MathDB
Problems
Contests
National and Regional Contests
Mexico Contests
Mexico National Olympiad
1997 Mexico National Olympiad
1997 Mexico National Olympiad
Part of
Mexico National Olympiad
Subcontests
(6)
6
1
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infinite representations of 1 =1/5+Σ 1/a_i , from i=1,..., n & 5<a_1<...<a_n
Prove that number
1
1
1
has infinitely many representations of the form
1
=
1
5
+
1
a
1
+
1
a
2
+
.
.
.
+
1
a
n
1 =\frac{1}{5}+\frac{1}{a_1}+\frac{1}{a_2}+ ...+\frac{1}{a_n}
1
=
5
1
+
a
1
1
+
a
2
1
+
...
+
a
n
1
, where
n
n
n
and
a
i
a_i
a
i
are positive integers with
5
<
a
1
<
a
2
<
.
.
.
<
a
n
5 < a_1 < a_2 < ... < a_n
5
<
a
1
<
a
2
<
...
<
a
n
.
5
1
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points on sides of a triangle, intersections, extensions, ratio of areas wanted
Let
P
,
Q
,
R
P,Q,R
P
,
Q
,
R
be points on the sides
B
C
,
C
A
,
A
B
BC,CA,AB
BC
,
C
A
,
A
B
respectively of a triangle
A
B
C
ABC
A
BC
. Suppose that
B
Q
BQ
BQ
and
C
R
CR
CR
meet at
A
′
,
A
P
A', AP
A
′
,
A
P
and
C
R
CR
CR
meet at
B
′
B'
B
′
, and
A
P
AP
A
P
and
B
Q
BQ
BQ
meet at
C
′
C'
C
′
, such that
A
B
′
=
B
′
C
′
,
B
C
′
=
C
′
A
′
AB' = B'C', BC' =C'A'
A
B
′
=
B
′
C
′
,
B
C
′
=
C
′
A
′
, and
C
A
′
=
A
′
B
′
CA'= A'B'
C
A
′
=
A
′
B
′
. Compute the ratio of the area of
△
P
Q
R
\triangle PQR
△
PQR
to the area of
△
A
B
C
\triangle ABC
△
A
BC
.
4
1
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min No of planes defined by 6 points, not all coplanar &no 3 of them collinear
What is the minimum number of planes determined by
6
6
6
points in space which are not all coplanar, and among which no three are collinear?
3
1
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numbers 1 - 16 written in the cells of a 4x4 board
The numbers
1
1
1
through
16
16
16
are to be written in the cells of a
4
×
4
4\times 4
4
×
4
board. (a) Prove that this can be done in such a way that any two numbers in cells that share a side differ by at most
4
4
4
. (b) Prove that this cannot be done in such a way that any two numbers in cells that share a side differ by at most
3
3
3
.
2
1
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4 points defined be equal ratios in a triangle, collinearity wanted
In a triangle
A
B
C
,
P
ABC, P
A
BC
,
P
and
P
′
P'
P
′
are points on side
B
C
,
Q
BC, Q
BC
,
Q
on side
C
A
CA
C
A
, and
R
R
R
on side
A
B
AB
A
B
, such that
A
R
R
B
=
B
P
P
C
=
C
Q
Q
A
=
C
P
′
P
′
B
\frac{AR}{RB}=\frac{BP}{PC}=\frac{CQ}{QA}=\frac{CP'}{P'B}
RB
A
R
=
PC
BP
=
Q
A
CQ
=
P
′
B
C
P
′
. Let
G
G
G
be the centroid of triangle
A
B
C
ABC
A
BC
and
K
K
K
be the intersection point of
A
P
′
AP'
A
P
′
and
R
Q
RQ
RQ
. Prove that points
P
,
G
,
K
P,G,K
P
,
G
,
K
are collinear.
1
1
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find prime p so that 8p^4-3003 is prime
Determine all prime numbers
p
p
p
for which
8
p
4
−
3003
8p^4-3003
8
p
4
−
3003
is a positive prime number.