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IKMC 2005 - Grades 11-12 - Self-study

Question 1

For which of the following values of $x$ is $\frac{x^2}{x^3}$ smallest?
  1. 2
  2. 1
  3. −1
  4. −2
  5. −3

Question 2

How many numbers from 2 to 100 are the cube of an integer?
  1. 1
  2. 2
  3. 3
  4. 4
  5. 5

Question 3

Five cards numbered 1 to 5 are arranged as shown. In one move any two cards may be interchanged. Find the smallest number of moves required to arrange them in increasing order.

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

Question 4

If $888\cdot111=2\cdot(2\cdot n)^2$ and $n$ is a positive integer, find $n$.
  1. 8
  2. 11
  3. 22
  4. 111
  5. 444

Question 5

Eight kangaroos occupy the squares shown. A kangaroo may move directly to any empty square. Find the least number of moves needed so that every row and every column contains exactly two kangaroos.

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

Question 6

A square piece of paper has been cut into three pieces. Two are shown. What is the shape of the third piece?

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Question 7

The sum of four consecutive positive integers cannot equal which number?
  1. 2002
  2. 22
  3. 202
  4. 222
  5. 220

Question 8

A 3×3×3 cube weighs 810 grams. Three holes are drilled through it as shown, each a 1×1×3 rectangular parallelepiped. What is the weight of the remaining solid?

Question illustration
  1. 540 g
  2. 570 g
  3. 600 g
  4. 630 g
  5. 660 g

Question 9

A function satisfies $f(x+1)=2f(x)-2002$ for every integer $x$, and $f(2005)=2008$. Find $f(2004)$.
  1. 2004
  2. 2005
  3. 2008
  4. 2010
  5. 2016

Question 10

Three semicircles are shown. ABEF is a rectangle; every semicircle has radius 2 cm. E and F are the centres of the bottom semicircles. Find the shaded area in cm².

Question illustration
  1. 8
  2. 7
  3. $2\pi$
  4. $2\pi+1$
  5. $2\pi+2$

Question 11

Mom Kangaroo and Jumpy move around a stadium of perimeter 330 m. Each makes one leap per second: Mom 5 m and Jumpy 2 m. They start together at the same point and in the same direction. After 25 seconds Jumpy stops while Mom continues. How long does it take her to overtake Jumpy?

Question illustration
  1. 15 sec
  2. 24 sec
  3. 40 sec
  4. 51 sec
  5. 66 sec

Question 12

Henny paints every face of several wooden cubes black or white, using both colours on each cube. How many different colourings are possible?
  1. 8
  2. 16
  3. 32
  4. 52
  5. 64

Question 13

Opposite faces of a die always sum to 7. A die rolls as shown. At the starting point S its top face is 3. Which face is on top at the end point F?

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

Question 14

A box contains 60 red, blue and white tickets. Replacing all red tickets by blue would make twice as many blue as white. Replacing all white tickets by blue would make three times as many blue as red. How many tickets are blue?
  1. 10
  2. 15
  3. 20
  4. 25
  5. 30

Question 15

Let a and b be the lengths of the sides of a right-angled triangle. If d is its incircle diameter and D its circumcircle diameter, what is d+D?

Editorial note: the intended reading takes a,b to be the two perpendicular sides; the source does not explicitly say this.
  1. $a+b$
  2. $2(a+b)$
  3. $\frac12(a+b)$
  4. $\sqrt{ab}$
  5. $\sqrt{a^2+b^2}$

Question 16

Let $M$ be the set of real $x$ satisfying $2^{4^x}<4^{2^x}$. Find $M$. Editorial note: option E is visibly blank in this copy; it is retained as a missing printed option.
  1. $(-\infty,1)$
  2. $(0,1)$
  3. $(-\infty,1)\cap(1,\infty)$
  4. $(0,\infty)$
  5. [blank in source]

Question 17

The picture gives the graph relating y−x (vertical axis) to y+x (horizontal axis). Which is the graph relating y to x?

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Question 18

Two equal-volume bottles contain water and acid in volume ratios 2:1 and 4:1. All their contents are mixed. What is the resulting water-to-acid ratio?
  1. 3:1
  2. 6:1
  3. 11:4
  4. 5:1
  5. 8:1

Question 19

The diagram shows rectangle ABEF and triangle ABC. Angle ACF equals angle CBE. If FC=6 and CE=2, find the area of ABC.

Question illustration
  1. 12
  2. 16
  3. $8\sqrt2$
  4. $8\sqrt3$
  5. none of these

Question 20

Sherlock Holmes and Dr Watson travel by train from London to Plymouth, a distance of 120 km. Holmes notices that the angle between the hour and minute hands is 90° both at departure and arrival. Which could be the speed of the train?
  1. 120 km/h
  2. 110 km/h
  3. 100 km/h
  4. 60 km/h
  5. any of these