IKMC 2027 - Grades 7-8 - Self-study SET 4 - Thematic Batch 3

Question 1

A hiker travels from A to B by going 600 m uphill, then 720 m downhill, and then 1200 m on level ground. The trip from A to B takes 1 hour 40 minutes, and the return trip from B to A takes 1 hour 41 minutes. The speed on level ground is 30 m per minute. What are the uphill and downhill speeds, respectively?

Original printed choices: A) 24 m/min and 20 m/min; B) 48 m/min and 30 m/min; C) 48 m/min and 24 m/min; D) 24 m/min and 30 m/min; E) 30 m/min and 12 m/min

Editorial task: Solve the literal problem, identify the source discrepancy, and explain any intended correction.

Question 2

The shadow of Vietnam’s tallest building (as of 2017) is 16.8 m long. At the same time, an upright tree 8 m tall has a shadow 40 cm long. Find the height of the building.

1. 336m
2. 337m
3. 338m
4. 339m
5. 340m

Question 3

A group of 25 people consists of knights, who always tell the truth; liars, who always lie; and alternators, who alternate between truth and lies. When asked “Are you a knight?”, 17 say yes. Next, when asked “Are you an alternator?”, 12 say yes. Finally, when asked “Are you a liar?”, 8 say yes. How many knights are there?

1. 0
2. 4
3. 5
4. 13
5. 16

Question 4

A four-digit number has the form $\overline{GAGA}$, with different digits G and A. How many such numbers are there?

Editorial note: The source’s final use of “digits” is understood to mean four-digit numbers.

1. 16
2. 18
3. 20
4. 81
5. 90

Question 5

A plane through the midpoints of the three edges meeting at one vertex cuts a corner off a cube. The tetrahedron cut off has volume 1 cm³. What is the volume of the whole cube?

1. 8
2. 16
3. 24
4. 36
5. 48

Question 6

George is three times as old as Michael. Cameron is half as old as Michael. Which statement is true?

1. Cameron is five times as old as George.
2. Cameron is six times as old as George.
3. George is five times as old as Cameron.
4. George is six times as old as Cameron.
5. George and Cameron are the same age.

Question 7

Six squares are arranged along segment $AB$ as shown. Given $AB=24$ cm, find the length of the broken line $AA_1A_2\ldots A_{12}B$.

1. 48 cm
2. 72 cm
3. 96 cm
4. 56 cm
5. 106 cm

Question 8

The digits of a five-digit number sum to 44. What is their product?

1. $2^3\cdot3^8$
2. $2^3\cdot9^3$
3. $8\cdot4^9$
4. $8\cdot3^4$
5. None of the above

Question 9

All the vertices of a cube are cut off as shown. How many edges does the resulting solid have?

1. 26
2. 30
3. 36
4. 40
5. Another answer

Question 10

A cylindrical glass tube contains water and is tilted through 30° as shown. What percentage of the tube’s volume is occupied by water?

1. 11
2. 11.5
3. 12
4. 12.5
5. 13

Question 11

Grandmother’s garden is divided into four smaller plots as shown. The areas of three plots are given. What is the area of the remaining plot?

1. 6
2. 7
3. $\frac{20}{7}$
4. $\frac{28}{5}$
5. $\frac{35}{4}$

Question 12

Which expression gives the area of the shaded region in the figure?

1. $x^2$
2. $3x^2$
3. $6x^2$
4. $7x^2$
5. $9x^2$

Question 13

Fill the boxes in the diagram with the numbers 1 to 9, each used exactly once, so that every calculation is correct. Which number goes in the box marked X?

1. 9
2. 8
3. 7
4. 6
5. 5

Question 14

Bob folds a sheet of paper and then punches one hole through it. After unfolding it, he sees the sheet shown. Which diagram could show how he folded it?

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

Question 15

Jane has an 8-litre can, a 5-litre can and a 3-litre can. The 8-litre can is full of oil and the other two are empty. What is the minimum number of pours between the cans needed to obtain exactly 4 litres? She has no equipment other than these three cans.

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

Question 16

A shop has 200 pairs of red socks, 200 pairs of yellow socks and 200 pairs of black socks. For a sale, it places 600 socks in a bin: 300 right socks and 300 left socks. Find the largest N such that N matching pairs can always be selected. The socks within a pair must have the same colour; different pairs need not have the same colour.

Editorial note: A wearable pair consists of one left and one right sock of the same colour.

1. 90
2. 100
3. 110
4. 120
5. 130

Question 17

Quadrilateral $PQRS$ has $PQ=2006$, $QR=2008$, $RS=2007$ and $SP=2009$. Which interior angles must be smaller than $180^\circ$?

1. $P,Q,R$
2. $Q,R,S$
3. $P,Q,S$
4. $P,R,S$
5. $P,Q,R,S$

Question 18

Anna has the cube shown. She writes a different positive integer on each face so that numbers on adjacent faces differ by at least 3. What is the smallest possible sum of the six numbers?

1. 31
2. 33
3. 35
4. 37
5. 39

Question 19

The diagonals divide a quadrilateral into four smaller triangles. Three of their areas are shown. What is the area of the remaining triangle?

1. 7
2. 7.5
3. 8
4. 8.5
5. 9

Question 20

A box is formed as shown. What is its volume?

1. 25 cm³
2. 36 cm³
3. 30 cm³
4. 16 cm³
5. 24 cm³