math quiz to g10

2016 DIVISION FESTIVAL OF TALENTS

Category B:

MATH QUIZ (Team-Orals) (Judges’ Copy)

GRADE 10

15-second Questions [2 points each]

15.1. An arithmetic sequence has a common difference of 4. If its first term is 5, what is its nth term? 15.2. [Multiple choice] Mark tossed a fair coin ten times with each outcome always being a tail. Which of the following is true if he tosses the coin for the sixth time? A. The outcome will most likely be a head; B. The outcome will most likely be a tail; C. There is an equal chance that the outcome is a head as it is a tail. 15.3. What three arithmetic means are between -8 and 16? 1

4n + 1

C

-2, 4, 10 2

15.4. Two consecutive terms of a geometric sequence are 3 and 9. Find the term 1

that precedes 3 in this sequence.

𝟏 𝟐

15.5. Give the sum of the first six terms of the Fibonacci sequence whose first term is 1.

20

15.6. There are three arithmetic means between 18 and 62. Find the least of these three.

29

15.7. There will be three teams winning this contest. If your team won, in how many ways can you arrange all the members of the winning teams for picture-taking?

720

15.8. What is the degree of the polynomial expression, 𝑥 15 + 𝑥 3 𝑦 2 + 𝑥 4 + 𝑦 2 𝑥 6 ?

15

15.9. In the DFOT Math Quiz, the quizmaster positioned herself to a point inside the room corresponding to the coordinates (-4,3). If your team is placed at the coordinates (2, -7) what are the coordinates of the midpoint of the segment made by connecting you and the quizmaster? 2

15.10. If the sequence 1, , 𝑥 is harmonic, what is the value of 𝑥? 3

(-1, -2)

𝟏 𝟐

30-second Questions, [3 points each]

30.1. The second and fourth term of an arithmetic sequence are 𝒙 + 𝟐 and 𝟑𝒙 − 𝟐𝒚 + 𝟐. Find the sum of its first four terms. 30.2. In how many ways can 4 Math books, 2 English books, and 3 Science books be arranged across a shelf if books of the same subject need to be placed together?

1728

30.3. The sequence 2, 3, 5, 6, 7, 8, 10, … is an enumeration of the positive integers which are NOT perfect squares. What is the 150th term in this sequence?

𝟏𝟔𝟐

30.4. By how much does the sum of the first 15 positive odd integers exceed the sum of the first 10 positive even integers?

𝟏𝟏𝟓

2016 DIVISION FESTIVAL OF TALENTS - MATH QUIZ (TEAM-ORAL) GRADE

10

30.5. Six numbers from a list of nine integers are 4, 9, 3, 6, 7, and 3. What is the largest possible value of the median of all nine integers in this list?

7

60-second Questions, [5 points each]

60.1. If 𝑎 + 1 = 𝑏 − 2 = 𝑐 + 3 = 𝑑 − 4, arrange the numbers 𝑎, 𝑏, 𝑐, and 𝑑 from biggest to smallest.

d, a, b, c

60.2. The sum of the first ten terms of an arithmetic sequence is 160. The sum of the next ten terms of the sequence is 340. What is the first term of the sequence?

7.9

60.3. What is the largest positive integer 𝑘 such that 27! Is divisible by 2𝑘 ?

23

60.4. Given that 𝑥 + 2 and 𝑥 − 3 are factors of 𝑃(𝑥) = 𝑎𝑥 3 + 𝑎𝑥 2 + 𝑏𝑥 + 12, what is the remainder when 𝑃(𝑥) is divided by 𝑥 − 1?

𝟏𝟖

60.5. If all the permutations of the letters of the word SMART are arranged alphabetically, what is the rank of the word SMART

79(th)

Clincher Questions

C1. The ratio of the areas of two squares is 3:4. What is the ratio of the lengths of their corresponding diagonals?

√𝟑: 𝟐

C2. September 21, 2016 is a Wednesday. On what day of the week will 10 001 days from now fall?

Monday

C3. On the blackboard, 1 is initially written. Then each of ten students, one after another, erases the number he finds on the board, and write its double plus one. What number is erased by the tenth student?

𝟐𝟏𝟎 − 𝟏 or 1023

Do-or-Die Question

DOD1. In how many distinguishable ways can the letters of the word WINNER be arranged if consonants should be placed beside each other.

𝟑𝟔

2016 DIVISION FESTIVAL OF TALENTS - MATH QUIZ (TEAM-ORAL) GRADE

10

2016 DIVISION FESTIVAL OF TALENTS

Category B:

MATH QUIZ (Team-Orals) (Quizmaster’s Copy)

GRADE 10

15-second Questions [2 points each] 15.1. An arithmetic sequence has a common difference of 4. If its first term is 5, what is its nth term? 15.2. [Multiple choice] Mark tossed a fair coin ten times with each outcome always being a tail. Which of the following is true if he tosses the coin for the sixth time? A. The outcome will most likely be a head; B. The outcome will most likely be a tail; C. There is an equal chance that the outcome is a head as it is a tail. 15.3. What three arithmetic means are between -8 and 16? 1

2

15.4. Two consecutive terms of a geometric sequence are 3 and 9. Find the term 1

that precedes 3 in this sequence. 15.5. Give the sum of the first six terms of the Fibonacci sequence whose first term is 1. 15.6. There are three arithmetic means between 18 and 62. Find the least of these three. 15.7. There will be three teams winning this contest. If your team won, in how many ways can you arrange all the members of the winning teams for picture-taking? 15.8. What is the degree of the polynomial expression, 𝑥 15 + 𝑥 3 𝑦 2 + 𝑥 4 + 𝑦 2 𝑥 6? 15.9. In the DFOT Math Quiz, the quizmaster positioned herself to a point inside the room corresponding to the coordinates (-4,3). If your team is placed at the coordinates (2, -7) what are the coordinates of the midpoint of the segment made by connecting you and the quizmaster? 2

15.10. If the sequence 1, 3 , 𝑥 is harmonic, what is the value of 𝑥? 30-second Questions, [3 points each] 30.1. The second and fourth term of an arithmetic sequence are 𝒙 + 𝟐 and 𝟑𝒙 − 𝟐𝒚 + 𝟐. Find the sum of its first four terms. 30.2. In how many ways can 4 Math books, 2 English books, and 3 Science books be arranged across a shelf if books of the same subject need to be placed together?

2016 DIVISION FESTIVAL OF TALENTS - MATH QUIZ (TEAM-ORAL) GRADE

10

30.3. The sequence 2, 3, 5, 6, 7, 8, 10, … is an enumeration of the positive integers which are NOT perfect squares. What is the 150th term in this sequence? 30.4. By how much does the sum of the first 15 positive odd integers exceed the sum of the first 10 positive even integers? 30.5. Six numbers from a list of nine integers are 4, 9, 3, 6, 7, and 3. What is the largest possible value of the median of all nine integers in this list? 60-second Questions, [5 points each] 60.1. If 𝑎 + 1 = 𝑏 − 2 = 𝑐 + 3 = 𝑑 − 4, arrange the numbers 𝑎, 𝑏, 𝑐, and 𝑑 from biggest to smallest. 60.2. The sum of the first ten terms of an arithmetic sequence is 160. The sum of the next ten terms of the sequence is 340. What is the first term of the sequence? 60.3. What is the largest positive integer 𝑘 such that 27! Is divisible by 2𝑘 ? 60.4. Given that 𝑥 + 2 and 𝑥 − 3 are factors of 𝑃(𝑥) = 𝑎𝑥 3 + 𝑎𝑥 2 + 𝑏𝑥 + 12, what is the remainder when 𝑃(𝑥) is divided by 𝑥 − 1? 60.5. If all the permutations of the letters of the word SMART are arranged alphabetically, what is the rank of the word SMART. Clincher Questions C1.

The ratio of the areas of two squares is 3:4. What is the ratio of the lengths of their corresponding diagonals?

C2.

September 21, 2016 is a Wednesday. On what day of the week will 10 001 days fall from now?

C3.

On the blackboard, 1 is initially written. Then each of ten students, one after another, erases the number he finds on the board, and write its double plus one. What number is erased by the tenth student?

Do-or-Die Question DOD1. In how many distinguishable ways can the letters of the word WINNER be arranged if consonants should be placed beside each other.

2016 DIVISION FESTIVAL OF TALENTS - MATH QUIZ (TEAM-ORAL) GRADE

10