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IKMC 2015 - Grades 9-10 - Self-study

Question 1

Which number is closest to $20.15\times51.02$?
  1. 100
  2. 1000
  3. 10000
  4. 100000
  5. 1000000

Question 2

Mother hangs T-shirts in a line, then children put a single sock between any two T-shirts. There are 29 items in all. How many T-shirts?
  1. 10
  2. 11
  3. 13
  4. 14
  5. 15

Question 3

The shaded part of a square of side a is bounded by a semicircle and two quarter-circle arcs as shown. What is its area?

Question illustration
  1. $\pi a^2/8$
  2. $a^2/2$
  3. $\pi a^2/2$
  4. $a^2/4$
  5. $\pi a^2/4$

Question 4

Ann, Beth, Cindy buy 30 cookies and receive 10 each. They pay 80,50,20 cents respectively. If divided proportionally to payment, how many more cookies should Ann receive?
  1. 10
  2. 9
  3. 8
  4. 7
  5. 6

Question 5

Mister Hide remembers his buried treasure is at least 5 m from the hedge and at most 5 m from the pear-tree trunk. Which picture shows where to search?
  1. Question illustration
  2. Question illustration
  3. Question illustration
  4. Question illustration
  5. Question illustration

Question 6

What is the units digit of $2015^2+2015^0+2015^1+2015^5$?
  1. 1
  2. 5
  3. 6
  4. 7
  5. 9

Question 7

A class has 33 children whose favourite subjects are computing and PE. Three like both. Twice as many like only computing as only PE. How many like computing?
  1. 15
  2. 18
  3. 20
  4. 22
  5. 23

Question 8

Which is neither a square nor a cube?
  1. $6^{13}$
  2. $5^{12}$
  3. $4^{11}$
  4. $3^{10}$
  5. $2^9$

Question 9

Mr Candle buys 100 candles, burns one daily, and makes one new candle from the remains of seven burnt ones. After how many days must he buy again?
  1. 112
  2. 114
  3. 115
  4. 116
  5. 117

Question 10

A convex pentagon has n right angles. Which list gives all possible n?
  1. 1,2,3
  2. 0,1,2,3,4
  3. 0,1,2,3
  4. 0,1,2
  5. 1,2

Question 11

The picture shows a decision die in three positions. What is the probability of rolling YES? Source editorial note: “Possible to take away first die on the picture!”

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

Question 12

Each square has side 1. What is the minimum Start-to-Finish distance, moving only along sides or diagonals of individual squares?

Question illustration
  1. $2\sqrt5$
  2. $\sqrt{10}+\sqrt2$
  3. $2+2\sqrt2$
  4. $4\sqrt2$
  5. 6

Question 13

Winger inhabitants have at least two ears. Imi, Dimi, Trimi meet and see 8,7,5 ears respectively; none sees their own. How many ears has Trimi?
  1. 2
  2. 4
  3. 5
  4. 6
  5. 7

Question 14

A rectangular-prism container has square base side 10 cm and water depth h cm. A solid cube of edge 2 cm is placed in it. What is the minimum initial h for complete submersion?
  1. 1.92 cm
  2. 1.93 cm
  3. 1.90 cm
  4. 1.91 cm
  5. 1.94 cm

Question 15

Square ABCD has area 80. Points E, F, G, H on its sides satisfy AE=BF=CG=DH and AE=3 EB. What is the shaded area?

Question illustration
  1. 20
  2. 25
  3. 30
  4. 35
  5. 40

Question 16

Today the integer ages of father and son have product 2015. What is their difference?
  1. 26
  2. 29
  3. 31
  4. 34
  5. 36

Question 17

Loads a, b, c, d sit on the scales. Two are interchanged and the scales change as shown. Which loads?

Question illustration
  1. a and b
  2. b and d
  3. b and c
  4. a and d
  5. a and c

Question 18

Both roots of $x^2-85x+c=0$ are prime. What is the digit sum of c?
  1. 12
  2. 13
  3. 14
  4. 15
  5. 21

Question 19

How many three-digit positive integers have adjacent digits differing by 3?
  1. 12
  2. 14
  3. 16
  4. 20
  5. 27

Question 20

Which is a counterexample to: “If n is prime, exactly one of n−2 and n+2 is prime”?
  1. n=11
  2. n=19
  3. n=21
  4. n=29
  5. n=37