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Problems
Contests
National and Regional Contests
Mexico Contests
Mexico National Olympiad
1999 Mexico National Olympiad
1999 Mexico National Olympiad
Part of
Mexico National Olympiad
Subcontests
(6)
4
1
Hide problems
distances of 10 centers in 8x8 board of unit squares
An
8
×
8
8 \times 8
8
×
8
board is divided into unit squares. Ten of these squares have their centers marked. Prove that either there exist two marked points on the distance at most
2
\sqrt2
2
, or there is a point on the distance
1
/
2
1/2
1/2
from the edge of the board.
5
1
Hide problems
length of segment of intersections of pairs of external angle bisectors = s
In a quadrilateral
A
B
C
D
ABCD
A
BC
D
with
A
B
/
/
C
D
AB // CD
A
B
//
C
D
, the external bisectors of the angles at
B
B
B
and
C
C
C
meet at
P
P
P
, while the external bisectors of the angles at
A
A
A
and
D
D
D
meet at
Q
Q
Q
. Prove that the length of
P
Q
PQ
PQ
equals the semiperimeter of
A
B
C
D
ABCD
A
BC
D
.
1
1
Hide problems
wining strategy on a table with 1999 red and black counters
On a table there are
1999
1999
1999
counters, red on one side and black on the other side, arranged arbitrarily. Two people alternately make moves, where each move is of one of the following two types: (1) Remove several counters which all have the same color up; (2) Reverse several counters which all have the same color up. The player who takes the last counter wins. Decide which of the two players (the one playing first or the other one) has a wining strategy.
3
1
Hide problems
hexagon area = 1/3 triangle area, plus concurrency question
A point
P
P
P
is given inside a triangle
A
B
C
ABC
A
BC
. Let
D
,
E
,
F
D,E,F
D
,
E
,
F
be the midpoints of
A
P
,
B
P
,
C
P
AP,BP,CP
A
P
,
BP
,
CP
, and let
L
,
M
,
N
L,M,N
L
,
M
,
N
be the intersection points of
B
F
BF
BF
and
C
E
,
A
F
CE, AF
CE
,
A
F
and
C
D
,
A
E
CD, AE
C
D
,
A
E
and
B
D
BD
B
D
, respectively. (a) Prove that the area of hexagon
D
N
E
L
F
M
DNELFM
D
NE
L
FM
is equal to one third of the area of triangle
A
B
C
ABC
A
BC
. (b) Prove that
D
L
,
E
M
DL,EM
D
L
,
EM
, and
F
N
FN
FN
are concurrent.
2
1
Hide problems
1999 primes in an arithmetic progression < 12345 do not exist
Prove that there are no
1999
1999
1999
primes in an arithmetic progression that are all less than
12345
12345
12345
.
6
1
Hide problems
mexico 1999
A polygon has each side integral and each pair of adjacent sides perpendicular (it is not necessarily convex). Show that if it can be covered by non-overlapping
2
x
1
2 x 1
2
x
1
dominos, then at least one of its sides has even length.