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Problems
Contests
National and Regional Contests
Mexico Contests
Mexico National Olympiad
1998 Mexico National Olympiad
1998 Mexico National Olympiad
Part of
Mexico National Olympiad
Subcontests
(6)
6
1
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max no of equidistant planes from 5 points, without 4 of 5 coplanar
A plane in space is equidistant from a set of points if its distances from the points in the set are equal. What is the largest possible number of equidistant planes from five points, no four of which are coplanar?
5
1
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PJ // AC iff BC^2 = AC· QC
The tangents at points
B
B
B
and
C
C
C
on a given circle meet at point
A
A
A
. Let
Q
Q
Q
be a point on segment
A
C
AC
A
C
and let
B
Q
BQ
BQ
meet the circle again at
P
P
P
. The line through
Q
Q
Q
parallel to
A
B
AB
A
B
intersects
B
C
BC
BC
at
J
J
J
. Prove that
P
J
PJ
P
J
is parallel to
A
C
AC
A
C
if and only if
B
C
2
=
A
C
⋅
Q
C
BC^2 = AC\cdot QC
B
C
2
=
A
C
⋅
QC
.
4
1
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integers in form 1/a_1+2/a_2+...+9/a_9, where a_i digits
Find all integers that can be written in the form
1
a
1
+
2
a
2
+
.
.
.
+
9
a
9
\frac{1}{a_1}+\frac{2}{a_2}+...+\frac{9}{a_9}
a
1
1
+
a
2
2
+
...
+
a
9
9
where
a
1
,
a
2
,
.
.
.
,
a
9
a_1,a_2, ...,a_9
a
1
,
a
2
,
...
,
a
9
are nonzero digits, not necessarily different.
2
1
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locus with circles
Rays
l
l
l
and
m
m
m
forming an angle of
a
a
a
are drawn from the same point. Let
P
P
P
be a fixed point on
l
l
l
. For each circle
C
C
C
tangent to
l
l
l
at
P
P
P
and intersecting
m
m
m
at
Q
Q
Q
and
R
R
R
, let
T
T
T
be the intersection point of the bisector of angle
Q
P
R
QPR
QPR
with
C
C
C
. Describe the locus of
T
T
T
and justify your answer.
1
1
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infinitely pairs of consecutive lucky numbers in number theory
A number is called lucky if computing the sum of the squares of its digits and repeating this operation sufficiently many times leads to number
1
1
1
. For example,
1900
1900
1900
is lucky, as
1900
→
82
→
68
→
100
→
1
1900 \to 82 \to 68 \to 100 \to 1
1900
→
82
→
68
→
100
→
1
. Find infinitely many pairs of consecutive numbers each of which is lucky.
3
1
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OMM Math Olympiad 1997 #5.
Every side and diagonal of a regular octagon is color with red or black. Show that there is at least seven triangles whose vertices are vertices of the octagon and its three sides are of the same color.