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Problems
Contests
National and Regional Contests
Argentina Contests
Argentina National Olympiad
2017 Argentina National Olympiad
2017 Argentina National Olympiad
Part of
Argentina National Olympiad
Subcontests
(6)
4
1
Hide problems
D_2(n)=999D_3(n), number of divisors of n that are perfect squares / cubes
For a positive integer
n
n
n
we denote
D
2
(
n
)
D_2(n)
D
2
(
n
)
to the number of divisors of
n
n
n
which are perfect squares and
D
3
(
n
)
D_3(n)
D
3
(
n
)
to the number of divisors of
n
n
n
which are perfect cubes. Prove that there exists such that
D
2
(
n
)
=
999
D
3
(
n
)
.
D_2(n)=999D_3(n).
D
2
(
n
)
=
999
D
3
(
n
)
.
Note. The perfect squares are
1
2
,
2
2
,
3
2
,
4
2
,
…
1^2,2^2,3^2,4^2,…
1
2
,
2
2
,
3
2
,
4
2
,
…
, the perfect cubes are
1
3
,
2
3
,
3
3
,
4
3
,
…
1^3,2^3,3^3,4^3,…
1
3
,
2
3
,
3
3
,
4
3
,
…
.
2
1
Hide problems
one of the numbers k and 100-k is representable
In a row there are
51
51
51
written positive integers. Their sum is
100
100
100
. An integer is representable if it can be expressed as the sum of several consecutive numbers in a row of
51
51
51
integers. Show that for every
k
k
k
, with
1
≤
k
≤
100
1\le k \le 100
1
≤
k
≤
100
, one of the numbers
k
k
k
and
100
−
k
100-k
100
−
k
is representable.
1
1
Hide problems
2 player game with 13 positive integers
Nico picks
13
13
13
pairwise distinct
3
−
3-
3
−
digit positive integers. Ian then selects several of these 13 numbers, the ones he wants, and using only once each selected number and some of the operations addition, subtraction, multiplication and division (
+
,
−
,
×
,
:
+,-,\times ,:
+
,
−
,
×
,
:
) must get an expression whose value is greater than
3
3
3
and less than
4
4
4
. If he succeeds, Ian wins; otherwise, Nico wins. Which of the two has a winning strategy?
5
1
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1810-d < sum of less than 100 positive integers < 1810+d
We will say that a list of positive integers is admissible if all its numbers are less than or equal to
100
100
100
and their sum is greater than
1810
1810
1810
. Find the smallest positive integer
d
d
d
such that each admissible list can be crossed out some numbers such that the sum of the numbers left uncrossed out is greater than or equal to
1810
−
d
1810-d
1810
−
d
and less than or equal to
1810
+
d
1810+d
1810
+
d
.
6
1
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all diagonals of a convex n-gon divide their angles into 80 parts
Draw all the diagonals of a convex polygon of
10
10
10
sides. They divide their angles into
80
80
80
parts. It is known that at least
59
59
59
of those parts are equal. Determine the largest number of distinct values among the
80
80
80
angles of division and how many times each of those values occurs.
3
1
Hide problems
computational, perimeter 100, AP/PM = 7/3, incenter related
Let
A
B
C
ABC
A
BC
be a triangle of perimeter
100
100
100
and
I
I
I
be the point of intersection of its bisectors. Let
M
M
M
be the midpoint of side
B
C
BC
BC
. The line parallel to
A
B
AB
A
B
drawn by
I
I
I
cuts the median
A
M
AM
A
M
at point
P
P
P
so that
A
P
P
M
=
7
3
\frac{AP}{PM} =\frac73
PM
A
P
=
3
7
. Find the length of side
A
B
AB
A
B
.