MathDB
Problems
Contests
National and Regional Contests
Paraguay Contests
Paraguay Mathematical Olympiad
2007 Paraguay Mathematical Olympiad
2007 Paraguay Mathematical Olympiad
Part of
Paraguay Mathematical Olympiad
Subcontests
(5)
5
1
Hide problems
Paraguayan National Olympiad 2007, Level 3, Problem 5
Let
A
,
B
,
C
,
A, B, C,
A
,
B
,
C
,
be points in the plane, such that we can draw
3
3
3
equal circumferences in which the first one passes through
A
A
A
and
B
B
B
, the second one passes through
B
B
B
and
C
C
C
, the last one passes through
C
C
C
and
A
A
A
, and all
3
3
3
circumferences share a common point
P
P
P
. Show that the radius of each of these circumferences is equal to the circumradius of triangle
A
B
C
ABC
A
BC
, and that
P
P
P
is the orthocenter of triangle
A
B
C
ABC
A
BC
.
4
1
Hide problems
Paraguayan National Olympiad 2007, Level 3, Problem 4
Each number from the set
{
1
,
2
,
3
,
4
,
5
,
6
,
7
}
\{1, 2, 3, 4, 5, 6, 7\}
{
1
,
2
,
3
,
4
,
5
,
6
,
7
}
must be written in each circle of the diagram, so that the sum of any three aligned numbers is the same (e.g.,
A
+
D
+
E
=
D
+
C
+
B
A+D+E = D+C+B
A
+
D
+
E
=
D
+
C
+
B
). What number cannot be placed on the circle
E
E
E
?
3
1
Hide problems
Paraguayan National Olympiad 2007, Level 3, Problem 3
Let
A
B
C
D
ABCD
A
BC
D
be a square,
E
E
E
and
F
F
F
midpoints of
A
B
AB
A
B
and
A
D
AD
A
D
respectively, and
P
P
P
the intersection of
C
F
CF
CF
and
D
E
DE
D
E
. a) Show that
D
E
⊥
C
F
DE \perp CF
D
E
⊥
CF
. b) Determine the ratio
C
F
:
P
C
:
E
P
CF : PC : EP
CF
:
PC
:
EP
2
1
Hide problems
Paraguayan National Olympiad 2007, Level 3, Problem 2
Let
A
B
C
D
ABCD
A
BC
D
be a square, such that the length of its sides are integers. This square is divided in
89
89
89
smaller squares,
88
88
88
squares that have sides with length
1
1
1
, and
1
1
1
square that has sides with length
n
n
n
, where
n
n
n
is an integer larger than
1
1
1
. Find all possible lengths for the sides of
A
B
C
D
ABCD
A
BC
D
.
1
1
Hide problems
Paraguayan National Olympiad 2007, Level 3, Problem 1
A list with
2007
2007
2007
positive integers is written on a board, such that the arithmetic mean of all the numbers is
12
12
12
. Then, seven consecutive numbers are erased from the board. The arithmetic mean of the remaining numbers is
11.915
11.915
11.915
. The seven erased numbers have this property: the sixth number is half of the seventh, the fifth number is half of the sixth, and so on. Determine the
7
7
7
erased numbers.