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IKMC 2007 - Grades 7-8 - Self-study

Question 1

Calculate $\frac{2007}{2+0+0+7}$.
  1. 1003
  2. 223
  3. 213
  4. 123

Question 2

Rose plants are planted in a line on both sides of a 20 m path, 2 m apart. What is the greatest total number of plants?
  1. 22
  2. 20
  3. 12
  4. 11

Question 3

The robot starts at A2 in the arrow direction. It keeps going forward. When a black cell or boundary blocks it, it turns right. It stops only if it still cannot go forward after that turn. Where will it stop?

Question illustration
  1. B2
  2. A1
  3. E1
  4. Nowhere

Question 4

What is the sum of the dots on the invisible faces of the two dice?

Question illustration
  1. 15
  2. 12
  3. 7
  4. 27

Question 5

Two positive integers sum to 11. What is their greatest possible product?
  1. 24
  2. 28
  3. 30
  4. 32

Question 6

A small square is inscribed in a larger square as shown. Find the small square area.

Question illustration
  1. 16
  2. 28
  3. 34
  4. 36

Question 7

What is the least number of additional little squares that must be shaded so the picture has an axis of symmetry?

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

Question 8

A palindrome reads the same forwards and backwards, such as 13931. What is the difference between the smallest five-digit palindrome and the largest six-digit palindrome?
  1. 989989
  2. 989998
  3. 998998
  4. 999898

Question 9

Six equal circles touch one another and the large rectangle as shown. The small rectangle joins the centres of four circles. Its perimeter is 60 cm. What is the large rectangle perimeter?

Question illustration
  1. 160 cm
  2. 120 cm
  3. 100 cm
  4. 80 cm

Question 10

x is a strictly negative integer. Which expression is greatest?
  1. $-2x$
  2. $2x$
  3. $6x+2$
  4. $x-2$

Question 11

Squares are formed by segment AB of length 24 cm and the broken line $AA_1A_2\ldots A_{12}B$ shown. Find the length of the broken line.

Question illustration
  1. 48 cm
  2. 72 cm
  3. 96 cm
  4. 106 cm

Question 12

Six points lie on two parallel lines: four on the first and two on the second. How many triangles have vertices among these points?
  1. 6
  2. 12
  3. 16
  4. 18

Question 13

Initially $\frac23$ of customers buy A and $\frac13$ buy B. After advertising, $\frac14$ of the former A customers switch to B. What are the new proportions?
  1. $\frac14$ buy A, $\frac34$ buy B
  2. $\frac7{12}$ buy A, $\frac5{12}$ buy B
  3. $\frac12$ buy A, $\frac12$ buy B
  4. $\frac13$ buy A, $\frac23$ buy B

Question 14

To obtain $8^8$, to what power must we raise $4^4$?
  1. 3
  2. 2
  3. 4
  4. 8

Question 15

ABC and CDE are congruent equilateral triangles. If angle ACD=80°, find angle ABD.

Question illustration
  1. 25°
  2. 30°
  3. 35°
  4. 40°

Question 16

What percentage of the integers 1,2,…,100 are perfect squares?
  1. 1%
  2. 5%
  3. 25%
  4. 10%

Question 17

Five horizontal and four vertical segments make 12 rectangular cells; six horizontal and three vertical make 10. What is the maximum number of cells using at most 15 horizontal/vertical lines in such a table?

Question illustration
  1. 30
  2. 36
  3. 40
  4. 42

Question 18

How many routes with the minimum number of moves lead from A to B in the grid? A move may go to any neighbouring square, including diagonally.

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

Question 19

Choose three numbers from the grid, exactly one from each row and each column. What is the largest possible sum?

Question illustration
  1. 18
  2. 15
  3. 21
  4. 24

Question 20

O is the centre of square KLMN of side 2. Rays OA,OB,OC,OD reach its sides, with AOB=90° and COD=90° as shown. What is the shaded area?

Question illustration
  1. 1
  2. 2
  3. 2.5
  4. 2.25