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IKMC 2027 - Grades 7-8 - Self-study SET 2 - Thematic Batch 3

Question 1

Tom has 5 chickens and Mick has 7 ducks. They want to exchange 3 of each kind. How many different exchanges can they make?

Question illustration
  1. 100
  2. 240
  3. 350
  4. 45
  5. 105

Question 2

Five people A, B, C, D and E are represented by five points. Among any three people, there are two who know one another, represented by a red segment, and two who do not know one another, represented by a blue segment. Which diagram represents the following statement: “The whole group can sit around a circular table so that each person sits between all the people that person knows”?
  1. Question illustration
  2. Question illustration
  3. Question illustration
  4. Question illustration
  5. All the above answers are correct

Question 3

In the figure below, what is the measure of the smallest angle?

Question illustration
  1. $30^\circ$
  2. $45^\circ$
  3. $90^\circ$
  4. $70^\circ$
  5. Does not exist

Question 4

Which number below cannot be expressed as the sum of three consecutive integers?

Question illustration
  1. $12345^{678910}$
  2. $123456^{78910}$
  3. $1234567^{8910}$
  4. $12345678^{910}$
  5. $123456789^{10}$

Question 5

A school notice reports that 60% of the teachers cycle to school, amounting to 45 teachers. Only 12% of the teachers travel to school in their own cars. How many teachers is that?

Question illustration
  1. 4
  2. 6
  3. 9
  4. 10
  5. 12

Question 6

A pair of shoes costs 64.4 USD after a 20% discount and 15% tax are applied. What was its original price before the discount and tax?

Question illustration
  1. 80
  2. 75
  3. 74
  4. 72
  5. 70

Question 7

How many digits does $2^4\cdot5^4\cdot10^4$ have when written in decimal notation?

Question illustration
  1. 4
  2. 5
  3. 6
  4. 9
  5. None of the answers is correct

Question 8

A book has 100 pages, with exactly one sentence on each page. Page 1 says “This book contains exactly 1 false sentence.” Page 2 says “This book contains exactly 2 false sentences.” This continues to page 100, which says “This book contains exactly 100 false sentences.” On which page is the book’s only true sentence?

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  1. 1
  2. 2
  3. 50
  4. 99
  5. 100

Question 9

Nora notices that in 2016 her age is three times the number formed by the last two digits of her birth year. When she tells her older relative, she notices the same property holds for that relative’s age. What is the difference between their ages?

Editorial note: As in the source solution, use calendar-year ages and the intended birth centuries, with Nora born in the 2000s and her older relative in the 1900s.

Question illustration
  1. 45
  2. 65
  3. 75
  4. 80
  5. 85

Question 10

A cube has edge length 1 m. An ant wants to walk along its edges, starting at a vertex, without using any edge more than once. It may visit a vertex more than once. What is the greatest distance it can walk?

Question illustration
  1. 7 m
  2. 8 m
  3. 9 m
  4. 10 m
  5. 12 m

Question 11

Nine houses stand along one side of Deer Road. Each house contains at least two people, and no two neighbouring houses together contain more than ten people. What is the greatest total number of people in the nine houses?

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  1. 96
  2. 49
  3. 48
  4. 17
  5. 46

Question 12

$\frac{2018+2018+2018}{2018+2018+2018+2018}=$

Question illustration
  1. $\frac1{2018}$
  2. 1
  3. 0.2018
  4. $\frac34$
  5. 2018

Question 13

Points $A,B,C,D$ lie on the sides of square $KLMN$ as shown. Point $O$ is its centre. Given $OA\perp OB$, $OC\perp OD$ and $LM=2$, what is the grey area?

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  1. 1
  2. 2
  3. 2.5
  4. 2.25
  5. Cannot be determined

Question 14

In the figure, $\angle A_1CA_2=7^\circ$. Segments $CA_1,A_1A_2,A_2A_3,\ldots$ all have equal lengths. What is the maximum possible number of such segments?

Original printed choices: A) 10; B) 11; C) 12; D) 13; E) Infinitely many

Editorial task: Resolve the printed question, explain any source inconsistency, and distinguish the literal result from a corrected or intended interpretation.

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Question 15

At the Winnie-the-Pool honey shop, jars now cost 60% more than before, while honey costs 60% less than before. The current cost of the jar equals the current cost of its honey. How has the combined cost of jar and honey changed?

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  1. Unchanged
  2. Decreases by 30%
  3. Increases by 30%
  4. Decreases by 36%
  5. Decreases by 20%

Question 16

A rectangular portion is cut away from a rectangular prism, as shown. What fraction of the surface area is lost?

Question illustration
  1. Less than 12.5%
  2. 12.5%
  3. Between 12.5% and 25%
  4. 25%
  5. More than 25%

Question 17

Equilateral triangle $ACD$ is rotated counterclockwise about $A$ until it coincides with triangle $ABC$. Through what angle has it rotated?

Question illustration
  1. $60^\circ$
  2. $120^\circ$
  3. $180^\circ$
  4. $240^\circ$
  5. $300^\circ$

Question 18

The figure consists of unit cubes. How many more unit cubes must be added to obtain a large cube?

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  1. 49
  2. 60
  3. 65
  4. 110
  5. 125

Question 19

Right triangle ABC has sides AB = 6 cm, AC = 8 cm and BC = 10 cm. Point I lies inside the triangle and is equally distant from all three sides. Find its distance from AB.

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

Question 20

Points $M$ and $N$ lie on sides $AD$ and $DC$ of square $ABCD$. Segments $BN,BM,AN,CM$ divide it into eight regions of areas $S_1,S_2,\ldots,S_8$ as shown. Which sum below equals $S_8$?

Question illustration
  1. $S_2+S_4+S_6$
  2. $S_1+S_3+S_5+S_7$
  3. $S_1+S_4+S_7$
  4. $S_2+S_5+S_7$
  5. $S_3+S_4+S_5$