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IKMC 2008 - Grades 9-10 - Self-study

Question 1

Five boxes contain the letter cards shown. Aslam removes cards until each box has exactly one card and all five remaining letters differ. Which card remains in box 2?

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  1. A
  2. E
  3. I
  4. O
  5. U

Question 2

Amir and Ahmad race 200 m. Ahmad does it under half a minute, and Amir under one hundredth of an hour. Who is faster, and by how many seconds? Source wording “under” is retained; see the conditional interpretation in the solution.
  1. Ahmad by 36 s
  2. Amir by 24 s
  3. Ahmad by 6 s
  4. Amir by 4 s
  5. Equal times

Question 3

Akram wears the pictured 2008 T-shirt and stands on his hands in front of a mirror. Zahid stands upright beside him. Which illustrated inscription does Zahid see in the mirror?

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Question 4

Let $a=2-(-4)$, $b=(-2)(-3)$, $c=2-8$, $d=0-(-6)$ and $e=(-12)\div(-2)$. How many are not equal to 6?
  1. 0
  2. 1
  3. 2
  4. 4
  5. 5

Question 5

Each of the four squares shown has side 1 m. Find the length AB.

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  1. $5$
  2. $\sqrt{13}$
  3. $\sqrt5+\sqrt2$
  4. $\sqrt5$
  5. None of these

Question 6

What is the smallest number of letters to remove from KANGOUROU so that the remaining letters are in alphabetical order?
  1. 1
  2. 2
  3. 3
  4. 4
  5. 5

Question 7

Different letters stand for different digits in the pictured addition OK+KO=WOW. Which digit is K? Source note: the supplied Pakistan page omits the entire equation. It is recovered from the matching English Junior J 7 in the official Lithuania 2008 book.

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  1. 0
  2. 1
  3. 2
  4. 8
  5. 9

Question 8

Tom and Jerry cut two equal rectangles into two rectangles each. Tom’s pieces each have perimeter 40 cm; Jerry’s pieces each have perimeter 50 cm. What is the perimeter of an original rectangle?
  1. 40 cm
  2. 50 cm
  3. 60 cm
  4. 80 cm
  5. 100 cm

Question 9

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

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

Question 10

The first spelling test score is 1 out of 5. All later tests score 5 out of 5. How many more tests are needed for an average of 4 out of 5?
  1. 2
  2. 3
  3. 4
  4. 5
  5. 6

Question 11

Seven cards numbered 1 to 7 are in a box. The first sage takes 3 at random and the second takes 2; two remain. The first sage says, “I know that the sum of your two numbers is even.” What is the sum of the first sage’s numbers?
  1. 10
  2. 12
  3. 6
  4. 9
  5. 15

Question 12

Bilal has one card for each of 3,8,13,18,23,28,33,48,53,68. What is the least number of cards whose numbers sum to 100?
  1. 2
  2. 3
  3. 4
  4. 5
  5. Impossible

Question 13

One cube face is cut along its two diagonals as shown. Which of the five nets cannot fold into the cube?

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  1. 1 and 3
  2. 1 and 5
  3. 3 and 4
  4. 3 and 5
  5. 2 and 4

Question 14

Seven dwarfs were born on the same day in seven consecutive years. The three youngest have total age 42. What is the total age of the three oldest?
  1. 51
  2. 54
  3. 57
  4. 60
  5. 63

Question 15

The two regular hexagons shown are congruent. What fraction of the parallelogram is shaded?

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

Question 16

Six integers are marked on the equally spaced number line. At least two are divisible by 3 and at least two by 5. Which marked integers are divisible by 15?

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  1. A and F
  2. B and D
  3. C and E
  4. All six
  5. Only one

Question 17

How many primes p also make $p^4+1$ prime?
  1. None
  2. 1
  3. 2
  4. 3
  5. Infinitely many

Question 18

What is the maximum number of digits that can be erased from the 1000-digit number 20082008…2008 so that the remaining digit sum is 2008? Original source choices: A 260; B 510; C 520; D 1020; E 130. None is correct; enter the numerical answer.

Question 19

In the isosceles triangle ABC, AB=AC. Point P lies on AC, PQ is perpendicular to AB, angle BPC=120 degrees and angle ABP=50 degrees. Find angle PBC.

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  1. 5°
  2. 10°
  3. 15°
  4. 20°
  5. 25°

Question 20

How many pairs of real numbers have their sum, product and quotient equal?
  1. No pair
  2. 1 pair
  3. 2 pairs
  4. 4 pairs
  5. 8 pairs