Universidad de Puerto Rico Recinto Universitario de Mayag¨ uez
Universidad de Puerto Rico
´ OLIMPIADAS MATEMATICAS DE PUERTO RICO 2011-2012 FIRST PHASE ANSWER FORM: ELEMENTARY LEVEL (4th , 5th and 6th grades) Student’s information: Last name:
First name: 4th ,
Grade: Home phone: (
5th , )
6th
-
Gender:
F
M
Date of birth (dd/mm/yyyy):
Student’s e-mail:
Teacher’s e-mail:
School’s name: School’s city:
a 1 2 3 4 5 6 7 8 9 10
School:
Private
Instructions: Mark your answers with an x b c d e a b c 11 12 13 14 15 16 17 18 19 20
Public
d
e
Responses must be sent electronically through the web page www.ompr.pr before december 6th , 2011 or trough mail to the address below: Dr. Luis F. C´ aceres-Duque Departamento de Ciencias Matem´ aticas Call Box 9000 Mayag¨ uez, PR 00681-9000 1
Recinto Universitario de Mayag¨ uez Departamento de Ciencias Matem´ aticas
Olimpiadas Matem´aticas de Puerto Rico FIRST PHASE
Universidad de Puerto Rico
2011-2012
ELEMENTARY LEVEL (4th , 5th and 6th grades)
1. An elevator takes 6 seconds to go from the first floor to the third floor. How many seconds will it take to go from the first floor to the seventh floor? a) 12 b) 14 c) 18 d ) 20 e) 24 2. Arturo chooses a number. He then adds 1 to it. He subtracts 2 from the result. He multiplies the new number by 3 and then divides the number he got by 4. The final result is 6. What was the number Arturo chose? a) 6 b) 8 c) 9 d ) 10 e) 12 3. This year we celebrate my uncle’s 40th birthday. When I thought about his children, who have 5, 6 and 7 years, I asked myself: how many years should pass so that the sum of the ages of my uncle’s children is equal to my uncle’s age at that moment? a) 7 b) 11 c) 14 d ) 18 e) 21 4. What is the maximum number of different positive integers whose sum is 43? a) 5 b) 6 c) 7 d) 8 e) 9
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5. We have three cards with the numbers 8, 9 and 9. With them we can form, for example, the number 989 and also the number 986 (since an upside down 9 is a 6). How many different numbers can we form with the three cards? a) 6
9 8
b) 8 c) 9
9
d ) 10 e) 12 6. What star appears in position 2011 in the following sequence?
a) b) c) d) e) 7. Pedro wrote the numbers 1 thru 9 inside the circles of the figure, so that the sum of the three numbers along each line is divisible by 5. What number should be the one in the center? a) Only 2 b) Only 5 c) Only 8 d ) 2, 5 and 8 e) 1 thru 9 8. How many 3-digit numbers can be formed using the digits 0, 1, 2, 3 and 4 only once? a) 48 b) 60 c) 64 d ) 100 e) 125
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9. Lina uses 36 cubes of the same size to build a fence around a square space to keep animals. Afterwards, she decides not to play with animals and fills the space instead with cubes. How many cubes did Lina use altogether? a) 36 b) 49 c) 64 d ) 81 e) 100 10. If we make a list, in increasing order, of all four-digit numbers the sum of whose digits is 4, in what position can we find the number 2011? a) 7 b) 8 c) 9 d ) 10 e) 12 11. How many 3-digit positive integers are there, the product of whose digits is equal to 24? a) 12 b) 15 c) 18 d ) 21 e) 24 12. What is the area, in square centimeters, of the figure below if neighboring points in each row and in each column are 1 cm apart? a) 18.5 cm2 b) 19 cm2 c) 19.5 cm2 d ) 20 cm2 e) 20.5 cm2
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13. The following figure consists of ten little cubes which are glued together. Based on this same figure, the least amount of little cubes needed to construct a larger cube is: a) 17 b) 25 c) 54 d ) 64 e) None of the above. 14. In the following figure we can see that a circle and a quadrilateral can subdivide a plane into 7 parts. What is the maximum number of parts into which a circle and a quadrilateral can subdivide a plane? a) 7 3
1
b) 8
2
c) 9
5 7
d ) 10
6
4
e) 11 15. We have three points that define a triangle and want to draw a parallelogram adding a point for the fourth vertex. In how many ways can we choose the fourth point? a) 1 b) 2 c) 3 d) 4 e) It depends on the initial triangle. 16. Juliana will place three squares superimposing them over a square blanket whose side measures 90 cm. The side of the smaller ones is 40 cm long and the side of the larger one is 50 cm long. She wants that the superimposed parts are congruent squares (the shaded squares in the figure). How long should the side of the superimposed squares be? a) 10 cm b) 20 cm c) 50 cm d ) 90 cm e) 100 cm 5
17. How many three-digit numbers are there, the sum of whose digits is equal to the product of its digits? a) 1 b) 3 c) 6 d) 9 e) None of the above. 18. Triangle PQR is isosceles with PQ = QR. Segments PQ and RS are parallel. What is the measure of P RS? a) 52◦
Q
P
b) 76◦ c) 104◦ d ) 109◦
O
38
e) 142◦
S
R
19. There are 100 persons in a park, 50 of which are Italian, 60 are men and 90 are vegetarian. What is the least amount of persons who we can be sure are vegetarian Italian men? a) 0 b) 1 c) 10 d ) 40 e) 50 20. Fill the blank cells in the table with integer numbers so that the sum of any three neighboring cells in each row and in each column is always the same. Find the number that goes in the ?. a) 0
1
2
b) 1
? c) 2
1 d) 3
4 e) Impossible to fill.
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