Universidad de Puerto Rico Recinto Universitario de Mayag¨ uez
Universidad de Puerto Rico
´ OLIMPIADAS MATEMATICAS DE PUERTO RICO 2012-2013 FIRST PHASE ANSWER FORM Student’s information: Last name:
First name:
Grade: 3rd 4th 5th
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
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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
Olimpiadas Matem´aticas de Puerto Rico
9th, 10th and 11th grades
´ Universidad de Puerto Rico OLIMPIADAS MATEMATICAS Universidad de Puerto Rico Recinto Universitario de DE PUERTO RICO Mayag¨uez Primera fase 2012-2013
9th, 10th and 11th grades Ejercicios de 3 puntos 1. If we add the seven digits of a number, we get 6. What is the product of these digits? a. 0
b. 6
c. 7
d. 1·2·3·4·5·6·7
e. 5
2. A four-digit number has a 3 in the hundreds place, and the sum of the other three digits is 3. How many such natural numbers are there? a. 2
b. 3
c. 4
d. 5
e. 6
3. M and N are the midpoints of the equal sides of an isoceles triangle. The area of the missing quadrilateral piece is:
a. 3
b. 4
c. 5
d. 6
e. 7
4. When Alicia wants to send a message to Ra´ul, she uses the following system, known to Ra´ul: A = 01, B = 02, C = 03 . . . Z = 26 After transferring each letter to a number, she calculates 2 × number + 9. The message is now transferred in a number sequence that Alice sends to Ra´ul. This morning Ra´ul has received 25, 19, 45, 38 and deciphers this sequence. What is the message Ra´ul received? a. HERO
c. HEAR
b. HELP
d. HERS
e. Alice has made a mistake.
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Olimpiadas Matem´aticas de Puerto Rico
9th, 10th and 11th grades
5. The square ABCE has side length 4 cm and the same area as the triangle ECD. What is the distance from the point D to the line g?
a. 8 cm b. (4 +
c. 12 cm √ 2) cm
e. Depends on the location of point D.
√ d. 10 2 cm
6. Three sportsmen Kan, Ga and Roo took part in the Marathon race. Before the beginning of the race four spectators from the audience had discussed the sportsmen’s chances for the victory. The first: “Either Kan or Ga will win”. The second: “If Ga is the second, Roo will win”. The third: “If Ga is the third, Kan will not win”. The fourth: “Either Ga or Roo will be the second”. After the race it turned out that all the four statements were true. In what order did the sportsmen finish? a. Kan, Ga, Roo
b. Kan, Roo, Ga
c. Roo, Ga, Kan
d. Gu, Roo, Kan
e. Ga, Kan, Roo
4-point problems 7. A cuboid is made of three pieces (see the drawing). Each of the pieces consists of 4 cubes and is of one color. What does the white piece look like?
a.
b.
c.
d. 2
e.
Olimpiadas Matem´aticas de Puerto Rico
9th, 10th and 11th grades
8. If Luis stands on the table and Pedro stands on the floor, Luis is 80 cm taller than Pedro. If Pedro stands on the same table and Luis is on the floor, Pedro is one meter taller than Luis. How high is the table? a. 20 cm
b. 80 cm
c. 90 cm
d. 100 cm
e. 120 cm
9. Carlos has 5 cubes. When he arranges them from the smallest to the biggest, two neighbouring cubes always differ in height by 2 cm. The biggest cube is as high as a tower built of the two smallest cubes. How high is the tower built out of all the 5 cubes? a. 6 cm
b. 14 cm
c. 22 cm
d. 44 cm
e. 50 cm
10. Mar´ıa has written 2012 = mm (mk − k), for some positive integer values of m and k. What is the value of k? a. 2
b. 3
c. 4
d. 9
e. 11
11. There are some three-digit numbers with the following property: if you remove the first digit, you get a perfect square, and if you remove the last digit, you also get a perfect square. What is the sum of all the numbers with this curious property? a. 1013
b. 1177
c. 1465
d. 1993
e. 2016
12. In Juan’s room there are four clocks, all of them are either slow or fast. The first clock is wrong by 2 minutes, the second clock by 3 minutes, the third by 4 minutes and the fourth by 5 minutes. Once Juan wanted to know the exact time by his clocks and he saw the following: 6 minutes to 3, 3 minutes to 3, 2 minutes past 3 and 3 minutes past 3. The exact time is: a. 2:57
b. 2:58
c. 2:59
d. 3:00
e. 3:01
13. Two sides of a quadrilateral have lengths 1 and 4. One of the diagonals, which is 2 in length, divides it into two isosceles triangles. Then the perimeter of the quadrilateral is equal to: a. 8
b. 9
c. 10
d. 11
e. 12
14. The tango is danced in pairs, one man and one woman. At a dance evening no more than 50 people are present. At one moment 34 of the men are dancing with 45 of the women. How many people were dancing at that given moment? a. 20
b. 24
c. 30
3
d. 32
e. 46
Olimpiadas Matem´aticas de Puerto Rico
9th, 10th and 11th grades
5-point problems 15. Efr´en wrote out all two-digit numbers and for each of the numbers he found the product of its digits. After that the boy found the sum of all the products obtained. What is the number the boy obtained? a. 45
b. 452
c. 453
d. 245
e. 345
16. Alexandra creates a Coqu´ı computer game. The picture represents the map of the game. At the start, the Coqu´ı is at the School (S). According to the rules of the game, from any place, except from the Home (H), the Coqu´ı can jump to any of two neighboring places. However, when it reaches H, the game is over. Find the number of ways the Coqu´ı can jump from S to H in exactly 13 jumps.
a. 12
b. 32
c. 64
d. 144
e. 1024
17. A jeweler has 12 pieces of chain of two links. She wants to make one big chain out of them. To do so she has to open some links (and close them afterwards). What is the smallest number of links she has to open?
a. 7
b. 8
c. 9
d. 10
e. 11
18. Let A, B, C, D, E, F, G, H the eight consecutive vertices of a convex octagon. Choose randomly a vertex among C, D, E, F, G, H and draw the segment connecting it with vertex A; then again, among the same six vertices, choose randomly a vertex and draw the segment connecting it with vertex B. What is the probability that the octagon is divided by these two segments in exactly three regions? a.
1 6
b.
1 4
c.
4 9
d.
4
5 18
e.
1 3
Olimpiadas Matem´aticas de Puerto Rico
9th, 10th and 11th grades
19. In a rectangle of length 6 cm an “equilateral triangle” of touching circles is drawn. What is the shortest distance between the two grey circles?
a.
√
2 cm
b.
√
3 cm
√ c. 2 3 − 2 cm
d.
π cm 2
e. 2 cm
20. A big triangle is divided by three segments into four triangles and three quadrilaterals. The sum of the perimeters of the quadrilaterals is equal to 25 cm. The sum of the perimeters of the four triangles is equal to 20 cm. The perimeter of the big triangle is equal to 19 cm. What is the sum of the lengths of the three segments?
a. 11 cm
b. 12 cm
c. 13 cm
d. 15 cm
e. 16 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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