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IKMC 2027 - Grades 9-10 - Self-study SET 4 - Thematic Batch 3

Question 1

Eight envelopes contain 1,2,4,8,16,32,64 and 128 monetary units. Bob chooses some at random and gives all the rest to Alice. Bob’s total exceeds Alice’s by 31. How many envelopes did Bob take?

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

Question 2

Five otherwise identical hats carry letters: two K hats and three M hats. Alex, Maria and Julia each receive a random hat. They see the others’ letters but not their own. All are excellent reasoners. Alex says he cannot determine his letter; Maria then says the same; Julia then says she can determine hers. Which statement is correct?

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  1. Julia’s hat has K.
  2. Julia’s hat has M.
  3. Julia can determine her letter, but its value is not fixed.
  4. Julia certainly cannot determine her letter.
  5. None of the other four statements is certainly true.

Question 3

Positive numbers X, Y satisfy: 25% of X equals 20% of (X+Y). Which statement is true?

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  1. X = Y
  2. X = 4Y
  3. Y = 4X
  4. Y = 2X
  5. X = 2Y

Question 4

Let S={1,3,4,…,50}. How many subsets K={a, b, c} satisfy c=b+1=a+2 and have a+b+c a perfect square?

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

Question 5

In the diagram, ABCD is a square. E, G, F are the midpoints of AD, AC, AG respectively. What fraction of the large square is shaded?

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  1. 1/4
  2. 1/3
  3. 3/8
  4. 5/16
  5. 4/9

Question 6

How many five-digit positive integers have digit product exactly 1000?

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  1. 20
  2. 25
  3. 30
  4. 40
  5. 50

Question 7

In how many ways can two boys and four girls stand in a row if the two boys are not adjacent?

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  1. 8
  2. 10
  3. 12
  4. 240
  5. 480

Question 8

Jenny fills a 3×3 grid with numbers and computes its row and column sums. An ink spill hides much of the work. The diagram preserves two row sums and two column sums. Let A be the third-column sum and B the third-row sum. Find A−B.

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  1. 1
  2. 3
  3. 13
  4. 21
  5. Cannot be determined

Question 9

ABC is a right isosceles triangle with AB=AC=4 cm. Outside the triangle are a semicircle with diameter AC and a circle with diameter BC, as shown. What is the shaded area?

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  1. π
  2. 4
  3. 4√2
  4. 2π
  5. Cannot be determined

Question 10

The six rectangles shown are labelled by their areas in square centimetres. The indicated side of the first upper rectangle is 6 cm. What is the indicated width of the last rectangle in the lower row?

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  1. 4.5 cm
  2. 5 cm
  3. 5.2 cm
  4. 5.5 cm
  5. 6 cm

Question 11

Sam fills the blank boxes with the integers 2 through 6, each once, so that the sum in any two adjacent boxes is prime. How many fillings are possible?

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

Question 12

In the shown drawing style, a 1×1×1 cube needs at least 9 straight segments and a 2×2×2 cube at least 15. How many are needed to illustrate a 10×10×10 cube?

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  1. 30
  2. 57
  3. 60
  4. 63
  5. 72

Question 13

A two-digit number is divided by its digit sum. What is the greatest possible remainder?

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  1. 13
  2. 14
  3. 15
  4. 16
  5. 17

Question 14

A box contains four green apples, three yellow apples, two green pears and six yellow pears. Alex draws fruit one at a time without looking. What is the least number needed to guarantee an apple and a pear of the same colour?

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  1. 8
  2. 9
  3. 10
  4. 11
  5. 12

Question 15

After a storm, only four trees remain along a roadside. Consecutive gaps are 7.5 m,9 m and 21 m. What is the minimum number of additional trees needed to make an equally spaced row? Ignore trunk diameters.

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  1. 11
  2. 12
  3. 21
  4. 22
  5. 25

Question 16

Some aliens are green and the others purple. All green aliens live on Mars. Which statement must be true from these facts?

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  1. All aliens live on Mars.
  2. Only green aliens live on Mars.
  3. Some purple aliens live on Venus.
  4. All purple aliens live on Venus.
  5. No green aliens live on Venus.

Question 17

Numbers x, y satisfy x:y=3:2 and x+3y=27. What is the smaller number?

Question illustration
  1. 3
  2. 54/11
  3. 6
  4. 9
  5. 18

Question 18

Three circles of radius 1 cm are mutually tangent as shown. What is the shaded area in square centimetres?

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  1. √2/3
  2. √3 − π/2
  3. √π − 1
  4. π/6
  5. √2 − π/6

Question 19

The two upper scales balance. Objects of the same shape and colour have equal weights. Which collection can be placed on the right pan of the lower scale to guarantee balance?

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Question 20

A cinema offers “n% off a bill for n tickets” for every positive integer n≤N. The manager notes that the maximum saving on one order is equivalent to four free tickets. What is N?

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  1. 15
  2. 20
  3. 25
  4. 50
  5. 75