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IKMC 2022 - Grades 9-10 - Self-study

Question 1

Carola forms the four-digit number 2022 using matches from a box that originally contained 30 matches. She has already formed its first two digits, as shown. How many matches remain when she finishes 2022?

Question illustration
  1. 20
  2. 19
  3. 10
  4. 9
  5. 5

Question 2

An equilateral triangle of side 12 has the same perimeter as a square of side x. What is x?
  1. 9
  2. 12
  3. 16
  4. 24
  5. 36

Question 3

The teacher folds the left side of the paper over the thick line. How many shapes on the left fit exactly on top of a shape on the right?

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

Question 4

Katrin arranges 2 × 1 tables for small, medium and large meetings as shown in top view. How many tables are used for the large meeting?

Question illustration
  1. 10
  2. 11
  3. 12
  4. 14
  5. 16

Question 5

A square of four entries is taken from a multiplication table. Only 77 is visible, as shown. The integers x and y are positive and x>y. What is x?

Question illustration
  1. 6
  2. 7
  3. 8
  4. 10
  5. 11

Question 6

I am less than my half and greater than my double. The sum of me and my square is zero. Who am I?
  1. −2
  2. −1
  3. 0
  4. 1
  5. 2

Question 7

In the rectangle shown, the midpoints of the two longer sides are joined to all four vertices. What fraction of the rectangle is shaded?

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

Question 8

The diagram shows Nadya’s time on four smartphone apps last week. This week she halved the time on two apps and kept the time unchanged on the other two. Which diagram could represent this week?

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

Question 9

Five candidates have received the following votes after 90% of the votes are counted: Alex 14, Bella 11, Calvin 10, Diane 8, Eddy 2. How many still have a chance to win?
  1. 1
  2. 2
  3. 3
  4. 4
  5. 5

Question 10

Five squares and two right-angled triangles are arranged as shown. The numbers 3, 8 and 22 are areas in m² of three squares. What is the area of the square with the question mark?

Question illustration
  1. 14 m²
  2. 15 m²
  3. 16 m²
  4. 17 m²
  5. 18 m²

Question 11

The figure shows three equal large circles and four equal small circles. All centres and points of contact lie on one straight line. Each small radius is 1. What is the shaded area?

Question illustration
  1. $\pi$
  2. $2\pi$
  3. $3\pi$
  4. $4\pi$
  5. $6\pi$

Question 12

Apini moves from hexagon X to hexagon Y, only between hexagons sharing an edge. How many routes pass through each of the seven white hexagons exactly once?

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

Question 13

Six siblings have ages that are six consecutive whole numbers. Each answers “How old is your oldest sibling? ” Which number could not be the sum of their six answers?
  1. 95
  2. 125
  3. 167
  4. 205
  5. 233

Question 14

Eva puts 2022 tiles in a line. Adam removes every sixth tile. Beata then removes every fifth of the remaining tiles. Calle then removes every fourth of those still remaining. Doris removes all the rest. How many does Doris remove?
  1. 0
  2. 337
  3. 674
  4. 1011
  5. 1348

Question 15

Three children guess their grandmother’s age as 75, 78 and 81. One guess is wrong by 1 year, one by 2 years and one by 4 years. What is her age?
  1. 76
  2. 77
  3. 79
  4. 80
  5. Cannot be determined exactly

Question 16

The large rectangle ABCD is divided into twelve identical small rectangles as shown. What is AD/DC?

Question illustration
  1. $8/9$
  2. $5/6$
  3. $7/8$
  4. $2/3$
  5. $9/8$

Question 17

A rabbit and a hedgehog race around a 550 m circular track at constant speeds 10 m/s and 1 m/s. They start together in opposite directions. When they meet, the hedgehog immediately turns and follows the rabbit’s direction. How long after the rabbit does the hedgehog reach the finish?
  1. 45 sec
  2. 50 sec
  3. 55 sec
  4. 100 sec
  5. 505 sec

Question 18

Square PQRS has side 1. U is the midpoint of RS and W is the centre. Segments TW, UW and VW split the square into three equal-area regions as shown. What is SV?

Question illustration
  1. $1/2$
  2. $2/3$
  3. $3/4$
  4. $4/5$
  5. $5/6$

Question 19

Three paths cross the city park as shown. One tree already stands in the middle. What is the smallest number of additional trees needed so each path has equally many trees on both sides?

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

Question 20

Veronica wears five rings as shown. She removes them one at a time. In how many different orders can she do this?

Question illustration
  1. 16
  2. 20
  3. 24
  4. 30
  5. 45