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

1. MEDAL REQUIREMENTS

In a running contest, the top five contestants completed the race in:

1 h 24 min   12 sec
1 h 25 min   10 sec
1 h 26 min     8 sec
1 h 30 min   53 sec
1 h 33 min   37 sec

Medals were handed out to those whose finishing time was less than the average time of the top five contestants + 25% of this time.

How fast did one have to run to get a medal?  

Your answer:

To get a medal one had to run faster than:

2. BUDGIES AND GUINEA PIGS

In a pet store there were a number of budgies and guinea pigs. The total number of heads and legs of these animals were 31. How many budgies and guinea pigs could there possibly have been?

The problem has two solutions, and you have to find both answers to score 5 points. 

Just one correct answer gives 3 points an one correct and one wrong answer gives 2 points.

3. SUMS AND PRODUCTS

The sum is 12. How big can the product possibly be?

There are many ways in which to write 12 as a sum of positive integers.
8 + 4 and 3 + 2 + 7 are just two of many.

If we multiply the summands of these two sums we get
8 * 4 = 32
3 * 2 * 7 = 42

Find the sum which gives the biggest possible product.

Write down your answer.

(You may choose as many as 6 was to represent 12 as a sum)  

If you find the biggest you get 5 points. The second biggest gives you 3 points, and the third biggest gives you 2 points. Any other answer gives no score.

4. “HEXANIMALS”

In this problem you attach hexagons (of the same size) to form various connected figures that we call “hexanimals”. When two hexagons are attached, the adjacent sides must cover each other completely. Two “hexanimals” are considered equal they can cover each other completely after a possible flip or rotation.

It is possible to create three different “hexanimals” with three hexagons in each figure. They are shown below.

How many different “hexanimals” can be made with four hexagons in each figure?

5. ROLLS

Espen , Per and Sigurd are having a meeting. For lunch they have rolls with cheese, jam and honey. There are equally many rolls of each kind. 

Espen, Per and Sigurd all have the same number of rolls. Espen has twice as many with cheese as Per, but only 1/3 as many with honey as Sigurd. They all get less than 5 rolls each.

How many rolls with jam did Espen get?

6. PRIME NUMBERS

There are prime numbers with the property that all the numbers you get by permuting the digits are primes as well. The prime number 199 is such a prime, because 919 and 991 are also prime numbers. Between 100 and 200 there are in addition to 199 two more primes with this property. Find these two prime numbers. You must find both numbers to score.

7. ALMOST A GIRLS' CLASS

In  the 9th. Grade at Eloks school  there are 99 girls and 1 boy. They are all gathered in the school's auditorium.

How many girls must leave the auditorium in order for there to be exactly 98% girls left in the auditorium?

8. CHOCOLATE BARS

Anne, Bente, Cecilie and Dina have 7 chocolate bars to share. Instead of trying to share them equally they start discussing in how many ways they can share the chocolate bars. Needless to say it is not the same if for example Anne gets 1 chocolate bar and the three others get 2 each or if Bente gets 1 chocolate bar and the other three gets 2 each.

In how many ways can the four girls share the 7 chocolate bars between themselves.

Each girl is to get some chocolate, and it is not allowed to break the chocolate bars into smaller pieces.

Solutions:

Problem 1

Medal requirements

One had to run faster than 1 h 50 min 0 sec:

The sum of the the 5 best results is 7 h 20 min 0 sec = 440 min, that gives an average of 88 min. 
25 % in addition gives 110 min, that is equal to 1 h 50 min

Problem 2

Budgies and guinea pigs

The 2 solutions:

2 budgies and 5 guinea pigs
7 budgies and 2 guinea pigs

Problem 3

Sums and products

The biggest possible product: 3 * 3 * 3 * 3 = 81

Problem 4

”Hexanimals”

Correct answer is 7.

Problem 5

Rolls 

If Espen gets 1 with honey and 2 with cheese, Per will get 1 with cheese and Sigurd 3 with honey. Then there are 4 rolls of each kind, so Sigurd gets 1 with cheese, Per gets 3 with jam, and

Espen gets 1 roll with jam.

Problem 6

Prime numbers

113 and 131

Problem 7

Almost a girls' class

50 girls

Problem 8

Chocolate bars

10 + 6 + 3 + 1 = 20

All possible combinations are listed here: