Universidad de Puerto Rico Recinto Universitario de Mayag¨ uez
Universidad de Puerto Rico
´ OLIMPIADAS MATEMATICAS DE PUERTO RICO 2011-2012 FIRST PHASE ANSWER FORM: INTERMEDIATE LEVEL (7th , 8th and 9th grades) Student’s information: Last name:
First name: 7th ,
Grade: Home phone: (
8th , )
9th
-
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
INTERMEDIATE LEVEL (7th , 8th and 9th grades)
1. If RP Q = 20◦ and RQU = 120◦ , what is the measure, in degrees, of SRT ? a) 60◦
S
T
b) 140◦ c) 80◦
R
d ) 100◦
Q
P
e) 120◦
U
2. Carlos has 6 stamps: two of 3 cents, two of 5 cents and two of 9 cents. Using at most 4 stamps one can obtain all amounts between 8 and 26 cents, except one. Which amount cannot be obtained? a) 26 cents b) 25 cents c) 18 cents d ) 20 cents e) 17 cents 3. Let ABCDE be a regular pentagon and DFGE a square. What is the measure of angle GAE? a) 9◦
B
b) 12◦
C
A
c) 6◦
D
E
d ) 10◦ e) 4◦
F
2
G
4. Andrea wrote numbers 1 thru 9 in the circles of the following figure so that the sum of the numbers along each line is divisible by 4. Which number or numbers must be in the center? a) Only 1 b) Only 5 c) Only 9 d ) 1, 5 and 9 e) 1 thru 9 5. The figure consists of squares. The areas of four of the squares are shown, in square units. Since x and y are also squares, their respective areas are: a) 16 and 25 b) 16 and 36
x
c) 25 and 36
y 1
1
d ) 25 and 64 9
4
e) 25 and 100 6. How many integers in the set {100, 101, 102, . . . , 999}, do not contain none of the digits 1, 2, 3 or 4? a) 150 b) 180 c) 240 d ) 300 e) None of the above. 7. Sof織覺a has 7 sticks with lengths 2, 4, 6, 7, 8, 9 and 10 centimeters. Out of all the rectangles that she can form using the 7 sticks, which one has the largest area? a) 140 cm2 b) 126 cm2 c) 118 cm2 d ) 102 cm2 e) 130 cm2
3
8. Louis has won a raffle and as a prize he will be given each day twice what he received the previous day during 8 days. If the first day he received 4 dollars, how much money will he receive in the end? a) 4 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 b) 2 + 22 + 23 + 24 + 25 + 26 + 27 + 28 c) 28 d ) 22 + 23 + 24 + 25 + 26 + 27 + 28 + 29 e) None of the above. 1 9. In a calculator, the A key transforms the number x on the screen to , and the B key multiplies x the number on the screen by 2. If the number 2 is on the screen and we hit 499 times the sequence AB, what number will be shown on the screen? a) 1 b) 2−498 c) 2−500 d ) 2499 e) 2500 10. PQRS is a rectangle with PQ = 4m, PS = 3m y QS = 5m. PT is the perpendicular from P to QS. What is the length, in meters, of PT? a) 2.1
P
Q
4m
b) 2.2 c) 2.3
5m
3m T
d ) 2.4 e) 2.5
S
R
11. Ver´onica, Ana and Carmen went fishing and caught less than 100 fishes. Ver´onica caught exactly three times the number of fishes caught by Ana and four times the number of fishes caught by Carmen. The largest number of fishes that Ver´onica could have caught was: a) 48 b) 50 c) 60 d ) 66 e) 72
4
12. How many 9 digits has the expansion of 10100 − 2011? a) 96 b) 97 c) 98 d ) 99 e) 100 13. Let ABCD be a rectangle whose length is double its width. If the width is 4 units and M and N are the midpoints of segments AD and BC, respectively, what is the area of polygon MOPD?
M
A
a) 4
D
b) 6
O
c) 8
P
d ) 16 e) 32
B
N
C
14. I went to a bookstore and I realized that the cost of each one of the books with soft cover was $5 while the cost of each one of the books with hardcover was $7. I spent exactly $86 in books of both kinds. What is the largest number of books I could have bought? a) 10 b) 14 c) 16 d ) 18 e) 20 15. We have a square with side 1m and we want to divide it with two segments of the same length x, as shown in the figure. If the three parts that are obtained must have the same area, what is the value of x? √
a)
13 3
b)
2 3 m √ 5 3 m
c) d) e)
m x
x
13 9 m √ 10 3 m
5
16. The sum of 20 integer numbers is 200. Out of these, what is the largest amount of numbers that can be greater than 20? a) 20 b) 19 c) 10 d) 9 e) 8 17. How many numbers from 1 to 50 have exactly 4 divisors? a) 11 b) 12 c) 13 d ) 15 e) None of the above. 18. The point (k, 17) lies on the line that joins points (1, 5) and (4, 11). The value of k is: a) 37 b) 14 c) 8 d) 6 e) 7 19. Consider all three-digit numbers, whose digits are all different, that can be formed with digits 0, 1, 2, 3 and 5. How many of these numbers are multiples of 6? a) 4 b) 7 c) 10 d ) 15 e) 20 20. A man conducting a survey goes to a house and is received by a woman: - Number of children? - the man asked. - Three daughters - she answered. - Ages? - the man inquired. - The product of the ages is 36 and the sum is equal to the number of the house, which you already know - she answered. The man leaves, but returns in a little while and tells the woman that there is missing information. The woman thinks for a moment and replies: - You are right. The oldest likes chocolate. What are the ages of the daughters? a) 2, 2 and 9 b) 3, 3 and 4
c) 1, 3 and 12 d ) 2, 3 and 6
6
e) 1, 3 and 9