MathDB
Problems
Contests
National and Regional Contests
Argentina Contests
Argentina National Olympiad
2005 Argentina National Olympiad
2005 Argentina National Olympiad
Part of
Argentina National Olympiad
Subcontests
(6)
6
1
Hide problems
sincere in 2k+1 people
Let
k
≥
1
k\geq 1
k
≥
1
be an integer. In a group of
2
k
+
1
2k+1
2
k
+
1
people, some are sincere (they always tell the truth) and the rest are unpredictable (sometimes they tell the truth and sometimes they lie). It is known that the unpredictable ones are at most
k
k
k
. Someone outside the group must determine who is sincere and who is unpredictable through a sequence of steps. In each step he chooses two people
A
A
A
and
B
B
B
from the group and asks
A
A
A
is
B
B
B
sincere? Show that after
3
k
3k
3
k
steps the stranger will be able to classify with certainty the
2
k
+
1
2k+1
2
k
+
1
people in the group. (Before asking each question, the answers to the previous questions are known.)Clarification: Each of the
2
k
+
1
2k+1
2
k
+
1
people in the group knows which ones are sincere and which ones are unpredictable.
4
1
Hide problems
n=sum of a perfect square plus a perfect cube
We will say that a positive integer is a winner if it can be written as the sum of a perfect square plus a perfect cube. For example,
33
33
33
is a winner because
33
=
5
2
+
2
3
33=5^2+2^3
33
=
5
2
+
2
3
. Gabriel chooses two positive integers, r and s, and Germán must find
2005
2005
2005
positive integers
n
n
n
such that for each
n
n
n
, the numbers
r
+
n
r+n
r
+
n
and
s
+
n
s+n
s
+
n
are winners. Prove that Germán can always achieve his goal.
3
1
Hide problems
a is irrational if 1/a=a-[a]
Let
a
a
a
be a real number such that
1
a
=
a
−
[
a
]
\frac{1}{a}=a-[a]
a
1
=
a
−
[
a
]
. Show that
a
a
a
is irrational. Clarification: The brackets indicate the integer part of the number they enclose.
2
1
Hide problems
two-letter alphabet on Babba Island
On Babba Island they use a two-letter alphabet,
a
a
a
and
b
b
b
, and every (finite) sequence of letters is a word. For each set
P
P
P
of six words of
4
4
4
letters each, we denote
N
P
N_P
N
P
to the set of all words that do not contain any of the words of
P
P
P
as a syllable (subword). Prove that if
N
P
N_P
N
P
is finite, then all its words are of length less than or equal to
10
10
10
, and find a set
P
P
P
such that
N
P
N_P
N
P
is finite and contains at least one word of length
10
10
10
.
1
1
Hide problems
a+b+c+d=502, a^2-b^2+c^2-d^2=50
Let
a
>
b
>
c
>
d
a>b>c>d
a
>
b
>
c
>
d
be positive integers satisfying
a
+
b
+
c
+
d
=
502
a+b+c+d=502
a
+
b
+
c
+
d
=
502
and
a
2
−
b
2
+
c
2
−
d
2
=
502
a^2-b^2+c^2-d^2=502
a
2
−
b
2
+
c
2
−
d
2
=
502
. Calculate how many possible values of
a
a
a
are there.
5
1
Hide problems
computational, ratio AP/CP wanted , AB/BC=2/3, tangent to circle
Let
A
M
AM
A
M
and
A
N
AN
A
N
be the lines tangent to a circle
Γ
\Gamma
Γ
drawn from a point
A
A
A
(
M
(M
(
M
and
N
N
N
belong to the circle). A line through
A
A
A
cuts
Γ
\Gamma
Γ
at
B
B
B
and
C
C
C
with
B
B
B
between
A
A
A
and
C
C
C
, and
A
B
B
C
=
2
3
\frac{AB}{BC} =\frac23
BC
A
B
=
3
2
. If
P
P
P
is the intersection point of
A
B
AB
A
B
and
M
N
MN
MN
, calculate
A
P
C
P
\frac{AP}{CP}
CP
A
P
.