MathDB
Problems
Contests
National and Regional Contests
Mexico Contests
Mexico National Olympiad
1987 Mexico National Olympiad
1987 Mexico National Olympiad
Part of
Mexico National Olympiad
Subcontests
(8)
8
1
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3 lines in space passing through a point ...
(a) Three lines
l
,
m
,
n
l,m,n
l
,
m
,
n
in space pass through point
S
S
S
. A plane perpendicular to
m
m
m
intersects
l
,
m
,
n
l,m,n
l
,
m
,
n
at
A
,
B
,
C
A,B,C
A
,
B
,
C
respectively. Suppose that
∠
A
S
B
=
∠
B
S
C
=
4
5
o
\angle ASB = \angle BSC = 45^o
∠
A
SB
=
∠
BSC
=
4
5
o
and
∠
A
B
C
=
9
0
o
\angle ABC = 90^o
∠
A
BC
=
9
0
o
. Compute
∠
A
S
C
\angle ASC
∠
A
SC
. (b) Furthermore, if a plane perpendicular to
l
l
l
intersects
l
,
m
,
n
l,m,n
l
,
m
,
n
at
P
,
Q
,
R
P,Q,R
P
,
Q
,
R
respectively and
S
P
=
1
SP = 1
SP
=
1
, find the sides of triangle
P
Q
R
PQR
PQR
.
7
1
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(n^2+n-1) / (n^2+2n) is irreducible for every n
Show that the fraction
n
2
+
n
−
1
n
2
+
2
n
\frac{n^2+n-1}{n^2+2n}
n
2
+
2
n
n
2
+
n
−
1
is irreducible for every positive integer n.
6
1
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(n^3 -n)(5^{8n+4} +3^{4n+2}) is a multiple of 3804
Prove that for every positive integer n the number
(
n
3
−
n
)
(
5
8
n
+
4
+
3
4
n
+
2
)
(n^3 -n)(5^{8n+4} +3^{4n+2})
(
n
3
−
n
)
(
5
8
n
+
4
+
3
4
n
+
2
)
is a multiple of
3804
3804
3804
.
5
1
Hide problems
for no point M 2 triangles and a rectangle have the same area
In a right triangle
A
B
C
ABC
A
BC
, M is a point on the hypotenuse
B
C
BC
BC
and
P
P
P
and
Q
Q
Q
the projections of
M
M
M
on
A
B
AB
A
B
and
A
C
AC
A
C
respectively. Prove that for no such point
M
M
M
do the triangles
B
P
M
,
M
Q
C
BPM, MQC
BPM
,
MQC
and the rectangle
A
Q
M
P
AQMP
A
QMP
have the same area.
4
1
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product of all positive integers < 100 with exactly 3 positive divisors
Calculate the product of all positive integers less than
100
100
100
and having exactly three positive divisors. Show that this product is a square.
3
1
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locus of projection so that a triangle is right
Consider two lines
ℓ
\ell
ℓ
and
ℓ
′
\ell '
ℓ
′
and a fixed point
P
P
P
equidistant from these lines. What is the locus of projections
M
M
M
of
P
P
P
on
A
B
AB
A
B
, where
A
A
A
is on
ℓ
\ell
ℓ
,
B
B
B
on
ℓ
′
\ell '
ℓ
′
, and angle
∠
A
P
B
\angle APB
∠
A
PB
is right?
2
1
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number of positive divisors of 20!
How many positive divisors does number
20
!
20!
20
!
have?
1
1
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when the sum of two irreducible fractions is an integer then ...
Prove that if the sum of two irreducible fractions is an integer then the two fractions have the same denominator.