IKMC 2027 - Grades 11-12 - Self-study SET 6 - Thematic Batch 3

Question 1

A clock hangs from the ceiling by its minute hand, which remains vertical while the clock body rotates as the clock runs normally. How many complete turns does the hour hand make in one day?

1. 22
2. 23
3. 24
4. 25
5. 26

Question 2

Kengi counts on adjacent fingers: 1 on the thumb, 2 on the index finger, 3 on the middle finger, 4 on the ring finger, 5 on the little finger, 6 on the ring finger, 7 on the middle finger, and so on. Which finger corresponds to 2020?

1. Thumb
2. Index finger
3. Middle finger
4. Ring finger
5. Little finger

Question 3

Ann, Betty, Kate walk from S to T. Betty is twice as fast as Ann, and Ann twice as fast as Kate. Kate takes two hours longer than Betty. How long does Ann take?

1. 1 hour
2. 1 hour 20 minutes
3. 1 hour 30 minutes
4. 1 hour 40 minutes
5. 2 hours

Question 4

A standard die is rolled twice. What is the probability that the second result exceeds the first by exactly two dots?

1. 1/9
2. 1/6
3. 1/4
4. 1/3
5. 2/9

Question 5

The figure below is said to show a rectangular plot being enlarged: one side increases by 20% and another by 50°. The result is a square. The shaded region is stated to have area 30 m². What is the area of the garden?

Original printed choices: A) 60 m²; B) 65 m²; C) 70 m²; D) 75 m²; E) 80 m²

Editorial note: Explain whether the printed question and choices are consistent. Give the literal mathematical result and distinguish any intended correction.

Question 6

Let (x, y, z) solve sin y −sin x =x −y, sin y −sin z =z −y, and x −y+z =π. As k ranges over all integers, how many different values can sin (kz) take?

1. 1
2. 2
3. 4
4. Infinitely many values
5. Undefined because the given system has no solution

Question 7

What is the perimeter, in centimetres, of the triangle shown?

1. 3 + $\sqrt{6}$ + $\sqrt{3}$
2. 3 + $\sqrt{6}$ + 2$\sqrt{3}$
3. 4 + $\sqrt{3}$
4. 4 + $\sqrt{6}$
5. 3 + $\sqrt{3}$ + $\sqrt{2}$

Question 8

A function f: R→R satisfies f (x+1)=2 f (x)−1 and f (2018)=2. What is f (0)?

1. 1 + 1/$2^{2018}$
2. 1/2 + $2017^{-1}$
3. 1
4. 2
5. 1 + $2^{2018}$

Question 9

Integers a, b satisfy |a|$\lt$4 and 3$\lt$|b|≤7. What is the greatest possible value of C =−2 a −3 b?

1. −3
2. −1
3. 27
4. 28
5. 29

Question 10

How many 2016-digit numbers have the property that every adjacent pair of digits forms a two-digit number divisible by 17 or 23?

1. 1
2. 2
3. 3
4. 5
5. More than 5

Question 11

How many primes below 50 cannot be expressed as p+q+1, where p and q are primes?

1. 2
2. 3
3. 4
4. 5
5. 6

Question 12

In parallelogram ABCD, E lies on AD and F on BC with ED =AD/3 and FC =BC/3. EF meets BD at J and AC at K. Triangle IJK has area 1, where the diagram defines I =AC∩BD. What is the parallelogram’s area?

1. 36
2. 27
3. 25
4. 32
5. 34

Question 13

A circle touches two sides of the pictured rectangle and passes through a vertex. One tangency point is at distances 5 and 4 from the two bottom vertices. What is the rectangle’s area?

1. 63
2. 72
3. 81
4. 27 π
5. Another answer

Question 14

On an 8-by-8 chessboard with alternating black and white squares, how many ways can one white and one black square be chosen so that they share neither a row nor a column?

1. 1024
2. 960
3. 896
4. 768
5. 64

Question 15

The pictured solid has 12 regular pentagonal faces; its other faces are equilateral triangles and squares. Each pentagon is surrounded by five squares, and each triangle by three squares. John writes 0 on triangles, 5 on pentagons, and −1 on squares. What is the sum of all labels?

1. 0
2. 30
3. 40
4. 60
5. 120

Question 16

A regular octahedron is drawn inside a cube of side 1 m, with its vertices at the centres of the cube’s faces. What is the octahedron’s volume?

1. 1/3
2. 1/4
3. 1/5
4. 1/6
5. 1/8

Question 17

Let 0$\lt$p $\lt$1 and q>1. Which of the five listed numbers is largest?

1. p · q
2. p + q
3. p/q
4. p
5. q

Question 18

ABCDEF is a regular hexagon. G is the midpoint of AB, and H, I are the intersections of GD, GE with FC respectively. What is area (GIF) divided by area (IHDE)?

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

Question 19

Seven islanders sit in a circle. Knights always tell the truth and liars always lie. Each person says,“I am sitting between two liars.” How many are knights?

1. 3
2. 4
3. 5
4. 6
5. Cannot be determined

Question 20