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IKMC 2027 - Grades 11-12 - Self-study SET 5 - Thematic Batch 3

Question 1

Five people leave their five hats at reception. Each later receives one randomly chosen hat. What is the probability that nobody receives their own hat?

Question illustration
  1. 11/30
  2. 15/2
  3. 17/4
  4. 31/30
  5. 3/5

Question 2

A positive integer N leaves remainder 5 when divided by 6. What remainder does N² leave when divided by 12?

Question illustration
  1. 1
  2. 4
  3. 6
  4. 13
  5. Cannot be uniquely determined

Question 3

Julia has nine cards numbered 1 through 9. She randomly selects five and arranges them in increasing order. What is the probability that card 5 is in the middle?

Question illustration
  1. 1/9
  2. 2/9
  3. 1/4
  4. 2/7
  5. 1/3

Question 4

A very large square is covered by round tiles in the pictured pattern. The tiles are very small compared with the square, so boundary effects may be ignored. What percentage is covered? Use π =3.14 and $\sqrt{3}$=1.73.

Question illustration
  1. Less than 84%
  2. 84% to 87%
  3. 87% to 90%
  4. 90% to 93%
  5. Greater than 93%

Question 5

Which equation has the two roots (−3±$\sqrt{41}$)/8?

Question illustration
  1. 4 x² + 3 x + 2 = 0
  2. 4 x² + 3 x − 2 = 0
  3. 4 x² − 3 x + 2 = 0
  4. 4 x² − 3 x − 2 = 0
  5. x² + 3 x − 8 = 0

Question 6

A positive integer n satisfies 1+2+…+n =2016. What is n?

Question illustration
  1. 46
  2. 56
  3. 63
  4. 66
  5. 73

Question 7

What is the value of the infinite continued fraction $1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{1+\cdots}}}$?

Question illustration
  1. 1/2
  2. ($\sqrt{5}$ + 1)/2
  3. $\sqrt{2}$/2
  4. ($\sqrt{6}$ − $\sqrt{2}$)/2
  5. $\sqrt{3}$/2

Question 8

Two sides of a triangle have lengths 4 and 5. How many positive integer lengths can the third side have?

Question illustration
  1. 1
  2. 5
  3. 7
  4. 8
  5. 9

Question 9

In the diagram AB =AD+DB. What is the length AC?

Question illustration
  1. 28
  2. 30
  3. 35
  4. 36
  5. 40

Question 10

A laboratory has two kangaroos and eight koalas. Six animals are selected at random to go to the International Space Station. What is the probability that both kangaroos are selected?

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

Question 11

In a certain country, the milk amount per meal in millilitres is calculated by dividing a baby’s weight in kilograms by 0.7, multiplying by 100, then dividing by the daily number of meals. If the baby drinks ten meals of 60 ml daily, what is the baby’s weight in kilograms?

Question illustration
  1. 0.42
  2. 4.2
  3. 8.5
  4. 14.3
  5. 85

Question 12

Alex fills the tower so that the three bottom cells are natural numbers greater than 1, and each upper cell is the product of the two immediately below it. Which listed number cannot be the top entry?

Question illustration
  1. 220
  2. 105
  3. 90
  4. 84
  5. 56

Question 13

Julia draws a regular pentagon and all its diagonals as shown. The grey region in figure A has area S. What is the grey area in figure B?

Question illustration
  1. 2 S
  2. 1/2 S
  3. $\sqrt{5}$ S
  4. (2 + $\sqrt{5}$)/2 S
  5. 3 S

Question 14

A circle with centre O has diameter AB =20. Circle S is tangent to the two circles with diameters AO, BO and internally tangent to the large circle, as shown. What is its radius?

Question illustration
  1. 5/2
  2. 10/3
  3. 5
  4. $\sqrt{5}$
  5. 2$\sqrt{5}$

Question 15

Pentagon ABCDE is inscribed in a circle. Angles BAE, CBA, CDE are 110°,120°,87° respectively. What is angle ANE in the diagram, where N is the intersection of AC and EB?

Question illustration
  1. 42°
  2. 44°
  3. 45°
  4. 48°
  5. 52°

Question 16

A class of 30 mathematics students collects information about their school during a trip. The average time is 25 minutes for a boy,19 minutes for a girl, and 21 minutes for the whole class. How many boys are in the class?

Question illustration
  1. 10
  2. 12
  3. 15
  4. 18
  5. 20

Question 17

For which values of k does 2 x²−kx+x+8=0 have a repeated root?

Question illustration
  1. k =9 or k =−7
  2. k =7 or k =−9
  3. k =1 or k =0
  4. k =2 or k =3
  5. k =5 or k =4

Question 18

The polynomial P (x)=x³+x²−2 x−1 has three distinct real roots a, b, c. What is a³+b³+c³?

Question illustration
  1. −4
  2. −44
  3. −1
  4. 4
  5. 6

Question 19

Sam draws a right triangle with legs 5 and 7 on the pictured chessboard. What is the total area of the black regions inside it?

Question illustration
  1. 8 3/4
  2. 8
  3. 9
  4. 8 6/7
  5. 18 18/35

Question 20

The prism shown has two triangular faces and three square faces. Its vertices receive the numbers 1 through 6 once each so that the four-vertex sums on the square faces are equal. The positions of 1 and 5 are given. Which number belongs at x?

Question illustration
  1. 2
  2. 3
  3. 4
  4. 6
  5. Cannot be uniquely determined