MathDB
Problems
Contests
National and Regional Contests
Mexico Contests
Mexico National Olympiad
2014 Mexico National Olympiad
2014 Mexico National Olympiad
Part of
Mexico National Olympiad
Subcontests
(6)
5
1
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Easy inequality on a + b + c = 3
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be positive reals such that
a
+
b
+
c
=
3
a + b + c = 3
a
+
b
+
c
=
3
. Prove:
a
2
a
+
b
c
3
+
b
2
b
+
c
a
3
+
c
2
c
+
a
b
3
≥
3
2
\frac{a^2}{a + \sqrt[3]{bc}} + \frac{b^2}{b + \sqrt[3]{ca}} + \frac{c^2}{c + \sqrt[3]{ab}} \geq \frac{3}{2}
a
+
3
b
c
a
2
+
b
+
3
c
a
b
2
+
c
+
3
ab
c
2
≥
2
3
And determine when equality holds.
6
1
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Relationship n and its number of divisors
Let
d
(
n
)
d(n)
d
(
n
)
be the number of positive divisors of a positive integer
n
n
n
(including
1
1
1
and
n
n
n
). Find all values of
n
n
n
such that
n
+
d
(
n
)
=
d
(
n
)
2
n + d(n) = d(n)^2
n
+
d
(
n
)
=
d
(
n
)
2
.
4
1
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Tangent line
Problem 4Let
A
B
C
D
ABCD
A
BC
D
be a rectangle with diagonals
A
C
AC
A
C
and
B
D
BD
B
D
. Let
E
E
E
be the intersection of the bisector of
∠
C
A
D
\angle CAD
∠
C
A
D
with segment
C
D
CD
C
D
,
F
F
F
on
C
D
CD
C
D
such that
E
E
E
is midpoint of
D
F
DF
D
F
, and
G
G
G
on
B
C
BC
BC
such that
B
G
=
A
C
BG = AC
BG
=
A
C
(with
C
C
C
between
B
B
B
and
G
G
G
). Prove that the circumference through
D
D
D
,
F
F
F
and
G
G
G
is tangent to
B
G
BG
BG
.
3
1
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Concurrency in two circles
Let
Γ
1
\Gamma_1
Γ
1
be a circle and
P
P
P
a point outside of
Γ
1
\Gamma_1
Γ
1
. The tangents from
P
P
P
to
Γ
1
\Gamma_1
Γ
1
touch the circle at
A
A
A
and
B
B
B
. Let
M
M
M
be the midpoint of
P
A
PA
P
A
and
Γ
2
\Gamma_2
Γ
2
the circle through
P
P
P
,
A
A
A
and
B
B
B
. Line
B
M
BM
BM
cuts
Γ
2
\Gamma_2
Γ
2
at
C
C
C
, line
C
A
CA
C
A
cuts
Γ
1
\Gamma_1
Γ
1
at
D
D
D
, segment
D
B
DB
D
B
cuts
Γ
2
\Gamma_2
Γ
2
at
E
E
E
and line
P
E
PE
PE
cuts
Γ
1
\Gamma_1
Γ
1
at
F
F
F
, with
E
E
E
in segment
P
F
PF
PF
. Prove lines
A
F
AF
A
F
,
B
P
BP
BP
, and
C
E
CE
CE
are concurrent.
2
1
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Reducing numbers
A positive integer
a
a
a
is said to reduce to a positive integer
b
b
b
if when dividing
a
a
a
by its units digits the result is
b
b
b
. For example, 2015 reduces to
2015
5
=
403
\frac{2015}{5} = 403
5
2015
=
403
. Find all the positive integers that become 1 after some amount of reductions. For example, 12 is one such number because 12 reduces to 6 and 6 reduces to 1.
1
1
Hide problems
Coloring cuates
Each of the integers from 1 to 4027 has been colored either green or red. Changing the color of a number is making it red if it was green and making it green if it was red. Two positive integers
m
m
m
and
n
n
n
are said to be cuates if either
m
n
\frac{m}{n}
n
m
or
n
m
\frac{n}{m}
m
n
is a prime number. A step consists in choosing two numbers that are cuates and changing the color of each of them. Show it is possible to apply a sequence of steps such that every integer from 1 to 2014 is green.