MathDB
Problems
Contests
National and Regional Contests
Mexico Contests
Regional Olympiad of Mexico West
2017 Regional Olympiad of Mexico West
2017 Regional Olympiad of Mexico West
Part of
Regional Olympiad of Mexico West
Subcontests
(6)
6
1
Hide problems
n divides m and all countrymen of m (adding zeros at the end)
A change in a natural number
n
n
n
consists of adding a pair of zeros between two digits or at the end of the decimal representation of
n
n
n
. A countryman of
n
n
n
is a number that can be obtained from one or more changes in
n
n
n
. For example.
40041
40041
40041
,
4410000
4410000
4410000
and
4004001
4004001
4004001
are all countrymen from
441
441
441
. Determine all the natural numbers
n
n
n
for which there is a natural number m with the property that
n
n
n
divides
m
m
m
and all the countrymen of
m
m
m
.
5
1
Hide problems
winning stategy, quadratics' game
Laura and Daniel play with quadratic polynomials. First Laura says a nonzero real number
r
r
r
. Then Daniel says a nonzero real number
s
s
s
, and then again Laura says another nonzero real number
t
t
t
. Finally. Daniel writes the polynomial
P
(
x
)
=
a
x
2
+
b
x
+
c
P(x) = ax^2 + bx + c
P
(
x
)
=
a
x
2
+
b
x
+
c
where
a
,
b
a,b
a
,
b
, and
c
c
c
are
r
,
s
r,s
r
,
s
, and
t
t
t
in some order Daniel chooses. Laura wins if the equation
P
(
x
)
=
0
P(x) = 0
P
(
x
)
=
0
has two different real solutions, and Daniel wins otherwise. Determine who has a winning strategy and describe that strategy.
4
1
Hide problems
locus of P such that (ABP)=(ACP)=(BCP) for given ABC
Let
△
A
B
C
\vartriangle ABC
△
A
BC
be a triangle. Determine all points
P
P
P
in the plane such that the triangles
△
A
B
P
\vartriangle ABP
△
A
BP
,
△
A
C
P
\vartriangle ACP
△
A
CP
and
△
B
C
P
\vartriangle BCP
△
BCP
all have the same area.
3
1
Hide problems
119 inhabitants who live in 120 apartments
In a building there are
119
119
119
inhabitants who live in
120
120
120
apartments (several inhabitants can live in the same apartment). We call an apartment overcrowded if
15
15
15
or more people live in it. Every day in some overcrowded apartment (if there is one) its inhabitants have a fight and yes they all go to live in a different apartment (which may or may not be already inhabited). Should you always terminate this process?
2
1
Hide problems
CD bisects SQ, 2 circles related
From a point
P
P
P
, two tangent lines are drawn to a circle
Γ
\Gamma
Γ
, which touch it at points
A
A
A
and
B
B
B
. A circle
Φ
\Phi
Φ
is drawn with center at
P
P
P
and passes through
A
A
A
and
B
B
B
and is taken a point
R
R
R
that is on the circumference
Φ
\Phi
Φ
and in the interior of
Γ
\Gamma
Γ
. The straight line
P
R
PR
PR
intersects
Γ
\Gamma
Γ
at the points
S
S
S
and
Q
Q
Q
. The straight lines
A
R
AR
A
R
and
B
R
BR
BR
meet
Γ
\Gamma
Γ
again at points
C
C
C
and
D
D
D
, respectively. Prove that
C
D
CD
C
D
passes through the midpoint of
S
Q
SQ
SQ
.
1
1
Hide problems
min nof of coins for 2017 pesos if banks gices only n^3 pesos
The Occidentalia bank issues coins with denominations of
1
1
1
peso,
8
8
8
pesos,
27
27
27
pesos... and any amount that is a perfect cube (
n
3
n^3
n
3
) of pesos. Determine what is the least amount
k
k
k
of coins needed to give
2017
2017
2017
pesos. For that amount, find all the possible ways to give
2017
2017
2017
pesos using exactly
k
k
k
currency.