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Problems
Contests
National and Regional Contests
Mexico Contests
Mexico National Olympiad
2007 Mexico National Olympiad
2007 Mexico National Olympiad
Part of
Mexico National Olympiad
Subcontests
(3)
3
2
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Sum of Square Roots Inequality
Given
a
a
a
,
b
b
b
, and
c
c
c
be positive real numbers with
a
+
b
+
c
=
1
a+b+c=1
a
+
b
+
c
=
1
, prove that
a
+
b
c
+
b
+
c
a
+
c
+
a
b
≤
2
\sqrt{a+bc}+\sqrt{b+ca}+\sqrt{c+ab}\le2
a
+
b
c
+
b
+
c
a
+
c
+
ab
≤
2
Midpoint and Strange Point
Let
A
B
C
ABC
A
BC
be a triangle with
A
B
>
B
C
>
C
A
AB>BC>CA
A
B
>
BC
>
C
A
. Let
D
D
D
be a point on
A
B
AB
A
B
such that
C
D
=
B
C
CD=BC
C
D
=
BC
, and let
M
M
M
be the midpoint of
A
C
AC
A
C
. Show that
B
D
=
A
C
BD=AC
B
D
=
A
C
and that
∠
B
A
C
=
2
∠
A
B
M
.
\angle BAC=2\angle ABM.
∠
B
A
C
=
2∠
A
BM
.
2
2
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<APB = < BPC Locus
Given an equilateral
△
A
B
C
\triangle ABC
△
A
BC
, find the locus of points
P
P
P
such that
∠
A
P
B
=
∠
B
P
C
\angle APB=\angle BPC
∠
A
PB
=
∠
BPC
.
Lightning Bugs
In each square of a
6
×
6
6\times6
6
×
6
grid there is a lightning bug on or off. One move is to choose three consecutive squares, either horizontal or vertical, and change the lightning bugs in those
3
3
3
squares from off to on or from on to off. Show if at the beginning there is one lighting bug on and the rest of them off, it is not possible to make some moves so that at the end they are all turned off.
1
2
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10 but not 11 Consecutive Divisors
Find all integers
N
N
N
with the following property: for
10
10
10
but not
11
11
11
consecutive positive integers, each one is a divisor of
N
N
N
.
Unit Fraction Expression
The fraction
1
10
\frac1{10}
10
1
can be expressed as the sum of two unit fraction in many ways, for example,
1
30
+
1
15
\frac1{30}+\frac1{15}
30
1
+
15
1
and
1
60
+
1
12
\frac1{60}+\frac1{12}
60
1
+
12
1
.Find the number of ways that
1
2007
\frac1{2007}
2007
1
can be expressed as the sum of two distinct positive unit fractions.