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

1. A SPECIAL SEQUENCE OF NUMBERS

Twelve integers are written in a sequence. The fourth integer is 4, and the twelvth integer is 12. The sum of three neighbouring numbers is 333, regardless what neighbouring 3 numbers you choose.

Fill in the empty boxes:  

2. FROM RECTANGLE TO SQUARE

Three rectangles of the same size and shape have been put together to form a greater rectangle. The area of the great rectangle is 168 cm².

Find the area of a square that has the same perimeter as the great rectangle.

3. TRIANGLES IN A HEXAGON

You randomly choose three corners of a regular hexagon (all sides have the same length and all angles are 120°).

What is the likeliness that the triangle made from these three corners is an equilateral or an isosceles triangle?  

Give the answer in per cent.

4. GO

Go is a Japanese game where the players one after another place white and black play stones in the crossings of the lines in order to ”conquer” the crossings within the area they surround.  

In the example 10 stones were used for ”conquering” 3 crossings:  

What is the greatest number of crossings that may be “conquered” when using 12 stones?

5. DICE THROWING

When three dice are thrown, what is more likely?

      A)  the sum of the spots (numbers) on the dice may be divided by 3
B)   the sum gives the rest 1 when divided by 3
C)    the sum gives the rest 2 when divided by 3  
D)    none of the alternatives is more likely than the others

Cross out the right alternative (remember: many of the possible sums may be achieved in different ways):

6. CARROT JUICE

A juice machine is used for pressing carrot juice. The first time you press the carrots, one fourth of the juice is pressed out. For each pressing you get out one fourth of the remaining juice.

How many times do you have to press the carrots to get out at least two thirds of the juice?

A)  2 times
B)   4 times
C)   6 times
D)    8 times  
E)    it is impossible to press out two thirds of the original amount of juice, 
      regardless how many times you try.

Cross out the right alternative:

7. WHERE DO YOU FIND KARI'S HOUSE?

The distance from Kari’s house to the Town Hall is exactly twice the distance from her house to the village church. Try to find Kari’s house. Where should you search to be quite sure to find Kari’s house?  

A)    at one very point on the line between the Town Hall and the church

B)    either at a point between the Town Hall and the church, or at a point on the line through the Town Hall and the church

C)    on a circle line that has its centre in the church

D)    on a circle line that has its centre some distance behind the church

E)    on a straight line that is 90° to the line between the Town Hall and the church

Cross out the right alternative:

8. PALINDROMIC NUMBERS

A palindromic number is a number that is the same either written from left to right or written from right to left. Two examples: 1331 and 123454321

A)    Find a palindromic number less than 100.000 that may be divided by 3 and where the product of the digits is 200.

B)    How many palindromic numbers is it possible to find when the product of the digits is 200 and the integer 1 is not allowed as a digit?

Solutions:

Problem 1

A special sequence of numbers  

Problem 2

From rectangle to square

175 cm² 

Problem 3

Triangles in a hexagon

40 %

Problem 4

GO

13

Problem 5

Dice throwing 

Alternative D

Problem 6

Carrot juice

Alternative B (4 times)

Problem 7

Where do you find Kari's house?

Alternative D (on a circle line that has its centre some distance behind the church)

Problem 8

Palindromic numbers

A) 585
B) Three numbers (585, 52225 and 25252)