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Contests
National and Regional Contests
Peru Contests
Peru MO (ONEM)
2007 Peru MO (ONEM)
2007 Peru MO (ONEM)
Part of
Peru MO (ONEM)
Subcontests
(4)
1
1
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sin A / (cos A-1) >=1, (3cos A-1) / sin A >=1,
Find all values of
A
A
A
such that
0
o
<
A
<
36
0
o
0^o < A < 360^o
0
o
<
A
<
36
0
o
and also
sin
A
cos
A
−
1
≥
1
\frac{\sin A}{\cos A - 1} \ge 1
c
o
s
A
−
1
s
i
n
A
≥
1
and
3
cos
A
−
1
sin
A
≥
1.
\frac{3\cos A - 1}{\sin A} \ge 1.
s
i
n
A
3
c
o
s
A
−
1
≥
1.
3
1
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max n of consecutive naturals such that they are all special
We say that a natural number of at least two digits
E
E
E
is special if each time two adjacent digits of
E
E
E
are added, a divisor of
E
E
E
is obtained. For example,
2124
2124
2124
is special, since the numbers
2
+
1
2 + 1
2
+
1
,
1
+
2
1 + 2
1
+
2
and
2
+
4
2 + 4
2
+
4
are all divisors of
2124
2124
2124
. Find the largest value of
n
n
n
for which there exist
n
n
n
consecutive natural numbers such that they are all special.
2
1
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AB/1=BC/3=AC/3 for painted points on a stariagh line, red or blue
Assuming that each point of a straight line is painted red or blue, arbitrarily, show that it is always possible to choose three points
A
,
B
A, B
A
,
B
and
C
C
C
in such a way straight, that are painted the same color and that:
A
B
1
=
B
C
2
=
A
C
3
.
\frac{AB}{1}=\frac{BC}{2}=\frac{AC}{3}.
1
A
B
=
2
BC
=
3
A
C
.
4
1
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equal angles wanted, starting with a rhombus ABCD with equil. ABD and BCD
Let
A
B
C
D
ABCD
A
BC
D
be rhombus
A
B
C
D
ABCD
A
BC
D
where the triangles
A
B
D
ABD
A
B
D
and
B
C
D
BCD
BC
D
are equilateral. Let
M
M
M
and
N
N
N
be points on the sides
B
C
BC
BC
and
C
D
CD
C
D
, respectively, such that
∠
M
A
N
=
3
0
o
\angle MAN = 30^o
∠
M
A
N
=
3
0
o
. Let
X
X
X
be the intersection point of the diagonals
A
C
AC
A
C
and
B
D
BD
B
D
. Prove that
∠
X
M
N
=
∠
D
A
M
\angle XMN = \angle\ DAM
∠
XMN
=
∠
D
A
M
and
∠
X
N
M
=
∠
B
A
N
\angle XNM = \angle BAN
∠
XNM
=
∠
B
A
N
.