IKMC 2027 - Grades 7-8 - Self-study SET 13 - Thematic Batch 3

Question 1

Consider these statements about the picture.
A. For some pair X,Y, X is grey, Y is white, and X is higher than Y.
B. For some pair X,Y, X is a square, Y is a circle, and X is higher than Y.
C. For some pair X,Y, X is small, Y is large, and X is higher than Y.
D. For some pair X,Y, X is white, Y is grey, and X is higher than Y.
Which statement is true?

Original printed choices: A) A; B) B; C) C; D) D; E) None is true

Editorial task: Determine every true statement under the printed existential wording and explain the conflict with a single-choice question.

Question 2

Viola practises long jump. Her average last week was 3.8 m. Today she first jumps 3.99 m, raising the overall average to 3.81 m. How far must her next jump be to raise the overall average to 3.82 m?

1. 3.97
2. 4.00
3. 4.01
4. 4.03
5. 4.04

Question 3

Three pirates and three dragons sit around a round table as shown. Either dragons always tell the truth and pirates always lie, or the reverse; we do not know which. Each is asked, “Did one of your neighbours steal the gold?” Their answers are shown. Who stole the gold?

Editorial convention: Exactly one of the six stole the gold. In the original figure, CÓ means yes and KHÔNG means no.

1. A
2. B
3. C
4. D
5. E

Question 4

A 1 × 1 × 1 cube is cut from each corner of a 4 × 4 × 4 cube. The picture shows the cube after just one corner has been removed. How many faces does the final solid have?

Editorial convention: A face is a maximal connected planar boundary region; the notched original faces each count once.

1. 22
2. 24
3. 26
4. 28
5. 30

Question 5

ABCD is an isosceles trapezoid. X is the midpoint of AB and $\angle CXD=90^\circ$. Find its perimeter if BX = 1.

1. 5
2. 6
3. 7
4. 8
5. Cannot be determined

Question 6

The girls buy plates costing 155 pounds each, and the boys buy cups costing 77 pounds each. In total they buy 108 plates and cups for 12,996 pounds. By what percentage does the number of plates exceed the number of cups?

1. 20%
2. 25%
3. 30%
4. 45%
5. 75%

Question 7

Which fraction is the largest?

1. $\frac{44444}{55555}$
2. $\frac{5555}{6666}$
3. $\frac{666}{777}$
4. $\frac{77}{88}$
5. $\frac89$

Question 8

A tank that is 25% empty contains 25 tons more oil than when it is 25% full. What is the tank’s capacity?

Editorial interpretation: Capacity is expressed as the mass of oil when full; the original “ton/tons” labels are retained in Vietnamese choices.

1. 75 tons
2. 100 ton
3. 37.5 tons
4. 80 tons
5. 50 tons

Question 9

A steamer travels 16.62 km downstream in 45 minutes and 14.4 km upstream in 48 minutes. What is the current’s speed, if the boat’s speed in still water is constant?

Original printed choices: A) 40.16 km/h; B) 20.08 km/h; C) 22.08 km/h; D) 18.16 km/h; E) 22.16 km/h

Editorial task: Calculate the requested current speed and explain whether a printed choice answers the question.

Question 10

In the figure, what is the ratio of the area of triangle ADE to the area of triangle ABC?

1. $\frac94$
2. $\frac73$
3. $\frac45$
4. $\frac{15}{10}$
5. $\frac{26}{9}$

Question 11

Michael labels the eight edges of quadrilateral pyramid V.ABCD with 2,3,6,7,8,9,10,15 so that the sum of labels at each vertex is the same. Six labels are erased, leaving the two shown. What is the label L on VA?

1. L = 2
2. L = 3
3. L = 6
4. L = 7
5. Cannot be determined

Question 12

How many squares can be formed by joining four of the ten marked grid points?

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

Question 13

Mixture 1 contains liquids A and B in the ratio 1:4. Mixture 2 contains A and B in the ratio 1:1. Mix mixtures 1 and 2 in the ratio 5:1 to make mixture 3. How many litres of A are in 120 litres of mixture 3?

1. 15
2. 18
3. 20
4. 24
5. 30

Question 14

Philip decides to go to the gym twice each week, once on a weekday and once on a weekend day. How many different schedules are possible?

Editorial convention: Weekdays are Monday–Friday; weekend days are Saturday and Sunday.

1. 4
2. 6
3. 7
4. 10
5. 12

Question 15

Two trains approach one another on parallel tracks. The first is 300 m long and travels at 70 km/h; the second is 200 m long and travels at 80 km/h. How long do they take to pass completely, from their front ends meeting to their rear ends meeting?

1. 8 seconds
2. 9 seconds
3. 10 seconds
4. 12 seconds
5. 15 seconds

Question 16

Five martial arts and their competitors are listed below.
Karate: Aley, Peter, Mita.
Boxing: Peter, Martin, Ben.
Aikido: Ben, Jack, Fred, Donald, Martin.
Wing Chun: Ron, Jack, Nick.
Kickboxing: Aley, Troy, Ron.
How can the five events be arranged in rounds so that nobody competes in two events simultaneously? Each round contains one event or several simultaneous events.

1. Round 1: Karate, Aikido, Kickboxing; Round 2: Boxing, Wing Chun
2. Round 1: Karate, Aikido; Round 2: Boxing, Wing Chun; Round 3: Kickboxing
3. Round 1: Karate, Wing Chun; Round 2: Boxing, Kickboxing; Round 3: Aikido
4. Round 1: Kickboxing, Aikido; Round 2: Boxing; Round 3: Karate, Wing Chun
5. All of B, C and D

Question 17

A semicircle is described in the source as having radius AB. Draw the zigzag inside it so that its segments make the marked angle $\alpha$ with AB, as shown. Find $\alpha$.

Editorial correction: Although the Vietnamese text twice says “radius AB”, the figure makes AB the diameter. All marked acute angles are taken equal to α, as intended by the diagram.

1. $60^\circ$
2. $72^\circ$
3. $75^\circ$
4. $80^\circ$
5. Another answer

Question 18

Find angle ABC, where A,B,C are the midpoints of the cube edges shown.

1. 75°
2. 90°
3. 105°
4. 120°
5. 135°

Question 19

ABC is equilateral. Choose D so that B is the midpoint of AD. Choose E in the plane with DE = AB and with CE as large as possible. What is $\angle BED$?

1. $45^\circ$
2. $30^\circ$
3. $20^\circ$
4. $15^\circ$
5. $10^\circ$

Question 20

Let a,b be distinct real numbers. What is the solution x of $\frac{x-a}{x-b}=\frac{x-b}{x-a}$?

1. $\frac{a-b}{2}$
2. $\frac{a^2+b^2}{a+b}$
3. $\frac{a^2+b^2}{2(a+b)}$
4. $a+b$
5. $\frac{a+b}{2}$