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Problems
Contests
National and Regional Contests
Argentina Contests
Argentina National Olympiad
2014 Argentine National Olympiad, Level 3
2014 Argentine National Olympiad, Level 3
Part of
Argentina National Olympiad
Subcontests
(6)
6.
1
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1000 consecutive integers among some sums
Determine whether there exists positive integers
a
1
<
a
2
<
⋅
⋅
⋅
<
a
k
a_{1}<a_{2}< \cdot \cdot \cdot <a_{k}
a
1
<
a
2
<
⋅
⋅
⋅
<
a
k
such that all sums
a
i
+
a
j
a_{i}+a_{j}
a
i
+
a
j
, where 1
≤
i
<
j
≤
k
\leq i < j \leq k
≤
i
<
j
≤
k
, are unique, and among those sums, there are
1000
1000
1000
consecutive integers.
5.
1
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A special number does not divide an expression
An integer
n
≥
3
n \geq 3
n
≥
3
is called special if it does not divide
(
n
−
1
)
!
(
1
+
1
2
+
⋅
⋅
⋅
+
1
n
−
1
)
\left ( n-1 \right )!\left ( 1+\frac{1}{2}+\cdot \cdot \cdot +\frac{1}{n-1} \right )
(
n
−
1
)
!
(
1
+
2
1
+
⋅
⋅
⋅
+
n
−
1
1
)
. Find all special numbers
n
n
n
such that
10
≤
n
≤
100
10 \leq n \leq 100
10
≤
n
≤
100
.
4.
1
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Two 50-term sums and irreducible fractions
Consider the following
50
50
50
-term sums:
S
=
1
1
⋅
2
+
1
3
⋅
4
+
.
.
.
+
1
99
⋅
100
S=\frac{1}{1\cdot 2}+\frac{1}{3\cdot 4}+...+\frac{1}{99\cdot 100}
S
=
1
⋅
2
1
+
3
⋅
4
1
+
...
+
99
⋅
100
1
,
T
=
1
51
⋅
100
+
1
52
⋅
99
+
.
.
.
+
1
99
⋅
52
+
1
100
⋅
51
T=\frac{1}{51\cdot 100}+\frac{1}{52\cdot 99}+...+\frac{1}{99\cdot 52}+\frac{1}{100\cdot 51}
T
=
51
⋅
100
1
+
52
⋅
99
1
+
...
+
99
⋅
52
1
+
100
⋅
51
1
.Express
S
T
\frac{S}{T}
T
S
as an irreducible fraction.
3.
1
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Two circumferences inside an angle
Two circumferences of radius
1
1
1
that do not intersect,
c
1
c_1
c
1
and
c
2
c_2
c
2
, are placed inside an angle whose vertex is
O
O
O
.
c
1
c_1
c
1
is tangent to one of the rays of the angle, while
c
2
c_2
c
2
is tangent to the other ray. One of the common internal tangents of
c
1
c_1
c
1
and
c
2
c_2
c
2
passes through
O
O
O
, and the other one intersects the rays of the angle at points
A
A
A
and
B
B
B
, with
A
O
=
B
O
AO=BO
A
O
=
BO
. Find the distance of point
A
A
A
to the line
O
B
OB
OB
.
2.
1
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1000 ones and iterative processes
Given several numbers, one of them,
a
a
a
, is chosen and replaced by the three numbers
a
3
,
a
3
,
a
3
\frac{a}{3}, \frac{a}{3}, \frac{a}{3}
3
a
,
3
a
,
3
a
. This process is repeated with the new set of numbers, and so on. Originally, there are
1000
1000
1000
ones, and we apply the process several times. A number
m
m
m
is called good if there are
m
m
m
or more numbers that are the same after each iteration, no matter how many or what operations are performed. Find the largest possible good number.
1.
1
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201 positive integers and averages
201
201
201
positive integers are written on a line, such that both the first one and the last one are equal to
19999
19999
19999
. Each one of the remaining numbers is less than the average of its neighbouring numbers, and the differences between each one of the remaining numbers and the average of its neighbouring numbers are all equal to a unique integer. Find the second-to-last term on the line.