Universidad de Puerto Rico Recinto Universitario de Mayag¨ uez

´ OLIMPIADAS MATEMATICAS DE PUERTO RICO 2011-2012 FIRST PHASE ANSWER FORM: HIGH SCHOOL LEVEL (10th , 11th and 12th grades) Student’s information: Last name:

First name: 10th ,

11th , )

12th

-

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

Olimpiada Matem´aticas de Puerto Rico FIRST PHASE

2011-2012

HIGH SCHOOL LEVEL (10th , 11th and 12th grades)

1. Mr. and Mrs. L´ opez have two children. When they travel in their family car, two persons sit in front and the other two sit in the rear. If either Mr. L´opez or Mrs. L´opez has to sit in driver’s seat, how many possible ways are there for them to sit? a) 4 b) 12 c) 16 d ) 24 e) 48 2. How many 2-digit numbers are the product of two different prime numbers? a) 14 b) 27 c) 37 d ) 31 e) 29 3. The numbers 1 thru 10000 were written on a board, and then all that are divisible by 5 and 11 were erased. What number of the resulting list can we find in position 2011? a) 2011 b) 2555 c) 6012 d ) 2764 e) None of the above. 4. In the diagram, x is equal to: a) 140 b) 122 xo

c) 80 d ) 90

o

140

e) 98

2

o

122

5. What is the value of the sum of the digits of the number 3216 路 12525 ? a) 5 b) 16 c) 25 d ) 32 e) 125 6. Luisa writes three 3-digit numbers on a board. She then finds the sum of the three numbers and gets 1575. Juan changes the units digit instead of the tenths digit in the three numbers and finds his sum. How many different results can Juan get? a) 3 b) 1 c) 6 d) 9 e) 11 7. Janette has two sisters, both younger than she. The product of the ages of the three of them is 396 and their sum is 23. How old is Janette? a) 6 b) 7 c) 11 d ) 12 e) 18 8. If x and y are positive real numbers, which of the following numbers is larger? a) xy b) x2 + y 2 c) (x + y)2 d ) x2 + y(x + y) e)

x3 + y 3 x+y

9. There are 200 candies in a bowl, 99 % of which are red. How many red candies do we have to take out so that 98 % of the remaining candies are red? a) 1 b) 2 c) 98 d ) 100 e) 101

3

10. In the diagram, triangles PQT, QTS and QRS are isosceles, and P QR is a right angle. Angles PQT and RQS measure 2x◦ , and angle QTS measures 5x◦ . The value of x is: a) 10

Q

P 2x

b) 12

o

2x o

c) 14

5x o

T

d ) 15 S

e) 20

R

11. What is the last digit of 6 × 82009 ? a) 0 b) 2 c) 4 d) 6 e) 8 12. PQRS is a rectangle. X is the midpoint of PQ, YQ is one third of QR, and RZ is a fourth of RS. What fraction of the area of PQRS is the quadrilateral XYZS? a) b) c) d) e)

1 2 7 12 2 3 3 4 3 5

S

Z

R

Y P

Q

X

13. A partially full bottle of water is shown (figure 1), with its dimensions in centimeters. The bottle is sealed and then it is turned upside down (figure 2). The height h, in centimeters, of the air in the upside-down bottle is: 1

a) 2 b) 2

1 3 7 25

2

c) 2 d) 2

1 5

4

3 h 2

5

3 e) 2 25

figure 1

4

figure 2

14. For real numbers x and y, we define x♠y = (x + y)(x − y). What is the value of 3♠(4♠5)? a) -72 b) -27 c) -24 d ) 24 e) 72 15. Mary is traveling eastward bound with a velocity of 3 miles per hour. Nicole is also traveling east, but with a velocity of 5 miles per hour. If Nicole is at this moment 1 mile west of Mary, how many minutes will it take Nicole to catch up with Mary? a) 30 b) 50 c) 60 d ) 90 e) 120 16. How many 3-digit even integer numbers have the property that their digits, reading them from left to right, are in strictly increasing order? a) 21 b) 34 c) 51 d ) 72 e) 150 17. An object on the plane moves from a point with integer coordinates to another with integer coordinates. In each step, the object can be moved one unit to the right, one unit to the left, one unit up, or one unit down. If the object starts at the origin and takes a path of ten steps, how many different points can be the ending point? a) 120 b) 121 c) 221 d ) 230 e) None of the above. 18. In a triangle whose sides have integer lengths, the length of one side is equal to three times the length of a second side, and the length of the third side is 15. What is the largest possible perimeter the triangle can have? a) 43 b) 44 c) 45

d ) 46 e) 47 5

19. Which of the following can not be the last digit of the sum of the squares of seven consecutive numbers? a) 3 b) 5 c) 6 d) 7 e) 8 20. Carlos has 8 tokens numbered 1 thru 8. He divides them in two piles such that each pile has at least two tokens and none of the numbers is equal to the average of any two numbers in the same pile. Which of the following terns of numbers can not be in the same pile? a) 1, 8, 2 b) 1, 2, 6 c) 4, 3, 7 d ) 8, 4, 3 e) 1, 2, 5

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Lastname: Firstname: Grade: 10 th , 11 th , 12 th Gender: F M OLIMPIADASMATEM´ATICASDEPUERTORICO2011-2012 School’scity: School: Private Publ...