MathDB
Problems
Contests
National and Regional Contests
Mexico Contests
Regional Olympiad of Mexico Center Zone
2019 Regional Olympiad of Mexico Center Zone
2019 Regional Olympiad of Mexico Center Zone
Part of
Regional Olympiad of Mexico Center Zone
Subcontests
(6)
2
1
Hide problems
f (x + y) <=f (xy) for x,y reals
Find all functions
f
:
R
→
R
f: \mathbb {R} \rightarrow \mathbb {R}
f
:
R
→
R
such that
f
(
x
+
y
)
≤
f
(
x
y
)
f (x + y) \le f (xy)
f
(
x
+
y
)
≤
f
(
x
y
)
for every pair of real
x
x
x
,
y
y
y
.
1
1
Hide problems
a ^ 2 + b ^ 2 + 3, 2b ^ 2 + c ^ 2 + 3, 2c ^ 2 + a ^ 2 + 3 not perfect squares
Let
a
a
a
,
b
b
b
, and
c
c
c
be integers greater than zero. Show that the numbers
2
a
2
+
b
2
+
3
,
2
b
2
+
c
2
+
3
,
2
c
2
+
a
2
+
3
2a ^ 2 + b ^ 2 + 3 \,\,, 2b ^ 2 + c ^ 2 + 3\,\,, 2c ^ 2 + a ^ 2 + 3
2
a
2
+
b
2
+
3
,
2
b
2
+
c
2
+
3
,
2
c
2
+
a
2
+
3
cannot be all perfect squares.
3
1
Hide problems
ZX = ZY wanted, <BAD=<DAC, 2 circumcircles, 2 angle bisectors, PX//AC, QY//AB
Let
A
B
C
ABC
A
BC
be an acute triangle and
D
D
D
a point on the side
B
C
BC
BC
such that
∠
B
A
D
=
∠
D
A
C
\angle BAD = \angle DAC
∠
B
A
D
=
∠
D
A
C
. The circumcircles of the triangles
A
B
D
ABD
A
B
D
and
A
C
D
ACD
A
C
D
intersect the segments
A
C
AC
A
C
and
A
B
AB
A
B
at
E
E
E
and
F
F
F
, respectively. The internal bisectors of
∠
B
D
F
\angle BDF
∠
B
D
F
and
∠
C
D
E
\angle CDE
∠
C
D
E
intersect the sides
A
B
AB
A
B
and
A
C
AC
A
C
at
P
P
P
and
Q
Q
Q
, respectively. Points
X
X
X
and
Y
Y
Y
are chosen on the side
B
C
BC
BC
such that
P
X
PX
PX
is parallel to
A
C
AC
A
C
and
Q
Y
QY
Q
Y
is parallel to
A
B
AB
A
B
. Finally, let
Z
Z
Z
be the point of intersection of
B
E
BE
BE
and
C
F
CF
CF
. Prove that
Z
X
=
Z
Y
ZX = ZY
ZX
=
Z
Y
.
4
1
Hide problems
AP passes through circumcenter of ABC, 2 reflections of a point wrt sides
Let
A
B
C
ABC
A
BC
be a triangle with
∠
B
A
C
>
9
0
∘
\angle BAC> 90 ^ \circ
∠
B
A
C
>
9
0
∘
and
D
D
D
a point on
B
C
BC
BC
. Let
E
E
E
and
F
F
F
be the reflections of the point
D
D
D
about
A
B
AB
A
B
and
A
C
AC
A
C
, respectively. Suppose that
B
E
BE
BE
and
C
F
CF
CF
intersect at
P
P
P
. Show that
A
P
AP
A
P
passes through the circumcenter of triangle
A
B
C
ABC
A
BC
.
6
1
Hide problems
A problem with perfect powers
Find all positive integers
m
m
m
with the next property: If
d
d
d
is a positive integer less or equal to
m
m
m
and it isn't coprime to
m
m
m
, then there exist positive integers
a
1
,
a
2
a_{1}, a_{2}
a
1
,
a
2
,. . .,
a
2019
a_{2019}
a
2019
(where all of them are coprimes to
m
m
m
) such that
m
+
a
1
d
+
a
2
d
2
+
⋅
⋅
⋅
+
a
2019
d
2019
m+a_{1}d+a_{2}d^{2}+\cdot \cdot \cdot+a_{2019}d^{2019}
m
+
a
1
d
+
a
2
d
2
+
⋅
⋅
⋅
+
a
2019
d
2019
is a perfect power.
5
1
Hide problems
A problem with series
A serie of positive integers
a
1
a_{1}
a
1
,
a
2
a_{2}
a
2
,. . . ,
a
n
a_{n}
a
n
is
a
u
t
o
−
d
e
l
i
m
i
t
e
d
auto-delimited
a
u
t
o
−
d
e
l
imi
t
e
d
if for every index
i
i
i
that holds
1
≤
i
≤
n
1\leq i\leq n
1
≤
i
≤
n
, there exist at least
a
i
a_{i}
a
i
terms of the serie such that they are all less or equal to
i
i
i
. Find the maximum value of the sum
a
1
+
a
2
+
⋅
⋅
⋅
+
a
n
a_{1}+a_{2}+\cdot \cdot \cdot+a_{n}
a
1
+
a
2
+
⋅
⋅
⋅
+
a
n
, where
a
1
a_{1}
a
1
,
a
2
a_{2}
a
2
,. . . ,
a
n
a_{n}
a
n
is an
a
u
t
o
−
d
e
l
i
m
i
t
e
d
auto-delimited
a
u
t
o
−
d
e
l
imi
t
e
d
serie.