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Contests
National and Regional Contests
Argentina Contests
Argentina National Olympiad
2016 Argentina National Olympiad
2016 Argentina National Olympiad
Part of
Argentina National Olympiad
Subcontests
(6)
2
1
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S(m)=1+1/3 +...+1/m
For an integer
m
≥
3
m\ge 3
m
≥
3
, let
S
(
m
)
=
1
+
1
3
+
…
+
1
m
S(m)=1+\frac{1}{3}+…+\frac{1}{m}
S
(
m
)
=
1
+
3
1
+
…
+
m
1
(the fraction
1
2
\frac12
2
1
does not participate in addition and does participate in fractions
1
k
\frac{1}{k}
k
1
for integers from
3
3
3
until
m
m
m
). Let
n
≥
3
n\ge 3
n
≥
3
and
k
≥
3
k\ge 3
k
≥
3
. Compare the numbers
S
(
n
k
)
S(nk)
S
(
nk
)
and
S
(
n
)
+
S
(
k
)
S(n)+S(k)
S
(
n
)
+
S
(
k
)
.
6
1
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particles moving between ends of a segment
Let
A
B
AB
A
B
be a segment of length
1
1
1
. Several particles start moving simultaneously at constant speeds from
A
A
A
up to
B
B
B
. As soon as a particle reaches
B
B
B
, turns around and goes to
A
A
A
; when it reaches
A
A
A
, begins to move again towards
B
B
B
, and so on indefinitely. Find all rational numbers
r
>
1
r>1
r
>
1
such that there exists an instant
t
t
t
with the following property: For each
n
≥
1
n\ge 1
n
≥
1
, if
n
+
1
n+1
n
+
1
particles with constant speeds
1
,
r
,
r
2
,
…
,
r
n
1,r,r^2,…,r^n
1
,
r
,
r
2
,
…
,
r
n
move as described, at instant
t
t
t
, they all lie at the same interior point of segment
A
B
AB
A
B
.
5
1
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min p, smallest prime divisor of n when s=a+b=a^2+b^2=m/n
Let
a
a
a
and
b
b
b
be rational numbers such that
a
+
b
=
a
2
+
b
2
a+b=a^2+b^2
a
+
b
=
a
2
+
b
2
. Suppose the common value
s
=
a
+
b
=
a
2
+
b
2
s=a+b=a^2+b^2
s
=
a
+
b
=
a
2
+
b
2
is not an integer, and let's write it as an irreducible fraction:
s
=
m
n
s=\frac{m}{n}
s
=
n
m
. Let
p
p
p
be the smallest prime divisor of
n
n
n
. Find the minimum value of
p
p
p
.
3
1
Hide problems
2player game, marking a box on 101x101 grid
Agustín and Lucas, by turns, each time mark a box that has not yet been marked on a
101
×
101
101\times 101
101
×
101
grid board. Augustine starts the game. You cannot check a box that already has two checked boxes in its row or column. The one who can't make his move loses. Decide which of the two players has a winning strategy.
1
1
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arithmetic progression of 2016 terms whose product is perfect power
Find an arithmetic progression of
2016
2016
2016
natural numbers such that neither is a perfect power but its multiplication is a perfect power.Clarification: A perfect power is a number of the form
n
k
n^k
n
k
where
n
n
n
and
k
k
k
are both natural numbers greater than or equal to
2
2
2
.
4
1
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<ABD=29^o,<ADB=41^o,<ACB=82^o ,<ACD=58^o (2016 Argentina OMA L3 p4)
Find the angles of a convex quadrilateral
A
B
C
D
ABCD
A
BC
D
such that
∠
A
B
D
=
2
9
o
\angle ABD = 29^o
∠
A
B
D
=
2
9
o
,
∠
A
D
B
=
4
1
o
\angle ADB = 41^o
∠
A
D
B
=
4
1
o
,
∠
A
C
B
=
8
2
o
\angle ACB = 82^o
∠
A
CB
=
8
2
o
and
∠
A
C
D
=
5
8
o
\angle ACD = 58^o
∠
A
C
D
=
5
8
o