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Contests
National and Regional Contests
Paraguay Contests
Paraguay Mathematical Olympiad
2008 Paraguay Mathematical Olympiad
2008 Paraguay Mathematical Olympiad
Part of
Paraguay Mathematical Olympiad
Subcontests
(5)
5
1
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Paraguayan National Olympiad 2008, Level 3, Problem 5
Let
m
,
n
,
p
m,n,p
m
,
n
,
p
be rational numbers such that
m
+
n
+
p
\sqrt{m} + \sqrt{n} + \sqrt{p}
m
+
n
+
p
is a rational number. Prove that
m
,
n
,
p
\sqrt{m}, \sqrt{n}, \sqrt{p}
m
,
n
,
p
are also rational numbers
4
1
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Paraguayan National Olympiad 2008, Level 3, Problem 4
Let
Γ
\Gamma
Γ
be a circumference and
A
A
A
a point outside it. Let
B
B
B
and
C
C
C
be points in
Γ
\Gamma
Γ
such that
A
B
AB
A
B
and
A
C
AC
A
C
are tangent to
Γ
\Gamma
Γ
. Let
P
P
P
be a point in
Γ
\Gamma
Γ
. Let
D
D
D
,
E
E
E
and
F
F
F
be points in
B
C
BC
BC
,
A
C
AC
A
C
and
A
B
AB
A
B
respectively, such that
P
D
⊥
B
C
PD \perp BC
P
D
⊥
BC
,
P
E
⊥
A
C
PE \perp AC
PE
⊥
A
C
, and
P
F
⊥
A
B
PF \perp AB
PF
⊥
A
B
. Show that
P
D
2
=
P
E
⋅
P
F
PD^2 = PE \cdot PF
P
D
2
=
PE
⋅
PF
3
1
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Paraguayan National Olympiad 2008, Level 3, Problem 3
Let
A
B
C
ABC
A
BC
be a triangle, where
A
B
=
A
C
AB = AC
A
B
=
A
C
and
B
C
=
12
BC = 12
BC
=
12
. Let
D
D
D
be the midpoint of
B
C
BC
BC
. Let
E
E
E
be a point in
A
C
AC
A
C
such that
D
E
⊥
A
C
DE \perp AC
D
E
⊥
A
C
. Let
F
F
F
be a point in
A
B
AB
A
B
such that
E
F
∥
B
C
EF \parallel BC
EF
∥
BC
. If
E
C
=
4
EC = 4
EC
=
4
, determine the length of
E
F
EF
EF
.
2
1
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Paraguayan National Olympiad 2008, Level 3, Problem 2
Find for which values of
n
n
n
, an integer larger than
1
1
1
but smaller than
100
100
100
, the following expression has its minimum value:
S
=
∣
n
−
1
∣
+
∣
n
−
2
∣
+
…
+
∣
n
−
100
∣
S = |n-1| + |n-2| + \ldots + |n-100|
S
=
∣
n
−
1∣
+
∣
n
−
2∣
+
…
+
∣
n
−
100∣
1
1
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Amount of numbers that have exclusively these prime factors
How many positive integers
n
<
500
n < 500
n
<
500
exist such that its prime factors are exclusively
2
2
2
,
7
7
7
,
11
11
11
, or a combination of these?