Universidad de Puerto Rico Recinto Universitario de Mayag¨ uez

´ OLIMPIADAS MATEMATICAS DE PUERTO RICO 2012-2013 FIRST PHASE ANSWER FORM Student’s information: Last name:

First name:

6th 7th 8th

Home phone: (

)

Student’s e-mail:

9th 10th 11th -

Gender: F

M

Teacher’s e-mail:

School’s name: School’s city:

School: Private

a 1 2 3 4 5 6 7 8 9 10

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 may 8, 2012 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

´ Universidad de Puerto Rico OLIMPIADAS MATEMATICAS Universidad de Puerto Rico Recinto Universitario de DE PUERTO RICO Mayag¨uez Primera fase 2012-2013

6th, 7th and 8th grades 3-point problems 1. There are 5 cities in Wonderland. Any two cities are connected by only one road, either visible or invisible. On the map of Wonderland, there are only seven visible roads. Alicia has magical glasses: when she sees the map through these glasses she can only see the roads that are invisible otherwise. How many invisible roads can she see?

a. 9

b. 8

c. 7

d. 3

e. 2

2. Which three of the numbered puzzle pieces should you add to the picture to complete the square?

a. 1, 3, 4

b. 1, 3, 6

c. 2, 3, 5

d. 2, 3, 6

e. 2, 5, 6

3. Each of 9 paths in the park is 100 m long. Jessica wants to go from A to B without taking any path more than once. What is the length of the longest way she can choose? a. 400 m b. 600 m c. 700 m d. 800 m e. 900 m 1

4. JosÂ´e has 8 dice with the letters A, B, C and D, the same letter on all sides of each die. He builds a block with the 8 dice. Two adjacent dice have always different letters. What letter is on the die that cannot be seen on the picture?

a. A

b. B

c. C

d. D

e. Cannot be determined.

5. In a vinegar-wine-water mix there are vinegar and wine at a ratio of 1 to 2. Wine and water are at a ratio of 3 to 1. Which of the following statements is true? a. There is more vinegar than wine. b. There is more wine than vinegar and water together. c. There is more vinegar than wine and water together.

d. There is more water than vinegar and wine together. e. Vinegar is contained least.

6. The natural numbers are coloured red, blue or green: number 1 is red, 2 is blue, 3 is green, 4 is red, 5 is blue, 6 is green, and so on. What color can be the number of the sum of a red number and a blue number? a. Only green.

c. Only blue.

b. Red or blue.

d. Only red.

e. Impossible to say.

7. The perimeter of the figure below, built up of identical squares, is equal to 42 cm. What is the area of the figure?

a. 8 cm2

b. 9 cm2

c. 24 cm2

2

d. 72 cm2

e. 128 cm2

4-point problems 8. The figure shows a starry pentagon. What is the value of angle A? a. 35◦ b. 42◦ c. 51◦ d. 65◦ e. 109◦ 9. We arrange the twelve numbers from 1 to 12 in a circle such that any neighbouring numbers always differ by either 1 or 2. Which of the following numbers have to be neighbours? a. 3 and 4 b. 5 and 6 c. 6 and 7 d. 8 and 10 e. 9 and 10 10. The numbers of the three houses are formed by three digits, two digits and one digit: abc, bc and c. Knowing that their sum equals 912, find the value of b. a. 0

b. 2

c. 3

d. 4

e. 5

11. The numbers 2, 5, 7 and 12 are written on one side of four cards (one number on one card), and on the other side the words “divisible by 7”, “prime”, “odd”, “greater than 100” (each on one card). It is known that the number written on each card DOES NOT CORRESPOND TO the word or phrase on its other side. What number is written on the card with the phrase “greater than 100”? a. 2

b. 5

c. 7

d. 12

e. Cannot be determined.

12. A dragon has 5 heads. Every time a head is chopped off, five new heads grow. If we chop off six heads one by one, how many heads will the dragon have? a. 25

b. 28

c. 29

3

d. 30

e. 33

13. There were 12 children in a birthday party. The children were aged 6, 7, 8, 9 and 10 years. Four of them were 6 years old. In the group the most common age is 8 years old. What was the average age of the 12 children? a. 6

b. 6.5

c. 7

d. 7.5

e. 8

5-point problems 14. Barbara wants to complete the following diagram by inserting three numbers, one in each empty cell. She wants the sum of the first three numbers to be 100, the sum of the three in the middle to be 200 and the sum of the last three numbers to be 300. What number should Barbara insert in the center of the diagram?

a. 50

b. 60

c. 70

d. 80

e. 100

15. A regular octagon is folded in half exactly three times until a triangle is obtained.

Then the apex is cut off in a right angle as shown in the picture. If the paper is unfolded what will it look like?

a.

b.

c.

d.

e.

16. Three equilateral triangles of the same size are cut from the corners of a big equilateral triangle with sides of 6 cm. The three small triangles together have the same perimeter as the remaining grey hexagon. What is the length of the sides of the small triangles?

a. 1 cm

b. 1.2 cm

c. 1.25 cm

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d. 1.5 cm

e. 2 cm

17. Mrs Barreto grows tomatoes and lettuce. This year she has changed the rectangular tomatoes bed to a square by lengthening one of its sides by 3 metres. Consequently, the area of the lettuce bed became smaller by 15 m2 . What was the area of the tomatoes bed previously?

a. 5 m2

b. 9 m2

c. 10 m2

d. 15 m2

e. 18 m2

18. Both figures are formed from the same five pieces. The rectangle is 5 cm × 10 cm and the other parts are quarters of two different circles. The difference between their perimeters is:

a. 2.5 cm

b. 5 cm

c. 10 cm

d. 20 cm

e. 30 cm

19. Pedro wants to cut a rectangle of size 6 × 7 into squares with integer sides. What is the minimal number of squares he can get? a. 4

b. 5

c. 7

d. 8

e. 42

20. The side of a magical talking square is 8 cm. If he tells the truth, its side becomes 2 cm shorter. If he lies, its perimeter doubles. From the last four sentences, two were true and two were false, but we don’t know in which order. What is the maximum possible perimeter of the square after the four sentences? a. 28 cm

b. 80 cm

c. 88 cm

d. 112 cm

e. 120 cm

Responses must be sent electronically through the web page www.ompr.pr before may 8, 2012 or trough mail to the address below: Dr. Luis F. C´aceres-Duque Departamento de Ciencias Matem´aticas Call Box 9000 ¨ PR 00681-9000 Mayaguez,

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