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IKMC 2027 - Grades 9-10 - Self-study SET 7 - Thematic Batch 3

Question 1

Peter must deliver a parcel from his company to his manager’s home by a scheduled time. At 40 km/h he would arrive 40 minutes early; at 30 km/h he would arrive 30 minutes late. How far is the journey?

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  1. 120 km
  2. 140 km
  3. 160 km
  4. 180 km
  5. 200 km

Question 2

What are the last two digits of $6^{2020}$?

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  1. 36
  2. 16
  3. 96
  4. 76
  5. 56

Question 3

Alex has a solid made from four rectangular prisms as shown. Each small prism shares a face with another. Three indicated faces have areas 6 cm², 15 cm² and 35 cm². What is the area of face x?

Question illustration
  1. 12 cm²
  2. 14 cm²
  3. 15 cm²
  4. 26 cm²
  5. Cannot be determined

Question 4

Sam has fourteen cards numbered 1,2,…,14. He partitions them into groups whose sums are all equal. What is the largest possible number of groups?

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  1. 2
  2. 3
  3. 5
  4. 7
  5. 9

Question 5

In the diagram, what is the ratio of the grey circle’s area to the large circle’s area?

Question illustration
  1. 1/6
  2. 1/9
  3. 1/10
  4. 1/8
  5. None of the other four options is correct.

Question 6

Three consecutive positive integers have squares summing to 770. What is the largest integer?

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  1. 15
  2. 16
  3. 17
  4. 18
  5. 19

Question 7

How many positive even divisors does 2016 have?

Question illustration
  1. 10
  2. 25
  3. 30
  4. 36
  5. 49

Question 8

Circle C and square S have equal areas. Square S′ is inscribed in C, and circle C′ is inscribed in S. What is the ratio of the area of S′ to that of C′?

Question illustration
  1. 1 : 1
  2. 4 : π
  3. 8 : π²
  4. 1 : 2
  5. 2√2 : π

Question 9

A standard die has six faces. What is the fewest number of rolls needed to guarantee that some face has appeared at least three times?

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  1. 15
  2. 7
  3. 12
  4. 13
  5. 18

Question 10

How many positive integers A have exactly one four-digit number among A and A+20?

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  1. 0
  2. 19
  3. 20
  4. 39
  5. 40

Question 11

What is the greatest possible number of acute interior angles in a convex hexagon?

Question illustration
  1. 2
  2. 3
  3. 4
  4. 5
  5. 6

Question 12

Jenny fills a 3×3 grid so that the sums in all four 2×2 subgrids are equal. Three corner entries are shown. What number must be put in the remaining corner?

Question illustration
  1. 1
  2. 4
  3. 5
  4. 0
  5. 3

Question 13

An airline gives the same free checked-baggage allowance for every ticket and charges proportionally for excess weight. Mr and Mrs Brown check a combined 60 kg and pay 11 dollars extra. Mr Robin also checks 60 kg but pays 33 dollars extra. The Browns have twice Robin’s free allowance because they have two tickets. What is the free allowance per passenger, in kilograms?

Question illustration
  1. 11 kg
  2. 18 kg
  3. 20 kg
  4. 24 kg
  5. 39 kg

Question 14

A kiln’s temperature increases by 75°C per hour from an initial 25°C. After how many hours does it reach 850°C?

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  1. 5 hours
  2. 11 hours
  3. 10 hours
  4. 6 hours
  5. 8 hours

Question 15

A farm has more than 17 horses. Exactly two are black, at most twelve are brown, and the rest are grey. Which statement below must be true?

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  1. Brown horses outnumber the black and grey horses combined.
  2. There are exactly six black and grey horses combined.
  3. There are more grey horses than black horses.
  4. There are fewer black horses than either other colour.
  5. There are more brown horses than either other colour.

Question 16

Three kangaroos have pairwise distinct positive integer weights in kilograms, totalling 100 kg. Kanga is the lightest. What is Kanga’s largest possible weight?

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  1. 30
  2. 31
  3. 32
  4. 33
  5. 34

Question 17

Alice builds a large cube from n³ identical unit cubes, where n is an integer greater than 3. In general, how many unit cubes can be seen from outside the large cube?

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  1. 6 n² − 12n + 8
  2. 6 n² − 12n
  3. 6 n²
  4. 4 n²
  5. 6 n² − 8

Question 18

Julia has seven distinct stones and colours each blue or red. How many colourings have more red stones than blue stones?

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  1. 32
  2. 64
  3. 128
  4. 256
  5. 512

Question 19

Triangle ABC has area 40 cm². In the diagram, AD=DE=EB, AI=IH=HG=GC, and F is the midpoint of BC. What fraction of triangle ABC is shaded?

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  1. 3/8
  2. 2√2/15
  3. √2/2
  4. 7/16
  5. None of the other four options is correct.

Question 20

Peter rolls four dice. The first three all show six dots on top. What is the probability that the fourth roll also shows six?

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  1. 1/6
  2. 1/36
  3. 1/216
  4. 1/1296
  5. 1/21