MathDB
Problems
Contests
National and Regional Contests
Mexico Contests
Regional Olympiad of Mexico Center Zone
2010 Regional Olympiad of Mexico Center Zone
2010 Regional Olympiad of Mexico Center Zone
Part of
Regional Olympiad of Mexico Center Zone
Subcontests
(6)
4
1
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there are 2 numbers in A whose difference is a or b
Let
a
a
a
and
b
b
b
be two positive integers and
A
A
A
be a subset of
{
1
,
2
,
…
,
a
+
b
}
\{1, 2,…, a + b \}
{
1
,
2
,
…
,
a
+
b
}
that has more than
a
+
b
2
\frac {a + b} {2}
2
a
+
b
elements. Show that there are two numbers in
A
A
A
whose difference is
a
a
a
or
b
b
b
.
1
1
Hide problems
angle bisector wanted, 2 circumcircles related
In the acute triangle
A
B
C
ABC
A
BC
,
∠
B
A
C
\angle BAC
∠
B
A
C
is less than
∠
A
C
B
\angle ACB
∠
A
CB
. Let
A
D
AD
A
D
be a diameter of
ω
\omega
ω
, the circle circumscribed to said triangle. Let
E
E
E
be the point of intersection of the ray
A
C
AC
A
C
and the tangent to
ω
\omega
ω
passing through
B
B
B
. The perpendicular to
A
D
AD
A
D
that passes through
E
E
E
intersects the circle circumscribed to the triangle
B
C
E
BCE
BCE
, again, at the point
F
F
F
. Show that
C
D
CD
C
D
is an angle bisector of
∠
B
C
F
\angle BCF
∠
BCF
.
5
1
Hide problems
r + p^4 = q^4 diophantine
Find all integer solutions
(
p
,
q
,
r
)
(p, q, r)
(
p
,
q
,
r
)
of the equation
r
+
p
4
=
q
4
r + p ^ 4 = q ^ 4
r
+
p
4
=
q
4
with the following conditions:
∙
\bullet
∙
r
r
r
is a positive integer with exactly
8
8
8
positive divisors.
∙
\bullet
∙
p
p
p
and
q
q
q
are prime numbers.
2
1
Hide problems
p-4 cannot be 4th power of a prime, for prime p>5
Let
p
>
5
p>5
p
>
5
be a prime number. Show that
p
−
4
p-4
p
−
4
cannot be the fourth power of a prime number.
3
1
Hide problems
inequality
Let
a
a
a
,
b
b
b
and
c
c
c
be real positive numbers such that
1
a
+
1
b
+
1
c
=
1
\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = 1
a
1
+
b
1
+
c
1
=
1
Prove that:
a
2
+
b
2
+
b
2
≥
2
a
+
2
b
+
2
c
+
9
a^2+b^2+b^2 \ge 2a+2b+2c+9
a
2
+
b
2
+
b
2
≥
2
a
+
2
b
+
2
c
+
9
6
1
Hide problems
APF = BPD in equilateral triangle
Let
A
B
C
ABC
A
BC
be an equilateral triangle and
D
D
D
the midpoint of
B
C
BC
BC
. Let
E
E
E
and
F
F
F
be points on
A
C
AC
A
C
and
A
B
AB
A
B
respectively such that
A
F
=
C
E
AF=CE
A
F
=
CE
.
P
=
B
E
P=BE
P
=
BE
∩
\cap
∩
C
F
CF
CF
. Show that
∠
\angle
∠
A
P
F
=
APF=
A
PF
=
∠
\angle
∠
B
P
D
BPD
BP
D