MathDB
Problems
Contests
National and Regional Contests
Mexico Contests
Mexico National Olympiad
1994 Mexico National Olympiad
1994 Mexico National Olympiad
Part of
Mexico National Olympiad
Subcontests
(6)
6
1
Hide problems
impossible to tile a 10 x 10 board with so many pieces of this type
Show that we cannot tile a
10
x
10
10 x 10
10
x
10
board with
25
25
25
pieces of type
A
A
A
, or with
25
25
25
pieces of type
B
B
B
, or with
25
25
25
pieces of type
C
C
C
.
5
1
Hide problems
at least 1 of the feet of altitudes in 4 triangles of convex quadr. lies on side
A
B
C
D
ABCD
A
BC
D
is a convex quadrilateral. Take the
12
12
12
points which are the feet of the altitudes in the triangles
A
B
C
,
B
C
D
,
C
D
A
,
D
A
B
ABC, BCD, CDA, DAB
A
BC
,
BC
D
,
C
D
A
,
D
A
B
. Show that at least one of these points must lie on the sides of
A
B
C
D
ABCD
A
BC
D
.
4
1
Hide problems
sequence with relative primes , pages of a book numbered 2 to 400
A capricious mathematician writes a book with pages numbered from
2
2
2
to
400
400
400
. The pages are to be read in the following order. Take the last unread page (
400
400
400
), then read (in the usual order) all pages which are not relatively prime to it and which have not been read before. Repeat until all pages are read. So, the order would be
2
,
4
,
5
,
.
.
.
,
400
,
3
,
7
,
9
,
.
.
.
,
399
,
.
.
.
2, 4, 5, ... , 400, 3, 7, 9, ... , 399, ...
2
,
4
,
5
,
...
,
400
,
3
,
7
,
9
,
...
,
399
,
...
. What is the last page to be read?
3
1
Hide problems
equal angles starting with a parallelogram with perpenducular
A
B
C
D
ABCD
A
BC
D
is a parallelogram. Take
E
E
E
on the line
A
B
AB
A
B
so that
B
E
=
B
C
BE = BC
BE
=
BC
and
B
B
B
lies between
A
A
A
and
E
E
E
. Let the line through
C
C
C
perpendicular to
B
D
BD
B
D
and the line through
E
E
E
perpendicular to
A
B
AB
A
B
meet at
F
F
F
. Show that
∠
D
A
F
=
∠
B
A
F
\angle DAF = \angle BAF
∠
D
A
F
=
∠
B
A
F
.
2
1
Hide problems
rearranging the 12 numbers on a clock, 3 adjacent numbers with sum >=21
The
12
12
12
numbers on a clock face are rearranged. Show that we can still find three adjacent numbers whose sum is
21
21
21
or more.
1
1
Hide problems
1,2,4, 5,7,9,10,12,14,16,17, ... sequence 1 odd, 2 even, 3 odds, 4 even, ...
The sequence
1
,
2
,
4
,
5
,
7
,
9
,
10
,
12
,
14
,
16
,
17
,
.
.
.
1, 2, 4, 5, 7, 9 ,10, 12, 14, 16, 17, ...
1
,
2
,
4
,
5
,
7
,
9
,
10
,
12
,
14
,
16
,
17
,
...
is formed as follows. First we take one odd number, then two even numbers, then three odd numbers, then four even numbers, and so on. Find the number in the sequence which is closest to
1994
1994
1994
.