MathDB
Problems
Contests
National and Regional Contests
Mexico Contests
Mexico National Olympiad
2009 Mexico National Olympiad
2009 Mexico National Olympiad
Part of
Mexico National Olympiad
Subcontests
(3)
3
2
Hide problems
Cubes and 2s
Let
a
a
a
,
b
b
b
, and
c
c
c
be positive numbers satisfying
a
b
c
=
1
abc=1
ab
c
=
1
. Show that
a
3
a
3
+
2
+
b
3
b
3
+
2
+
c
3
c
3
+
2
≥
1
and
1
a
3
+
2
+
1
b
3
+
2
+
1
c
3
+
2
≤
1
\frac{a^3}{a^3+2}+\frac{b^3}{b^3+2}+\frac{c^3}{c^3+2}\ge1\text{ and }\frac1{a^3+2}+\frac1{b^3+2}+\frac1{c^3+2}\le1
a
3
+
2
a
3
+
b
3
+
2
b
3
+
c
3
+
2
c
3
≥
1
and
a
3
+
2
1
+
b
3
+
2
1
+
c
3
+
2
1
≤
1
Friends at a Party
At a party with
n
n
n
people, it is known that among any
4
4
4
people, there are either
3
3
3
people who all know one another or
3
3
3
people none of which knows another. Show that the
n
n
n
people can be separated into two rooms, so that everyone in one room knows one another and no two people in the other room know each other.
2
2
Hide problems
Integers in Boxes
In boxes labeled
0
0
0
,
1
1
1
,
2
2
2
,
…
\dots
…
, we place integers according to the following rules:
∙
\bullet
∙
If
p
p
p
is a prime number, we place it in box
1
1
1
.
∙
\bullet
∙
If
a
a
a
is placed in box
m
a
m_a
m
a
and
b
b
b
is placed in box
m
b
m_b
m
b
, then
a
b
ab
ab
is placed in the box labeled
a
m
b
+
b
m
a
am_b+bm_a
a
m
b
+
b
m
a
.Find all positive integers
n
n
n
that are placed in the box labeled
n
n
n
.
A Bunch of Perpendicular Lines
Consider a triangle
A
B
C
ABC
A
BC
and a point
M
M
M
on side
B
C
BC
BC
. Let
P
P
P
be the intersection of the perpendiculars from
M
M
M
to
A
B
AB
A
B
and from
B
B
B
to
B
C
BC
BC
, and let
Q
Q
Q
be the intersection of the perpendiculars from
M
M
M
to
A
C
AC
A
C
and from
C
C
C
to
B
C
BC
BC
. Show that
P
Q
PQ
PQ
is perpendicular to
A
M
AM
A
M
if and only if
M
M
M
is the midpoint of
B
C
BC
BC
.
1
2
Hide problems
Similar Triangles
In
△
A
B
C
\triangle ABC
△
A
BC
, let
D
D
D
be the foot of the altitude from
A
A
A
to
B
C
BC
BC
. A circle centered at
D
D
D
with radius
A
D
AD
A
D
intersects lines
A
B
AB
A
B
and
A
C
AC
A
C
at
P
P
P
and
Q
Q
Q
, respectively. Show that
△
A
Q
P
∼
△
A
B
C
\triangle AQP\sim\triangle ABC
△
A
QP
∼
△
A
BC
.
Plus or Minuses
Let
n
>
1
n>1
n
>
1
be an odd integer, and let
a
1
a_1
a
1
,
a
2
a_2
a
2
,
…
\dots
…
,
a
n
a_n
a
n
be distinct real numbers. Let
M
M
M
be the maximum of these numbers and
m
m
m
the minimum. Show that it is possible to choose the signs of the expression
s
=
±
a
1
±
a
2
±
⋯
±
a
n
s=\pm a_1\pm a_2\pm\dots\pm a_n
s
=
±
a
1
±
a
2
±
⋯
±
a
n
so that
m
<
s
<
M
m<s<M
m
<
s
<
M