
weighing in a two-course balance? b) Find the least necessary number of weighing that must be done to cover a false currency between
27
27
27
coins if all the others are true.
Finding all possible $n$ on a strange division condition!!
Find the sum of all positive integers
n
n
n
such that
n
+
11
n
−
1
\frac{n+11}{\sqrt{n-1}}
n
−
1
n
+
11
is an integer.
When a congruence plays Hide N Seek on a rhombus. Can u find it :D?
Inside a rhombus
A
B
C
D
ABCD
A
BC
D
with
∠
B
A
D
=
60
\angle BAD=60
∠
B
A
D
=
60
, points
F
,
H
,
G
F,H,G
F
,
H
,
G
are choosen on lines
A
D
,
D
C
,
A
C
AD,DC,AC
A
D
,
D
C
,
A
C
respectivily such that
D
F
G
H
DFGH
D
FG
H
is a paralelogram. Show that
B
F
H
BFH
BF
H
is a equilateral triangle.