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IKMC 2017 - Grades 11-12 - Self-study

Question 1

On the number wall shown the number on each tile is equal to the sum of the numbers on the two tiles directly below it. Which number is on the tile marked with “?” ?

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  1. 15
  2. 16
  3. 17
  4. 18
  5. 19

Question 2

Many model railways use the H0-scale 1:87. For his railway Benjamin owns a 2 cm high model of his brother in H0-scale. How tall is his brother in reality?
  1. 1.74 m
  2. 1.62 m
  3. 1.86 m
  4. 1.94 m
  5. 1.70 m

Question 3

In the diagram we see 10 islands that are connected by 15 bridges. What is the minimum number of bridges that need to be closed off so that there is no connection from A to B anymore?

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

Question 4

Two positive numbers $a$ and $b$ have the following property: 75% of $a$ equals 40% of $b$. Which relation follows?
  1. $15a=8b$
  2. $7a=8b$
  3. $3a=2b$
  4. $5a=12b$
  5. $8a=15b$

Question 5

Four of the following five pictures show pieces of the graph of the same quadratic function. Which piece does not belong?
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Question 6

The diagram shows a circle with centre $O$ and diameters $AB$ and $CX$. Let $OB=BC$. What fraction of the circle area is shaded?

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  1. $\frac25$
  2. $\frac13$
  3. $\frac27$
  4. $\frac38$
  5. $\frac4{11}$

Question 7

A $4\times1\times1$ cuboid consists of 2 white and 2 grey cubes as shown. Which of the following cuboids can be built entirely out of such $4\times1\times1$ cuboids?

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Question 8

Which quadrant contains no points of the graph of $f(x)=-3.5x+7$?

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  1. I
  2. II
  3. III
  4. IV
  5. Every quadrant contains at least one point.

Question 9

In each of the five boxes (A) to (E) there are red and blue balls. Benedict wants to take exactly one ball without looking, out of one of these boxes and hopes to get a blue ball. In which box is the probability of that happening greatest?
  1. 10 blue, 8 red
  2. 6 blue, 4 red
  3. 8 blue, 6 red
  4. 7 blue, 7 red
  5. 12 blue, 9 red

Question 10

The graph of which function has the most intersections with the graph of $f(x)=x$?
  1. $g_1(x)=x^2$
  2. $g_2(x)=x^3$
  3. $g_3(x)=x^4$
  4. $g_4(x)=-x^4$
  5. $g_5(x)=-x$

Question 11

Three circles with centres $A,B,C$ touch each other externally in pairs, as shown. Their radii are 3, 2 and 1. What is the area of triangle $ABC$?

Question illustration
  1. 6
  2. $4\sqrt3$
  3. $3\sqrt2$
  4. 9
  5. $2\sqrt6$

Question 12

The positive number $p$ is smaller than 1, and $q$ is greater than 1. Which number is greatest?
  1. $pq$
  2. $p+q$
  3. $\frac pq$
  4. $p$
  5. $q$

Question 13

Two cylinders A and B have equal volumes. The base radius of B is 10% greater than that of A. By what percentage is the height of A greater than the height of B?
  1. 5%
  2. 10%
  3. 11%
  4. 20%
  5. 21%

Question 14

Each face of the polyhedron shown is either a triangle or a square. Each square borders 4 triangles, and each triangle borders 3 squares. The polyhedron has 6 squares. How many triangles does it have?

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  1. 5
  2. 6
  3. 7
  4. 8
  5. 9

Question 15

The four faces of a regular tetrahedron are labelled with the four digits 2, 0, 1 and 7 (one digit on each face). For a game, four such tetrahedrons are used as fair dice. All four dice are thrown simultaneously. Three of the four faces of each die can then be seen from above. What is the probability that we can form the number 2017 using exactly one of the three visible digits of each die?
  1. $\frac1{256}$
  2. $\frac{63}{64}$
  3. $\frac{81}{256}$
  4. $\frac3{32}$
  5. $\frac{29}{32}$

Question 16

The polynomial $5x^3+ax^2+bx+24$ has integer coefficients $a,b$. Which number is definitely not a solution of $5x^3+ax^2+bx+24=0$?
  1. 1
  2. -1
  3. 3
  4. 5
  5. 6

Question 17

Julia has 2017 round discs available: 1009 black ones and 1008 white ones. Using them, she wants to lay the biggest square pattern (as shown) possible and starts by using a black disc in the left upper corner. Subsequently she lays the discs in such a way that the colours alternate in each row and column. How many discs are left over when she has laid the biggest square possible?

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  1. None
  2. 40 of each colour
  3. 40 black and 41 white
  4. 41 of each colour
  5. 40 white and 41 black

Question 18

Two consecutive positive whole numbers are written on a board. The sum of the digits of each number is divisible by 7. What is the minimum number of digits the smaller of the two numbers has to have?
  1. 3
  2. 4
  3. 5
  4. 6
  5. 7

Question 19

In convex quadrilateral $ABCD$, the diagonals are perpendicular. The side lengths are $AB=2017$, $BC=2018$, $CD=2019$ (diagram not to scale). How long is $AD$?

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  1. 2016
  2. 2018
  3. $\sqrt{2020^2-4}$
  4. $\sqrt{2018^2+2}$
  5. 2020

Question 20

Lilli tries to be a well-behaved kangaroo but she is having just too much fun not to lie every now and then. Therefore every third statement of hers is a lie and the rest is true. Sometimes she starts with a lie and sometimes with one or two true statements. Lilli thinks of a two-digit number and says to her friend: 1: “One digit of the number is a 2.” 2: “The number is greater than 50.” 3: “It is an even number.” 4: “The number is less than 30.” 5: “The number is divisible by 3.” 6: “One digit of the number is a 7.” How big is the sum of the digits of the number, Lilli is thinking of?
  1. 9
  2. 12
  3. 13
  4. 15
  5. 17