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Problems
Contests
National and Regional Contests
Paraguay Contests
Paraguay Mathematical Olympiad
2005 Paraguay Mathematical Olympiad
2005 Paraguay Mathematical Olympiad
Part of
Paraguay Mathematical Olympiad
Subcontests
(5)
5
1
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MX=MY wanted, concurrent chords - OMAPA Paraguay 2005.5
Given a chord
P
Q
PQ
PQ
of a circle and
M
M
M
the midpoint of the chord, let
A
B
AB
A
B
and
C
D
CD
C
D
be two chords that pass through
M
M
M
.
A
C
AC
A
C
and
B
D
BD
B
D
are drawn until
P
Q
PQ
PQ
is intersected at points
X
X
X
and
Y
Y
Y
respectively. Show that
X
X
X
and
Y
Y
Y
are equidistant from
M
M
M
.
4
1
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t=(8a+ 1)/b, b<7, a,b,t pos. integers - OMAPA Paraguay 2005.4
In the expression
t
=
8
a
+
1
b
t=\frac{8a+ 1}{b}
t
=
b
8
a
+
1
where
a
,
b
,
t
a, b, t
a
,
b
,
t
are positive integers, where
b
<
7
b <7
b
<
7
. Determine the values of
a
a
a
and
b
b
b
that allow to obtain
t
t
t
under the established conditions.
3
1
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46150 = sum of all 3digits palindromes - 8 ones - OMAPA Paraguay 2005.3
The complete list of the three-digit palindrome numbers is written in ascending order:
101
,
111
,
121
,
131
,
.
.
.
,
979
,
989
,
999.
101, 111, 121, 131,... , 979, 989, 999.
101
,
111
,
121
,
131
,
...
,
979
,
989
,
999.
Then eight consecutive palindrome numbers are eliminated and the numbers that remain in the list are added, obtaining
46.150
46.150
46.150
. Determine the eight erased palindrome numbers .
2
1
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5100= no of faces x no of edgrs of pyramid - OMAPA Paraguay 2005.2
If you multiply the number of faces that a pyramid has with the number of edges of the pyramid, you get
5.100
5.100
5.100
. Determine the number of faces of the pyramid.
1
1
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3 3-digit numbers with sum of digits 17 - OMAPA Paraguay 2005.1
With the digits
1
,
2
,
3
,
.
.
.
.
.
.
,
9
1, 2, 3,. . . . . . , 9
1
,
2
,
3
,
......
,
9
three-digit numbers are written such that the sum of the three digits is
17
17
17
. How many numbers can be written?