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IKMC 2012 - Grades 9-10 - Self-study

Question 1

The isosceles triangle has M, N at the midpoints of its equal sides. Two straight lines divide it into four regions with three areas 3,3,6 as shown. Find the fourth area.

Question illustration
  1. 3
  2. 4
  3. 5
  4. 6
  5. 7

Question 2

11.11−1.111 equals:
  1. 9.009
  2. 9.0909
  3. 9.99
  4. 9.999
  5. 10

Question 3

The cuboid is made of four pieces as shown. Each piece consists of four cubes and has a single colour. Which shape is the white piece?

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

Question 4

Alice encodes each letter as A=01, B=02, …, Z=26, then sends 2 ×number+9. Bob receives 25−19−45−38. What was the original message?
  1. HERO
  2. HELP
  3. HEAR
  4. HERS
  5. Alice made a mistake

Question 5

The square ABCE has side 4 cm and the same area as triangle ECD. What is the distance from D to line g?

Editorial note: the printed labels conflict with the diagram. For this package, read the square as ABCD and the requested distance as E to line g. The original wording and diagram are retained; the solution explains the literal conflict.

Question illustration
  1. 8 cm
  2. $(4+2\sqrt3)$ cm
  3. 12 cm
  4. $10\sqrt2$ cm
  5. Depends on D

Question 6

A seven-digit integer has digit sum 6. What is the product of its digits?
  1. 0
  2. 6
  3. 7
  4. 1 × 2 × 3 × 4 × 5 × 6 × 7
  5. 5

Question 7

ABC is right-angled with legs 6 cm and 8 cm. K, L, M are the midpoints of its sides. Find the perimeter of KLM.
  1. 10 cm
  2. 12 cm
  3. 15 cm
  4. 20 cm
  5. 24 cm

Question 8

In four expressions, replacing every 8 by the same other positive number leaves the result unchanged. Which expression lacks this property?
  1. (8+8−8):8
  2. 8+(8:8)−8
  3. 8: (8+8+8)
  4. 8−(8:8)+8
  5. 8 × (8:8):8

Question 9

Two sides of a quadrilateral have lengths 1 and 4. A diagonal of length 2 splits it into two isosceles triangles. Find its perimeter.
  1. 8
  2. 9
  3. 10
  4. 11
  5. 12

Question 10

Both 144 and 220 leave remainder 11 when divided by positive integer N. Find N.
  1. 7
  2. 11
  3. 15
  4. 19
  5. 38

Question 11

Adam on a table is 80 cm taller than Mike on the floor. Mike on the same table is 1 m taller than Adam on the floor. Find the table height.
  1. 20 cm
  2. 80 cm
  3. 90 cm
  4. 100 cm
  5. 120 cm

Question 12

Denis and Mary toss a coin. On heads Mary wins 2 candies from Denis; on tails Denis wins 3 from Mary. After 30 games both have their original candy counts. How many wins did Denis have?
  1. 6
  2. 12
  3. 18
  4. 24
  5. 30

Question 13

A rectangle of length 6 cm encloses an equilateral triangular arrangement of touching circles as shown. Find the shortest distance between the grey circles.

Question illustration
  1. 1
  2. $\sqrt2$
  3. $2\sqrt3-2$
  4. $\pi/2$
  5. 2

Question 14

In Billy’s room are four clocks, each slow or fast. The first is wrong by 2 minutes, the second by 3, the third by 4, and the fourth by 5. One day they read 6 minutes to 3,3 minutes to 3,2 minutes past 3, and 3 minutes past 3. What was the exact time? Editorial note: read these readings as an unordered list, not in numbered clock order.
  1. 3:00
  2. 2:57
  3. 2:58
  4. 2:59
  5. 3:01

Question 15

In the right triangle with sides 5,12,13 shown, find the radius of the inscribed semicircle.

Question illustration
  1. 7/3
  2. 10/3
  3. 12/3
  4. 13/3
  5. 17/3

Question 16

How many four-digit integers have hundreds digit 3 and the sum of the other three digits also 3?
  1. 2
  2. 3
  3. 4
  4. 5
  5. 6

Question 17

Fill the twelve cells with numbers 1 through 9, so all row sums are equal and all column sums are equal. Some numbers are given. Find the shaded entry.

Question illustration
  1. 1
  2. 4
  3. 6
  4. 8
  5. 9

Question 18

Kan, Ga, Roo finish in the top three. Four true predictions were: (1) Kan or Ga wins; (2) if Ga is second, Roo wins; (3) if Ga is third, Kan does not win; (4) Ga or Roo is second. Find the order.
  1. Kan, Ga, Roo
  2. Kan, Roo, Ga
  3. Roo, Ga, Kan
  4. Ga, Roo, Kan
  5. Ga, Kan, Roo

Question 19

The shape comprises squares of side 4 and 5 cm, a triangle of area 8 cm², and a shaded parallelogram. Find the parallelogram’s area.

Question illustration
  1. 15 cm²
  2. 16 cm²
  3. 18 cm²
  4. 20 cm²
  5. 21 cm²

Question 20

Positive integers m, k satisfy $2012=m^m(m^k-k)$. Find k.
  1. 2
  2. 3
  3. 4
  4. 9
  5. 11