IKMC 2027 - Grades 9-10 - Self-study SET 1 - Thematic Batch 3

Question 1

What is the probability that, when a die is rolled, its upper face has six dots?

Editorial note: State the missing assumptions and explain the result if a fair standard die is assumed.

Question 2

What are the last two digits of $2^{2016}$?

1. 36
2. 72
3. 68
4. 04
5. 52

Question 3

Let a, b and c be pairwise distinct positive integers such that a, ab and abc are all perfect squares. What is the smallest possible value of a+b+c?

1. 9
2. 11
3. 14
4. 15
5. 17

Question 4

A rectangular cuboid with dimensions a × b × c has space diagonal $\sqrt{2016}$. The numbers a, b and c are natural numbers. What is the largest possible value of a+b+c?

1. 36
2. 72
3. 120
4. 144
5. 180

Question 5

Tom writes five distinct positive one-digit integers on the board. No two have sum 10. Which of the following numbers must be among those he writes?

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

Question 6

For a special scene in a fantasy film, the crew films for 8 hours a day over 16 days. The scene lasts 30 seconds. If no take is interrupted, how many times is the scene filmed?

1. 3840
2. 7680
3. 11520
4. 15360
5. 23040

Question 7

Alex has twelve cards labelled with the integers 1 through 12. Julia draws three cards at random. What is the probability that the three numbers drawn are consecutive integers?

1. 1/12
2. 1/24
3. 5/11
4. 1/6
5. None of the other four options is correct

Question 8

Julia lists all 24 permutations of A, B, C, D in alphabetical order, beginning with ABCD and ending with DCBA. Which is the 12th arrangement?

1. BACD
2. BADC
3. BCDA
4. BCAD
5. BDCA

Question 9

A large ball has mass 100 kg, of which 99% is water. Water is removed until water accounts for only 98% of the ball. What is its mass then, in kilograms?

1. 50 kg
2. 80 kg
3. 90 kg
4. 95 kg
5. 98 kg

Question 10

Alex assembles a 5 × 5 × 5 cube from unit cubes as shown. Each unit cube has either six black faces or six white faces. Any two unit cubes sharing a face have different colours. How many more black unit cubes than white unit cubes are there?

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

Question 11

How many natural numbers N have the property that exactly 10% of the first N positive integers are divisible by 9?

1. 0
2. 5
3. 8
4. 9
5. Infinitely many such values of N

Question 12

A 1 × 1 square is placed inside a 3 × 7 rectangle, with its sides parallel to the rectangle. Some vertices are joined as shown. What is the total area of the grey regions?

1. 4
2. 4.5
3. 5
4. 7
5. 7.5

Question 13

A 4 × 4 square grid is called anti-magic if its entries are the integers 1 to 16, each used once, and the four row sums, four column sums and two diagonal sums form ten consecutive positive integers. Some entries are shown. Which number must be placed in the starred cell?

1. 1
2. 2
3. 8
4. 15
5. 16

Question 14

What is the sum of the smallest two-digit positive divisor and the largest two-digit positive divisor of 2016?

1. 107
2. 108
3. 109
4. 110
5. Other

Question 15

Let A and B be odd positive integers whose sum is divisible by 5. What is the last digit of $A^3+B^3$?

1. 0
2. 1
3. 4
4. 5
5. 9

Question 16

Ten positive integers, not necessarily distinct, have product $10^{10}$. What is their largest possible sum?

1. $10^{11}$
2. $10^{11}$ + 10
3. $10^{10}$ + 9
4. 9 × $10^9$
5. $10^{10}$

Question 17

A fleet carries 505 tourists to historical sites around Hanoi. Large buses have capacity 36 and small buses capacity 27. Guides do not travel on the tourists’ buses. What is the smallest possible total number of empty seats?

1. 0
2. 5
3. 8
4. 9
5. 12

Question 18

A king has 100 boxes. The first contains 1 coin, the second 2, the third 4, the fourth 8, and so on. He wants to give exactly 2016 coins to his gods. How many boxes must he open so that their contents total exactly 2016 coins?

1. 4
2. 6
3. 8
4. 10
5. The king has several possible choices

Question 19

What is the greatest possible number of months in one year that have five Sundays?

1. 2
2. 3
3. 4
4. 5
5. 6

Question 20

A workshop plans that four workers will assemble 300 essential-oil diffusers in 9 hours. If two more workers join them, how many hours earlier will the group finish?

1. 1.5 hours
2. 3 hours
3. 4.5 hours
4. 5 hours
5. 6 hours