MathDB
Problems
Contests
National and Regional Contests
Mexico Contests
Mexico National Olympiad
1992 Mexico National Olympiad
1992 Mexico National Olympiad
Part of
Mexico National Olympiad
Subcontests
(6)
6
1
Hide problems
rectangle, midpoints, right isoscles triangles, area wanted
A
B
C
D
ABCD
A
BC
D
is a rectangle.
I
I
I
is the midpoint of
C
D
CD
C
D
.
B
I
BI
B
I
meets
A
C
AC
A
C
at
M
M
M
. Show that the line
D
M
DM
D
M
passes through the midpoint of
B
C
BC
BC
.
E
E
E
is a point outside the rectangle such that
A
E
=
B
E
AE = BE
A
E
=
BE
and
∠
A
E
B
=
9
0
o
\angle AEB = 90^o
∠
A
EB
=
9
0
o
. If
B
E
=
B
C
=
x
BE = BC = x
BE
=
BC
=
x
, show that
E
M
EM
EM
bisects
∠
A
M
B
\angle AMB
∠
A
MB
. Find the area of
A
E
B
M
AEBM
A
EBM
in terms of
x
x
x
.
5
1
Hide problems
6 < \sqrt{2x+3}+ \sqrt{2y+3}+ \sqrt{2z+3}<= 3\sqrt5 if x+y+z=3 & x,y,z>0
x
,
y
,
z
x, y, z
x
,
y
,
z
are positive reals with sum
3
3
3
. Show that
6
<
2
x
+
3
+
2
y
+
3
+
2
z
+
3
≤
3
5
6 < \sqrt{2x+3} + \sqrt{2y+3} + \sqrt{2z+3}\le 3\sqrt5
6
<
2
x
+
3
+
2
y
+
3
+
2
z
+
3
≤
3
5
4
1
Hide problems
100 \ 1+ 11^{11} +111^{111} +1111^{1111} +...+1111111111^{1111111111}
Show that
1
+
1
1
11
+
11
1
111
+
111
1
1111
+
.
.
.
+
111111111
1
1111111111
1 + 11^{11} + 111^{111} + 1111^{1111} +...+ 1111111111^{1111111111}
1
+
1
1
11
+
11
1
111
+
111
1
1111
+
...
+
111111111
1
1111111111
is divisible by
100
100
100
.
3
1
Hide problems
7 points inside or on a regular hexagon, with (ABC)<1/6 hexagon area
Given
7
7
7
points inside or on a regular hexagon, show that three of them form a triangle with area
≤
1
/
6
\le 1/6
≤
1/6
the area of the hexagon.
2
1
Hide problems
(a,b,c,d) of positive integers with 0<a,b,c,d <p-1 satisfy ad = bc mod p
Given a prime number
p
p
p
, how many
4
4
4
-tuples
(
a
,
b
,
c
,
d
)
(a, b, c, d)
(
a
,
b
,
c
,
d
)
of positive integers with
0
≤
a
,
b
,
c
,
d
≤
p
−
1
0 \le a, b, c, d \le p-1
0
≤
a
,
b
,
c
,
d
≤
p
−
1
satisfy
a
d
=
b
c
ad = bc
a
d
=
b
c
mod
p
p
p
?
1
1
Hide problems
show that 4 faces of tetrahedron have equal area
The tetrahedron
O
P
Q
R
OPQR
OPQR
has the
∠
P
O
Q
=
∠
P
O
R
=
∠
Q
O
R
=
9
0
o
\angle POQ = \angle POR = \angle QOR = 90^o
∠
POQ
=
∠
POR
=
∠
QOR
=
9
0
o
.
X
,
Y
,
Z
X, Y, Z
X
,
Y
,
Z
are the midpoints of
P
Q
,
Q
R
PQ, QR
PQ
,
QR
and
R
P
.
RP.
RP
.
Show that the four faces of the tetrahedron
O
X
Y
Z
OXYZ
OX
Y
Z
have equal area.