MathDB
Problems
Contests
National and Regional Contests
Mexico Contests
Mexico National Olympiad
2010 Mexico National Olympiad
2010 Mexico National Olympiad
Part of
Mexico National Olympiad
Subcontests
(3)
3
2
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Tangent Circles
Let
C
1
\mathcal{C}_1
C
1
and
C
2
\mathcal{C}_2
C
2
be externally tangent at a point
A
A
A
. A line tangent to
C
1
\mathcal{C}_1
C
1
at
B
B
B
intersects
C
2
\mathcal{C}_2
C
2
at
C
C
C
and
D
D
D
; then the segment
A
B
AB
A
B
is extended to intersect
C
2
\mathcal{C}_2
C
2
at a point
E
E
E
. Let
F
F
F
be the midpoint of \overarc{CD} that does not contain
E
E
E
, and let
H
H
H
be the intersection of
B
F
BF
BF
with
C
2
\mathcal{C}_2
C
2
. Show that
C
D
CD
C
D
,
A
F
AF
A
F
, and
E
H
EH
E
H
are concurrent.
(pq)^r+(qr)^p+(rp)^q
Let
p
p
p
,
q
q
q
, and
r
r
r
be distinct positive prime numbers. Show that if
p
q
r
∣
(
p
q
)
r
+
(
q
r
)
p
+
(
r
p
)
q
−
1
,
pqr\mid (pq)^r+(qr)^p+(rp)^q-1,
pq
r
∣
(
pq
)
r
+
(
q
r
)
p
+
(
r
p
)
q
−
1
,
then
(
p
q
r
)
3
∣
3
(
(
p
q
)
r
+
(
q
r
)
p
+
(
r
p
)
q
−
1
)
.
(pqr)^3\mid 3((pq)^r+(qr)^p+(rp)^q-1).
(
pq
r
)
3
∣
3
((
pq
)
r
+
(
q
r
)
p
+
(
r
p
)
q
−
1
)
.
2
2
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Lights on a Board
In each cell of an
n
×
n
n\times n
n
×
n
board is a lightbulb. Initially, all of the lights are off. Each move consists of changing the state of all of the lights in a row or of all of the lights in a column (off lights are turned on and on lights are turned off).Show that if after a certain number of moves, at least one light is on, then at this moment at least
n
n
n
lights are on.
Right Angle in Acute Triangle...?
Let
A
B
C
ABC
A
BC
be an acute triangle with
A
B
≠
A
C
AB\neq AC
A
B
=
A
C
,
M
M
M
be the median of
B
C
BC
BC
, and
H
H
H
be the orthocenter of
△
A
B
C
\triangle ABC
△
A
BC
. The circumcircle of
B
B
B
,
H
H
H
, and
C
C
C
intersects the median
A
M
AM
A
M
at
N
N
N
. Show that
∠
A
N
H
=
9
0
∘
\angle ANH=90^\circ
∠
A
N
H
=
9
0
∘
.
1
2
Hide problems
abc = a+b+c+1
Find all triplets of natural numbers
(
a
,
b
,
c
)
(a,b,c)
(
a
,
b
,
c
)
that satisfy the equation
a
b
c
=
a
+
b
+
c
+
1
abc=a+b+c+1
ab
c
=
a
+
b
+
c
+
1
.
0,1,2 in boxes
Let
n
n
n
be a positive integer. In an
n
×
4
n\times4
n
×
4
table, each row is equal to\begin{tabular}{| c | c | c | c |} \hline 2 & 0 & 1 & 0 \\ \hline \end{tabular}A change is taking three consecutive boxes in the same row with different digits in them and changing the digits in these boxes as follows:
0
→
1
,
1
→
2
,
2
→
0
.
0\to1\text{, }1\to2\text{, }2\to0\text{.}
0
→
1
,
1
→
2
,
2
→
0
.
For example, a row \begin{tabular}{| c | c | c | c |}\hline 2 & 0 & 1 & 0 \\ \hline\end{tabular} can be changed to the row \begin{tabular}{| c | c | c | c |}\hline 0 & 1 & 2 & 0 \\ \hline\end{tabular} but not to \begin{tabular}{| c | c | c | c |}\hline 2 & 1 & 2 & 1 \\ \hline\end{tabular} because
0
0
0
,
1
1
1
, and
0
0
0
are not distinct.Changes can be applied as often as wanted, even to items already changed. Show that for
n
<
12
n<12
n
<
12
, it is not possible to perform a finite number of changes so that the sum of the elements in each column is equal.