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

Question 1

Sam’s garden has two taps. The first fills a 5-litre container in 20 seconds, while the second takes 30 seconds. At these rates, how many more litres per minute does the first tap deliver than the second?

Question illustration
  1. 3
  2. 5
  3. 6
  4. 8
  5. 10

Question 2

Cinema seats form a rectangular array. Marry sits in the fifth row from the front and the sixth from the back. Tony sits in the tenth column from the left and the eleventh from the right. How many seats are there?

Question illustration
  1. 200
  2. 210
  3. 220
  4. 231
  5. None of these

Question 3

Sam must score at least 28 points in total on three tests to graduate from an extracurricular course. Each test score is a whole number from 0 to 10. He meets the requirement. How many different ordered triples of test scores are possible?

Question illustration
  1. 4
  2. 7
  3. 8
  4. 9
  5. 10

Question 4

When all positive integers from 1 through 200 are listed, how many times does the digit 2 occur?

Question illustration
  1. 22
  2. 23
  3. 31
  4. 40
  5. 41

Question 5

Some students attend an art-club meeting. After five girls leave, the boys number twice the remaining girls. Then 15 boys leave, and the girls number three times the remaining boys. How many girls attended initially?

Question illustration
  1. 15
  2. 17
  3. 16
  4. 18
  5. 14

Question 6

A Rubik’s cube is turned as shown. Which statement is correct?

Question illustration
  1. Both volume and total surface area increase.
  2. Volume increases and total surface area is unchanged.
  3. Volume is unchanged and total surface area increases.
  4. Volume is unchanged and total surface area decreases.
  5. The right-hand solid has smaller volume than the left-hand solid.

Question 7

For which listed value of n is $(2n)^{n^2}=2^{256}$?

Question illustration
  1. 8
  2. 32
  3. 4
  4. 64
  5. 16

Question 8

Tony performs the addition shown and then replaces some digits by P, Q, R, S. What is the sum of the four replaced digits?

Question illustration
  1. 14
  2. 15
  3. 16
  4. 24
  5. Other

Question 9

Tom watched the seven games in the quarterfinals, semifinals and final of an elimination noughts-and-crosses tournament. He reports six results: Bella beat Anna; Gina beat Celine; Celine beat Donna; Celine beat Bella; Gina beat Holly; Emma beat Farah. What was the missing result?

Question illustration
  1. Gina beat Bella
  2. Celine beat Ann
  3. Emma beat Celine
  4. Bella beat Holly
  5. Gina beat Emma

Question 10

Maria starts weaving on January 1 and makes 25 metres of cloth each day. At the end of each day she gives her sister one tenth of all the cloth she then has. On which date does the amount she keeps first exceed 100 metres?

Question illustration
  1. January 5
  2. January 6
  3. January 9
  4. January 10
  5. Cannot be determined

Question 11

Square ABCD has side 19 cm. G lies on CD. Outside ABCD, construct square CEFG with side 14 cm. What is the area of triangle AEG?

Question illustration
  1. 70 cm²
  2. 98 cm²
  3. 180 cm²
  4. 196 cm²
  5. 361 cm²

Question 12

Experts from the Czech Republic, Slovakia and Poland meet at a conference. There are eight Polish experts, and the Slovak group is largest. There are 13 women, half the number of men. Which statement must be true?

Question illustration
  1. More than one third of the experts are women.
  2. The Polish group makes up more than 25% of the attendees.
  3. There are at least 16 Slovak experts.
  4. There are at most 14 Czech experts.
  5. There are at most seven Polish women.

Question 13

Positive integers a, b, c satisfy $a\le b\le c$ and $(a^b)^c=512$. What is the greatest possible b?

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

Question 14

How many two-digit numbers equal twice the product of their digits?

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

Question 15

Three horses have pairwise different positive integer masses in kilograms, totaling 100 kg. What is the greatest possible mass of the lightest horse?

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

Question 16

Joy’s program lists every even three-digit number with distinct digits chosen from 1,2,3,4. What is the sum of all these numbers?

Question illustration
  1. 3006
  2. 3116
  3. 3106
  4. 3016
  5. 3160

Question 17

In the diagram, the smaller square has area 16 cm² and the grey triangle has area 1 cm². What is the area of the larger square?

Question illustration
  1. 17 cm²
  2. 18 cm²
  3. 19 cm²
  4. 20 cm²
  5. 21 cm²

Question 18

Two intersecting circles lie in an a×b rectangle, where a<b. Each circle is tangent to three sides of the rectangle, as shown. In general, what is the distance between their centres?

Question illustration
  1. (b − a)π
  2. 1/4(a + b)
  3. 1/8(b² − a²)
  4. b − a
  5. All four other options are incorrect.

Question 19

Nonnegative integers m, n, p, q satisfy $3^m+5^n+6^p+7^q=47$. What is the greatest possible value of m+n+p+q?

Question illustration
  1. 2
  2. 4
  3. 5
  4. 6
  5. 7

Question 20

Bob and Tom race on a 300 m track. Bob gives Tom a 200 m head start. They start together and run at constant speeds, and Bob never passes Tom before the finish. At most how many times Tom’s speed can Bob’s speed be?

Question illustration
  1. 1.25
  2. 1.5
  3. 2
  4. 3
  5. 5