Problems and solutions (the solutions can be found at the end of the file)

Working time: 80 minutes

1.     RECTANGLE ON A SQUARE

The rectangle on the figure has been placed so that its corners divides the length of the sides of the square in the scale 1 : 2, i.e. a is twice the length b.

How much of the area of the square has been covered by the rectangle?

 ANSWER (cross out): :

 2.     WHO IS DOING WHICH SPORTS?

Anna, Eva and Hans are active in two of these sports each::

Football, handball, alpine skiing, basketball, tennis, golf

None of them is active in the same sport as any of the others.

• The alpine skier and the tennis player went to the cinema with Anna

• Eva is the neighbour of the tennis player 

• Hans beat Eva and the basketball player in monopoly

• The football player had lunch with the alpine skier 

• The basketball player is a relative of the handball player

• The football player got an sms from the basketball player

Which two sports are the sports of Anna, Eva and Hans?

ANSWER:

Cross out 2 sports for each person:: 

3.     PETER'S 100 KRONER

Kari asks Peter to lend her 100 kroner.

”No, I had 100 kroner, but now I have spent a part of it,” Peter says.

Kari asks how much he has spent.

”I have spent exactly ¼  of the amount I have left,” Peter answers.

How much has Peter got left?

ANSWER:

4.     NEW SYMBOLS IN MATHEMATICS

Here are examples showing what happens to some integers when we use ☻on them: 

 3 ☻ 9             10 ☻ 100                1 ☻ 1

Here are examples showing what happens to some integers when we use ▲ on them:

1 ▲ 5              20 ▲ 43          91 ▲ 185

When ☻and ▲ are used in combination, we get: 

3 ☻▲ 21

and

n ☻▲ 53 

n is a positive integer. Which integer is it? 

ANSWER:

5.     THE ORANGE PYRAMID

A number of oranges create a kind of a pyramid. The ground level of the “pyramid” has the shape of a rectangle measuring 5 x 8 oranges.

Each orange above the ground level is resting in a “hole” formed by 4 oranges in the level below. The top level of the pyramid is a simple row of oranges.

How many oranges are there in this “pyramid”?

ANSWER (cross out):

A:

120

B:

144

C:

80

D:

100

E:

62

6.     A VERY SPECIAL INTEGER

Make an integer of 8 digits with the digits 1, 1, 2, 2, 3, 3, 4, 4 so that there is one digit between the two 1s, two digits between the two 2s, three digits between the 3s and four digits between the 4s.

ANSWER (fill in):

7.     THE RULER

A ruler has the length of exactly 12 units. But there is only one distance marking on it: 1 unit from the end. You are to make 3 more markings on it so that it may be used for measuring all lengths (in integers) from 1 to 12 units. You may use the ruler just once for each measuring.

How many units from the end would you place the three markings, all seen from the same end of the ruler close to the already excisting marking?

This problem has more than one solution, but we ask for the solution with the new markings as close to the original marking as possible.

ANSWER (fill in 3x):

8.     CIRCLES TOUCHING EACH OTHER

The radius in the circles is 1 dm. The circle lines touch at one single point. Find the size (in dm²) of the white area. Give two decimals.

ANSWER (fill in before and after the decimal comma:

Solutions:

Problem 1

Rectangle on a square

The area of the rectangle is 4 square units, and the area of the square is 9 square units.

Problem 2

Who is doing which sports?

Correct answer:  

For instance: The first saying tells that neither is doing alpine skiing nor is playing tennis. The three first sayings give table 1, that shows that Anna is the basketball player and Hans the tennis player.

Table 1:

Table 2: 

Problem 3

Peter's 100 kroner

Correct answer:       Peter has 80 kroner left.

We say: Peter has x (kr) left. That gives x + ¼ x = 100.   And: x = 80

Problem 4

New symbols in Mathematics

Correct answer:  n = 5  

The operation ☻ means that the integer is to be squared, i.e.  n ☻ n²

The operation ▲means that the integer is to be multiplied by 2, and then 3 is to be added: n ▲ 2n + 3

When ☻and ▲ are used in combination, we get  n ☻▲ 53

This means that  n ☻ (53 – 3) : 2 = 50 : 2 = 25

that gives  n = 5  (as n is to be a positive integer).

Problem 5

The orange pyramid

Correct answer:       d) 100

The ”pyramid” is 5 levels high, and each level has 1 orange less in the length and 1 orange less in the width than in the rectangle below. Then the total number of oranges in the “pyramid” is:

Problem 6

A very special integer

Problem 7

The ruler

Correct answer:       2 units - 3 units - 8 units

Problem 8

Circles touching each other  

Correct answer:       0,86 dm².

The area is a square with area (2∙1)² = 4 (dm²)
minus two semicircles with a total area π (dm²)