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IKMC 2027 - Grades 7-8 - Self-study SET 15 - Thematic Batch 3

Question 1

Through the vertices A,B,C,D of parallelogram ABCD draw the perpendiculars to the same line, as shown. Given AA₁=4 cm, DD₁=5 cm and CC₁=7 cm, find BB₁.

Question illustration
  1. 9
  2. 11
  3. 12
  4. 16
  5. 21

Question 2

A sponge weighs 2 kg and is 90% water by mass. Some water evaporates, after which the sponge is 80% water. What is its new mass?

Question illustration
  1. 1.9 kg
  2. 1.8 kg
  3. 1.7 kg
  4. 1.1 kg
  5. 1 kg

Question 3

A tree grows according to this rule: each new shoot takes two months to develop, and from its third month it produces one new shoot every month. The diagram shows months 1–7, with 13 shoots in month 7. How many shoots are there in month 10?

Question illustration
  1. 55
  2. 56
  3. 57
  4. 58
  5. 59

Question 4

A shop reduces every price by 30%. A “favourite product” receives a further 20% reduction. If its original price is 1000 EUR, what is its price after both reductions?

Question illustration
  1. 700
  2. 500
  3. 560
  4. 600
  5. 400

Question 5

A kangaroo jumps between horizontally or vertically adjacent circles, never visiting any circle twice. How many routes take it from S to F in exactly seven jumps?

Question illustration
  1. 25
  2. 36
  3. 48
  4. 50
  5. 54

Question 6

Anna, Ben, Chris, Didi, Eva and Franz have an average monthly allowance of 10 EUR. Including Gitti makes the average for seven people 11 EUR. By what percentage does Gitti’s allowance exceed the average of the other six?

Question illustration
  1. 10%
  2. 50%
  3. 60%
  4. 70%
  5. 100%

Question 7

An ant wants to travel from K to L along the shortest possible route on the surface of the rectangular box shown, whose edges are a,b,c. Find the route length.

Editorial model: The ant may use every exterior face. No inequality between a,b,c is assumed from the sketch. The source restricts its calculation to one face pair.

Question illustration
  1. a + b + c
  2. √(a² + b² + c²)
  3. √(a² + b² + c² + 2ac)
  4. √(a² + b² + c² + 2bc)
  5. Depends on a,b,c

Question 8

Today, 16 March 2017, is described as the 1000th day since Andrea was born. On what date was Ann born?

Original printed choices: A) 16/3/2014; B) 16/6/2014; C) 19/6/2014; D) 20/6/2014; E) 21/6/2014

Editorial task: Explain the source’s elapsed-day convention and the inclusive alternative. Andrea and Ann are treated as the same intended person.

Question illustration

Question 9

Someone brought flowers to class. The guesses were: Smart Boy or Brainie Boy; Smart Boy or Sweet Tooth; Smart Boy or Beauty King; Brainie Boy or Sweet Tooth; Smile Boy or Sweet Tooth; Smile Boy or Brianie Boy. Exactly one guess is correct. Who brought the flowers?

Editorial reading: Brianie Boy is the same person as Brainie Boy.

Question illustration
  1. Smart Boy
  2. Smile Boy
  3. Sweet Tooth
  4. Brainie Boy
  5. Beauty King

Question 10

Maria forgot the five-digit code for her grandmother’s door lock. Its buttons are 1,2,3,4,5. From their wear, buttons 2 and 5 were unused, buttons 3 and 4 were used equally often, and each less often than button 1. How many possible codes must she consider?

Question illustration
  1. 60
  2. 40
  3. 20
  4. 12
  5. 6

Question 11

In cube ABCD.MNPQ, which segment is longest?

Original printed choices: A) CD; B) MD; C) BQ; D) ACDP; E)

Editorial task: Compare the well-defined segments and explain the malformed D and blank E choices without silently inventing missing text.

Question illustration

Question 12

Equilateral triangle ABC has area 40 cm². Draw right-isosceles triangle ACD with hypotenuse AC, then equilateral triangle CDE. Similarly draw right-isosceles triangle DEF and equilateral triangle EFG, as shown. What is the area of EFG?

Question illustration
  1. 30cm²
  2. 25cm²
  3. 20cm²
  4. 15cm²
  5. 10cm²

Question 13

Alice is now four years old, and her uncle is seven times her age. When Alice is six, how many times her age will her uncle be?

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

Question 14

Residents of asteroid Beta grow antennae instead of hair. No resident has more than 1000 antennae. There are more than 70,000 residents. Find the greatest N for which one can always guarantee N residents with the same antenna count.

Question illustration
  1. 70
  2. 71
  3. 100
  4. 250
  5. 1000

Question 15

If $(20/18)^{2018}=(81/100)^x$, find x.

Question illustration
  1. 1009
  2. −1009
  3. 2018
  4. −2018
  5. Another answer

Question 16

Fill the seven small circles with 1 through 7, each used once, so that the sums on the two large circles are equal. How many fillings are possible?

Question illustration
  1. 72
  2. 16
  3. 144
  4. 36
  5. 216

Question 17

Three small equilateral triangles are cut from a large equilateral triangle of perimeter 60 cm, as shown. The remaining shape F has perimeter 40 cm. What is the sum of the three small triangles’ perimeters?

Question illustration
  1. $80cm$
  2. $75cm$
  3. $66cm$
  4. $54cm$
  5. $60cm$

Question 18

A cube of edge 10 cm has a cylindrical hole of radius 5 cm drilled through it as shown. What is the remaining volume?

Question illustration
  1. 1000 − 250π
  2. 1000 − 230π
  3. 1000 − 300π
  4. 1000
  5. 550π

Question 19

Fold the triangle along the dashed line to obtain the figure shown. The original triangle’s area is 1.5 times the area covered by the folded figure, and the grey regions have total area 1. What was the original area?

Question illustration
  1. 2
  2. 3
  3. 4
  4. 5
  5. Cannot be determined

Question 20

A square of area 36 is shaded as shown. The grey regions have total area 27. Find a+b+c+d.

Question illustration
  1. 4
  2. 6
  3. 8
  4. 9
  5. Cannot be determined