IKMC 2027 - Grades 9-10 - Self-study SET 2 - Thematic Batch 3

Question 1

Tom observes that each time a ball hits the floor, it rebounds to one third of its previous maximum height. After its second contact with the floor, it rebounds to 9 cm. From what height was it initially dropped?

1. 1 cm
2. 3 cm
3. 9 cm
4. 27 cm
5. 81 cm

Question 2

Boxers A and B compete to decide a winner. A is twice as likely to win as B. Marry pays a stake of 10 monetary units to bet that A will win. Under fair odds, how much money should she receive back if her bet wins?

1. 5
2. 12.5
3. 15
4. 18
5. 20

Question 3

Alex and Julia have four equally sized pizzas. Two pizzas are cut into four slices each and the other two into five slices each. They each eat some whole slices. Alex eats 32.5% and Julia 46.25% of all their pizza. How many slices remain altogether?

1. 1
2. 2
3. 3
4. 4
5. 5

Question 4

How many positive integers n make $n^4-2n^3+2n^2-2n+1$ a prime number?

1. 0
2. 1
3. 2
4. 3
5. 4

Question 5

In the diagram, OCDE is a rectangle with E on OB, C on OA and D on the circle with centre O and radius OA. Given BE=4 cm and AC=2 cm, find OA in centimetres.

1. 6 cm
2. 7 cm
3. 8 cm
4. 9 cm
5. Another value

Question 6

Triangle ABC has ∠A=40°. Its incircle touches BC, CA and AB at D, E and F respectively. What is ∠EDF?

1. 40°
2. 50°
3. 60°
4. 70°
5. 80°

Question 7

Which of the following equals 16% of 25?

1. 25% of 16
2. 18% of 20
3. 15% of 26
4. 10% of 30
5. 36% of 12

Question 8

Calculate $(2016+2^5)\div2^6$.

1. 2⁴
2. 2⁵
3. 2⁶
4. 2¹⁰
5. The result is not a power of 2

Question 9

A circus sells adult tickets for 20 monetary units and child tickets for 10. Fewer than 400 tickets are sold on a certain day, for a total of 4650. What is the smallest possible number of adult tickets sold?

1. 233
2. 155
3. 66
4. 65
5. Cannot be determined

Question 10

An island has 2017 inhabitants, each either a knight who always tells the truth or a liar who always lies. They sit around a round table. Each says: “Of my two neighbours, one is a knight and the other a liar.” How many knights are there?

1. 0
2. 1
3. 670
4. 1344
5. 1343

Question 11

Robin has four identical cubes. He can assemble them into larger solids whose volume is four times that of one cube. How many different shapes can he make?

1. 4
2. 6
3. 8
4. 10
5. Cannot be counted

Question 12

Sue buys five packets of sweets and counts 102 sweets altogether. Which statement about the packets must be true?

1. Every packet contains at least 20 sweets.
2. At least two packets each contain at least 21 sweets.
3. At least one packet contains at least 21 sweets.
4. No packet contains more than 21 sweets.
5. At least one packet contains exactly 21 sweets.

Question 13

Six pairwise distinct positive integers have sum 23. How many different values can their product take?

1. 1
2. 2
3. 3
4. 4
5. 5

Question 14

A rectangular cuboid has dimensions 2 × 2 × 1. Two ants start at opposite ends of a space diagonal, as shown. Each travels one unit along one of the drawn lines. What is the probability that their final positions lie on a common face of the cuboid?

1. $\frac{2}{9}$
2. $\frac{1}{3}$
3. $\frac{4}{9}$
4. $\frac{5}{9}$
5. $\frac{2}{3}$

Question 15

Twelve guests leave their shoes on a rack before entering a party. There are six pairs of men’s shoes and six pairs of women’s shoes, six brown pairs and six black pairs. Some shoes are stolen. Among the stolen shoes are five men’s shoes and five brown shoes. What are the smallest and largest possible numbers of guests who have at least one shoe stolen?



Original choices (retained for reference):
A. 3 and 5
B. 3 and 6
C. 3 and 10
D. 6 and 6
E. 5 and 5

Question 16

Four workers begin a job intended to finish on schedule. After one day, two stop work because they are suspected of having SARS-CoV-2. The other two finish the job two days later than planned. All workers have equal, constant productivity. How many days was the job originally planned to take?

1. 3 days
2. 4 days
3. 5 days
4. 6 days
5. 7 days

Question 17

Thirty people stand in a circle facing its centre. Each turns randomly left or right. Those who are face to face with another person shout “Hello”. Ten people shout. Everyone then turns around and follows the same greeting rule. How many people shout the second time?

1. 10
2. 20
3. 8
4. 15
5. No unique answer can be determined

Question 18

The sequence is defined by $x_1=2$ and $x_{n+1}=x_n^{x_n}$ for every positive integer n. What is $x_4$?

1. $2^{2^3}$
2. $2^{2^4}$
3. $2^{2^{11}}$
4. $2^{2^{16}}$
5. $2^{2^{768}}$

Question 19

Alex says his elder brother was born in the twentieth century, exactly 15000 days ago. How old will his brother be on his next birthday?

1. 38 years
2. 39 years
3. 40 years
4. 41 years
5. 42 years

Question 20

Before learning negative integers, Alice invents notation for numbers below zero. Counting down, she writes …, 3, 2, 1, 0, 00, 000, 0000, … . In her notation, what is 000 + 0000?

1. 1
2. 0000
3. 00000
4. 000000
5. 0000000