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

Question 1

ABCD.EFGH is a cube of side 2 cm. Points P, Q and R are respectively the midpoints of edges AD, AG and BF. What is the area of triangle PQR?

Original printed choices: A) $\sqrt{3}$/2 cm²; B) 3$\sqrt{3}$ cm²; C) 3$\sqrt{3}$/2 cm²; D) 2$\sqrt{3}$ cm²; E) 1/$\sqrt{3}$ cm²

Editorial task: Solve the literal problem, identify any missing condition, and distinguish the intended interpretation.

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Question 2

Tom and Bob stand together on the shore of a circular lake. To reach the opposite point, Tom swims at constant speed v, while Bob must run one and a half laps of the shore. Which listed speed is the smallest that lets Bob arrive first?

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  1. 3 v
  2. 4.71 v
  3. 4.75 v
  4. 4.79 v
  5. 6.32 v

Question 3

Pete builds a large cube from small white cubes, then paints every outer face black. The number of small cubes with exactly one black face equals the number with every face white. How many small cubes did Pete use?

Original printed choices: A) 8; B) 9; C) 10; D) 11; E) 12

Editorial task: Solve the literal problem, identify any missing condition, and distinguish the intended interpretation.

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Question 4

Starting at P in the arrow's direction, walk along cube edges. At the end of each edge choose right or left, alternating these choices. After how many edges do you first return to P?

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  1. 1
  2. 4
  3. 6
  4. 9
  5. 12

Question 5

In the middle of the large cube, red small cubes form passages from top to bottom, front to back and side to side, as shown. What percentage of the small cubes are red?

Question illustration
  1. 25
  2. 30
  3. 40
  4. 50
  5. 75

Question 6

Tam chooses 35 distinct two-digit natural numbers. Find the greatest N for which it is always possible to find N chosen numbers having the same digit sum.

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  1. 3
  2. 5
  3. 10
  4. 14
  5. 15

Question 7

Three taps A, B and C have constant filling rates. Together they fill a tank in 1 hour; A and C fill it in 1.5 hours; B and C fill it in 2 hours. How many hours do A and B alone need? In every case the taps are opened simultaneously.

Question illustration
  1. 1
  2. 1.2
  3. 1.5
  4. 2
  5. 2.5

Question 8

Numbers a,b,c satisfy $\frac{a^3}{a^2+ab+b^2}+\frac{b^3}{b^2+bc+c^2}+\frac{c^3}{c^2+ca+a^2}=2014$. Find $\frac{a^3}{c^2+ac+c^2}+\frac{b^3}{a^2+ab+b^2}+\frac{c^3}{b^2+bc+c^2}$.

Original printed choices: A) $2014^3$; B) $2014^2$; C) 2014; D) 6042; E) Another value

Editorial task: Solve the literal problem, identify any missing condition, and distinguish the intended interpretation.

Question illustration

Question 9

The sum of ten consecutive integers is 35. Find their product.

Question illustration
  1. 0
  2. 10
  3. 125
  4. 150
  5. 201

Question 10

Given p+q=10, q+r=9 and r+s=11, find p+s.

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

Question 11

Kangaroo Day is the third Thursday of March. Today, 15 March 2018, is Kangaroo Day. On what nearest subsequent date will Kangaroo Day again fall on 15 March?

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  1. 15.03.2025
  2. 15.03.2027
  3. 15.03.2029
  4. 15.03.2031
  5. 15.03.2033

Question 12

Azmi, Burhan and Choo go shopping. Burhan spends 15% as much as Choo, and Azmi spends 60% more than Choo. Together they spend 55 RM. How much does Azmi spend?

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  1. 3 RM
  2. 30 RM
  3. 25 RM
  4. 26 RM
  5. 32 RM

Question 13

In triangle ABC, D lies on AC and E on BC, with DE parallel to AB. Given CE=4, DC=x−3, AD=3x−19 and EB=x−4, find x.

Question illustration
  1. 8
  2. 12
  3. 8 or 11
  4. $x\le2$
  5. $x\ge7$

Question 14

In triangle ABC, angle B is 20° and angle C is 40°. The angle bisector from A has length 2. Find BC−AB.

Question illustration
  1. 1
  2. 1.5
  3. 2
  4. 4
  5. Insufficient information

Question 15

Triangle ABC is right-angled at A, with AB=c, AX=p and XC=q as shown. Jenny and Vicky move at equal speeds in opposite directions, starting simultaneously at X and meeting at B. Express q in terms of p and c.

Editorial interpretation: The two routes follow the triangle boundary in opposite directions from X.

Question illustration
  1. $\frac p2+c$
  2. $\frac{pc}{2p+c}$
  3. $\sqrt{p^2+c^2}+\frac c2$
  4. $\frac{p+c}{2}$
  5. $c-p$

Question 16

Yadira has the squared sheet shown on the left. She wants to cut out pieces of the shape shown on the right. What is the greatest number of pieces she can obtain?

Question illustration
  1. 4
  2. 5
  3. 6
  4. 7
  5. 8

Question 17

Two upright palm trees stand on opposite riverbanks, 14 m apart. A fish lies on the line joining their bases. Birds on the two tops dive straight towards the fish simultaneously at equal speeds and arrive together. Their dive angles from the vertical are 30° and 60°. Find their initial separation, rounded to one decimal place.

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  1. 14.5 m
  2. 15.6 m
  3. 18 m
  4. 28 m
  5. 32.5 m

Question 18

A course has five tests with the same score scale. Let A be the average of the first two scores, B the average of A and the third score, C the average of B and the fourth score, and the final grade the average of C and the fifth score. What percentage of the final grade is contributed by the third test?

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  1. 10%
  2. 12.5%
  3. 20%
  4. 25%
  5. Insufficient information

Question 19

A class has 20 students, seated two at each desk. One third of the boys sit beside a girl, and one half of the girls sit beside a boy. How many boys are there?

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  1. 9
  2. 12
  3. 15
  4. 18
  5. 21

Question 20

The shown net consists of three squares of side 4 cm and two equilateral triangles. Find the volume of the solid formed by folding it.

Question illustration
  1. 16$\sqrt{3}$ cm³
  2. 32 cm³
  3. 64/3 cm³
  4. 32$\sqrt{3}$ cm³
  5. 64 cm³