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National and Regional Contests
Paraguay Contests
Paraguay Mathematical Olympiad
2019 Paraguay Mathematical Olympiad
2019 Paraguay Mathematical Olympiad
Part of
Paraguay Mathematical Olympiad
Subcontests
(5)
2
1
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pattern of figures by 2 sizes of equilaterals - OMAPA Paraguay 2019.2
Nair has puzzle pieces shaped like an equilateral triangle. She has pieces of two sizes: large and small. https://cdn.artofproblemsolving.com/attachments/a/1/aedfbfb2cb17bf816aa7daeb0d35f46a79b6e9.jpg Nair build triangular figures by following these rules:
∙
\bullet
∙
Figure
1
1
1
is made up of
4
4
4
small pieces, Figure
2
2
2
is made up of
2
2
2
large pieces and
8
8
8
small, Figure
3
3
3
by
6
6
6
large and
12
12
12
small, and so on.
∙
\bullet
∙
The central column must be made up exclusively of small parts.
∙
\bullet
∙
Outside the central column, only large pieces can be placed. https://cdn.artofproblemsolving.com/attachments/5/7/e7f6340de0e04d5b5979e72edd3f453f2ac8a5.jpg Following the pattern, how many pieces will Nair use to build Figure
20
20
20
?
5
1
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computational with incircle - OMAPA Paraguay 2019.5
A circle of radius
4
4
4
is inscribed in a triangle
A
B
C
ABC
A
BC
. We call
D
D
D
the touchpoint between the circle and side BC. Let
C
D
=
8
CD =8
C
D
=
8
,
D
B
=
10
DB= 10
D
B
=
10
. What is the length of the sides
A
B
AB
A
B
and
A
C
AC
A
C
?
4
1
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min n, when n^2 + 10 is divisible by n-5 - OMAPA Paraguay 2019.4
Find the largest positive integer
n
n
n
such that
n
2
+
10
n^2 + 10
n
2
+
10
is divisible by
n
−
5
n-5
n
−
5
.
3
1
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min of ABCD- DCBA , when those are digits - OMAPA Paraguay 2019.3
Let
A
B
C
D
‾
\overline{ABCD}
A
BC
D
be a
4
4
4
-digit number. What is the smallest possible positive value of
A
B
C
D
‾
−
D
C
B
A
‾
\overline{ABCD}- \overline{DCBA}
A
BC
D
−
D
CB
A
?
1
1
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mistakes when solving a trinomial - OMAPA Paraguay 2019.1
Elías and Juanca solve the same problem by posing a quadratic equation. Elijah is wrong when writing the independent term and gets as results of the problem
−
1
-1
−
1
and
−
3
-3
−
3
. Juanca is wrong only when writing the coefficient of the first degree term and gets as results of the problem
16
16
16
and
−
2
-2
−
2
. What are the correct results of the problem?