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National and Regional Contests
Paraguay Contests
Paraguay Mathematical Olympiad
2015 Paraguay Mathematical Olympiad
2015 Paraguay Mathematical Olympiad
Part of
Paraguay Mathematical Olympiad
Subcontests
(5)
5
1
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2015 Paraguayan Mathematical Olympiad: Problem 5
In the figure, the rectangle is formed by
4
4
4
smaller equal rectangles. If we count the total number of rectangles in the figure we find
10
10
10
. How many rectangles in total will there be in a rectangle that is formed by
n
n
n
smaller equal rectangles?
4
1
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2015 Paraguayan Mathematical Olympiad: Problem 4
The sidelengths of a triangle are natural numbers multiples of
7
7
7
, smaller than
40
40
40
. How many triangles satisfy these conditions?
3
1
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2015 Paraguayan Mathematical Olympiad: Problem 3
A cube is divided into
8
8
8
smaller cubes of the same size, as shown in the figure. Then, each of these small cubes is divided again into
8
8
8
smaller cubes of the same size. This process is done
4
4
4
more times to each resulting cube. What is the ratio between the sum of the total areas of all the small cubes resulting from the last division and the total area of the initial cube?
2
1
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2015 Paraguayan Mathematical Olympiad: Problem 2
Consider all sums that add up to
2015
2015
2015
. In each sum, the addends are consecutive positive integers, and all sums have less than
10
10
10
addends. How many such sums are there?
1
1
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2015 Paraguayan Mathematical Olympiad: Problem 1
Alexa wrote the first
16
16
16
numbers of a sequence:
1
,
2
,
2
,
3
,
4
,
4
,
5
,
6
,
6
,
7
,
8
,
8
,
9
,
10
,
10
,
11
,
…
1, 2, 2, 3, 4, 4, 5, 6, 6, 7, 8, 8, 9, 10, 10, 11, …
1
,
2
,
2
,
3
,
4
,
4
,
5
,
6
,
6
,
7
,
8
,
8
,
9
,
10
,
10
,
11
,
…
Then she continued following the same pattern, until she had
2015
2015
2015
numbers in total. What was the last number she wrote?