MathDB
Problems
Contests
National and Regional Contests
Mexico Contests
Regional Olympiad of Mexico Center Zone
2007 Regional Olympiad of Mexico Center Zone
2007 Regional Olympiad of Mexico Center Zone
Part of
Regional Olympiad of Mexico Center Zone
Subcontests
(6)
6
1
Hide problems
exactly half of N tickets have the digit 1 on them
Certain tickets are numbered as follows:
1
,
2
,
3
,
…
,
N
1, 2, 3, \dots, N
1
,
2
,
3
,
…
,
N
. Exactly half of the tickets have the digit
1
1
1
on them. If
N
N
N
is a three-digit number, determine all possible values of
N
N
N
.
5
1
Hide problems
1/BC - 2/ DC = 1 / CA if <ACB = 2 < CAB
Consider a triangle
A
B
C
ABC
A
BC
with
∠
A
C
B
=
2
∠
C
A
B
\angle ACB = 2 \angle CAB
∠
A
CB
=
2∠
C
A
B
and
∠
A
B
C
>
9
0
∘
\angle ABC> 90 ^ \circ
∠
A
BC
>
9
0
∘
. Consider the perpendicular on
A
C
AC
A
C
that passes through
A
A
A
and intersects
B
C
BC
BC
at
D
D
D
, prove that
1
B
C
−
2
D
C
=
1
C
A
\frac {1} {BC} - \frac {2} {DC} = \frac {1} {CA}
BC
1
−
D
C
2
=
C
A
1
4
1
Hide problems
power of 2 by reordering digits another power of 2
Is there a power of
2
2
2
that when written in the decimal system has all its digits different from zero and it is possible to reorder them to form another power of
2
2
2
?
3
1
Hide problems
2004 bicolor tiles on a circle
Let there be
2004
2004
2004
be bicolor tiles, white on one side and black on the other, placed in a circle. A move consists of choosing a black piece and turning over three pieces: the chosen one, the one on its left and the one on its right. If at the beginning there is only one black piece, will it be possible, repeating the movement described, to make all the pieces have the white face up?
2
1
Hide problems
concurrency wanted, circumcenter related
Consider the triangle
A
B
C
ABC
A
BC
with circumcenter
O
O
O
. Let
D
D
D
be the intersection of the angle bisector of
∠
A
\angle{A}
∠
A
with
B
C
BC
BC
. Show that
O
A
OA
O
A
, the perpendicular bisector of
A
D
AD
A
D
and the perpendicular to
B
C
BC
BC
passing through
D
D
D
are concurrent.
1
1
Hide problems
convicted person will be released at the top of a 100-step staircase.
A convicted person will be released when he reaches the top of a
100
100
100
-step staircase. But he cannot advance as he pleases, since he is obliged to go up one step each day of the odd-numbered months and go down one step each day of the even-numbered months. If it begins on January
1
1
1
,
2001
2001
2001
, what day will it be released?