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Problems
Contests
National and Regional Contests
Mexico Contests
Regional Olympiad of Mexico West
2023 Regional Olympiad of Mexico West
2023 Regional Olympiad of Mexico West
Part of
Regional Olympiad of Mexico West
Subcontests
(6)
6
1
Hide problems
2023 guinea pigs placed in a circle, where 2022 have a mirror, reflections
There are
2023
2023
2023
guinea pigs placed in a circle, from which everyone except one of them, call it
M
M
M
, has a mirror that points towards one of the
2022
2022
2022
other guinea pigs.
M
M
M
has a lantern that will shoot a light beam towards one of the guinea pigs with a mirror and will reflect to the guinea pig that the mirror is pointing and will keep reflecting with every mirror it reaches. Isaías will re-direct some of the mirrors to point to some other of the
2023
2023
2023
guinea pigs. In the worst case scenario, what is the least number of mirrors that need to be re-directed, such that the light beam hits
M
M
M
no matter the starting point of the light beam?
5
1
Hide problems
BFH is equilateral if ABCD is a 120-60 rhombus and DFGH is #
We have a rhombus
A
B
C
D
ABCD
A
BC
D
with
∠
B
A
D
=
6
0
∘
\angle BAD=60^\circ
∠
B
A
D
=
6
0
∘
. We take points
F
,
H
,
G
F,H,G
F
,
H
,
G
on the sides
A
D
,
D
C
AD,DC
A
D
,
D
C
and the diagonal
A
C
AC
A
C
, respectively, such that
D
F
G
H
DFGH
D
FG
H
is a parallelogram. Prove that
B
F
H
BFH
BF
H
is equilateral.
4
1
Hide problems
15 integers in (1 -2023) such that the sum of a few is not m^2, m^3, m^n
Prove that you can pick
15
15
15
distinct positive integers between
1
1
1
and
2023
2023
2023
, such that each one of them and the sum between some of them is never a perfect square, nor a perfect cube or any other greater perfect power.
3
1
Hide problems
decimal and floor function ineq
Let
x
>
1
x>1
x
>
1
be a real number that is not an integer. Denote
{
x
}
\{x\}
{
x
}
as its decimal part and
⌊
x
⌋
\lfloor x\rfloor
⌊
x
⌋
the floor function. Prove that
(
x
+
{
x
}
⌊
x
⌋
−
⌊
x
⌋
x
+
{
x
}
)
+
(
x
+
⌊
x
⌋
{
x
}
−
{
x
}
x
+
⌊
x
⌋
)
>
16
3
\left(\frac{x+\{x\}}{\lfloor x\rfloor}-\frac{\lfloor x\rfloor}{x+\{x\}}\right)+\left(\frac{x+\lfloor x\rfloor}{\{x\}}-\frac{\{x\}}{x+\lfloor x\rfloor}\right)>\frac{16}{3}
(
⌊
x
⌋
x
+
{
x
}
−
x
+
{
x
}
⌊
x
⌋
)
+
(
{
x
}
x
+
⌊
x
⌋
−
x
+
⌊
x
⌋
{
x
}
)
>
3
16
2
1
Hide problems
n guinea pigs placed on the vertices of a regular n-gon
We have
n
n
n
guinea pigs placed on the vertices of a regular polygon with
n
n
n
sides inscribed in a circumference, one guinea pig in each vertex. Each guinea pig has a direction assigned, such direction is either "clockwise" or "anti-clockwise", and a velocity between
1
k
m
/
h
1 km/h
1
km
/
h
,
2
k
m
/
h
2km/h
2
km
/
h
,..., and
n
k
m
/
h
n km/h
nkm
/
h
, each one with a distinct velocity, and each guinea pig has a counter starting from
0
0
0
. They start moving along the circumference with the assigned direction and velocity, everyone at the same time, when 2 or more guinea pigs meet a point, all of the guinea pigs at that point follow the same direction of the fastest guinea pig and they keep moving (with the same velocity as before); each time 2 guinea pigs meet for the first time in the same point, the fastest guinea pig adds 1 to its counter. Prove that, at some moment, for each
1
≤
i
≤
n
1\leq i\leq n
1
≤
i
≤
n
we have that the
i
−
i-
i
−
th guinea pig has
i
−
1
i-1
i
−
1
in its counter.
1
1
Hide problems
a_n=\frac{n}{d}-d where n= greatest divisor of n such that d\leq \sqrt{n}
For every positive integer
n
n
n
we take the greatest divisor
d
d
d
of
n
n
n
such that
d
≤
n
d\leq \sqrt{n}
d
≤
n
and we define
a
n
=
n
d
−
d
a_n=\frac{n}{d}-d
a
n
=
d
n
−
d
. Prove that in the sequence
a
1
,
a
2
,
a
3
,
.
.
.
a_1,a_2,a_3,...
a
1
,
a
2
,
a
3
,
...
, any non negative integer
k
k
k
its in the sequence infinitely many times.