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

Question 1

Five girls and some boys sit around a round table with 60 cookies. First, each girl gives one cookie from the table to every boy she knows. Then each boy gives one cookie from the table to every girl he does not know. All 60 cookies are used. How many boys are there?

1. 5
2. 6
3. 8
4. 10
5. 12

Question 2

A regular pentagon PQRST and a regular hexagon STUVWX share the side ST as shown. What is ∠QTX?

1. 126°
2. 120°
3. 112°
4. 108°
5. 102°

Question 3

Goldsmith David has a solid gold cube of side 5 cm and a solid gold cylinder of height 1 dm and base diameter 5 cm. He melts both to make gold spheres of diameter 2 cm. What is the greatest number of such spheres he can make?

Original options (preserved):
A. 7
B. 8
C. 9
D. 10
E. 11

Editorial note: Retain the stated diameters, find the correct maximum, and explain the discrepancy with the options. The rubric is editorial.

Question 4

A performing-arts group has two boys and eight girls. A teacher chooses one boy and four girls to prepare a performance for the opening ceremony of the International Kangaroo Mathematics Contest. How many choices are possible?

1. 48
2. 70
3. 96
4. 140
5. 280

Question 5

Alex cycles from home to school at an average speed of 10 km/h. At what average speed must he cycle home so that his average speed over both journeys is 15 km/h?

1. 25 km/h
2. 30 km/h
3. 40 km/h
4. 50 km/h
5. This is impossible

Question 6

A house has three rooms: two are empty and one contains treasure. Their signs say: Room 1: “This room contains treasure.” Room 2: “This room does not contain treasure.” Room 3: “There is certainly treasure in this house.” Exactly two signs are true. Which room contains the treasure?

1. Room 1
2. Room 2
3. Room 3
4. There is no treasure
5. Cannot be determined

Question 7

Last week, a school held a meeting for students whose birthdays are in September. What is the smallest number of them that must be selected to guarantee two sharing a birthday?

1. 2
2. 3
3. 30
4. 31
5. 32

Question 8

A school awards scholarships at five different levels. David says that S students receive scholarships, at least five students receive different levels, and the largest group that can be chosen with the same level has six students. What is the sum of the smallest and largest possible values of S?

1. 11
2. 30
3. 31
4. 40
5. 48

Question 9

Which of these three numbers is largest: 12% of 75, 75% of 12, or 30% of 30?

1. 12% of 75
2. 75% of 12
3. 30% of 30
4. 12% of 75 and 75% of 12
5. All three are equal

Question 10

Let $N=8^5\times5^8$. How many zero digits occur in the decimal representation of N?

1. 6
2. 7
3. 8
4. 9
5. Other

Question 11

Peter colours the cells of a 3×3 square grid, one colour per cell. The three cells in every row, every column and each main diagonal must all have different colours. What is the minimum number of colours?

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

Question 12

Alex walks to school and Julia cycles. Alex’s home is 1 km nearer to school, and Julia passes his home on her route. They leave home at the same time. Although Julia travels three times as fast as Alex, she does not pass him before school. What is the greatest possible distance from Julia’s home to school?

1. 3 km
2. 2 km
3. 1.75 km
4. 1.7 km
5. 1.5 km

Question 13

Alex chooses a positive integer, adds 5, and multiplies the result by 5. The final result has three digits. How many possibilities are there for the original number?

1. 179
2. 180
3. 181
4. 194
5. 195

Question 14

A sequence satisfies $a_0=2$, $a_1=6$ and $a_n^2=a_{n-1}a_{n+1}$ for every positive integer n. What is $a_3$?

1. 3
2. 18
3. 24
4. 48
5. 54

Question 15

A pen contains 24 chickens and some rabbits. Half the rabbits and two thirds of the chickens are moved elsewhere. The animals remaining have 28 legs in total. How many rabbits were in the pen initially?

1. 4
2. 6
3. 8
4. 9
5. 10

Question 16

Alex’s box contains four balls, exactly two red. Julia’s contains n balls, exactly four red. Each randomly draws two balls. Can their probabilities of drawing two red balls be equal?

1. Yes, when n=6
2. Yes, when n=7
3. Yes, when n=8
4. Yes, when n=9
5. This is impossible

Question 17

Albert writes an even number of distinct primes whose sum is odd. Bianca writes an odd number of distinct primes whose sum is even. Let A and B be their respective products. Which statement is true?

1. A is even and B is odd.
2. A is odd and B is even.
3. Both A and B are odd.
4. Both A and B are even.
5. No definite conclusion is possible.

Question 18

Maria colours every face of a regular octahedron so that faces sharing an edge have different colours. What is the minimum number of colours?

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

Question 19

What is the sum of the marked angles in the diagram?

1. 360°
2. 900°
3. 1080°
4. 1120°
5. 1440°

Question 20

Thirteen knights and liars sit around a round table. Knights always tell the truth and liars always lie. Each says, “Apart from me and my two immediate neighbours, everyone else is a liar.” How many knights are there?

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