The Australian Informatics Competition has 15 questions. The first six are traditional multiple choice format like the first three sample questions below. The last nine will be three sets of three questions such as the Referendum question below. These nine answers will each be a number in the range 0 to 999 which students can fill in on the supplied mark-sense sheet.

The paper has a time limit of one hour. These questions are chosen to help identify a talent for computer programming, and the event is an ideal activity for a class in mathematics.

In your house is an unusually active colony of beetles. You have been watching these beetles for some time, and you have observed the following facts.

• Each week, the number of beetles doubles;
• When the beetles first came into your house, there was an odd number of beetles.

Given that there are 544 beetles in your house now, how many weeks ago did the beetles first come?

A line of eight counters needs to be placed in a row. Each counter is labelled with the number 1 , 2 , 4 or 5 . If any two counters have the same label n , they must be separated by at least n -1 other counters.

For example, the row 4 2 1 1 5 2 1 2 is allowed. Note that the first two counters labelled 2 are separated by three other counters, and the last two counters labelled 2 are separated by one other counter. On the other hand, the row 4 2 1 4 1 2 1 2 is not allowed, since the two counters labelled 4 are only separated by two other counters.

The total for a row of counters is obtained by adding together the labels on all the counters. For instance, the two example rows above have totals 18 and 17 respectively. What is the largest total possible for a row of eight counters that obeys the rule above?

David wants to pack an aeroplane with bags. He has several large bags, medium bags and small bags. Each large bag weighs 7kg, each medium bag weighs 5kg and each small bag weighs 1kg. The aeroplane has a fixed weight limit w . David must meet this weight limit exactly, and would like to use as few bags as possible to do it.

David has devised the following set of rules for packing the plane.

1. Start by placing large bags into the aeroplane one at a time, until he is less than 7kg beneath the weight limit.
2. Continue by placing medium bags into the aeroplane until he is less than 5kg beneath the weight limit.
3. Finish by placing small bags into the aeroplane until he reaches the weight limit precisely.

For example, if the weight limit were w  = 20kg, then David would proceed as follows. He would place two large bags into the aeroplane, giving a total weight of 14kg and leaving him 6kg beneath the limit. He would then place a single medium bag into the aeroplane, giving a total weight of 19kg and leaving him 1kg beneath the limit. Finally he would add a single small bag, bringing him to 20kg precisely.

Unfortunately David's method does not always use as few bags as possible. Consider a weight limit of w  = 25kg. Here David would use three large bags (weighing 21kg) followed by four small bags (coming to 25kg total). Overall David has used seven bags, but he could have made 25kg using just five medium bags and nothing else.

Your task is to work out the smallest weight limit w for which David's method does not use as few bags as possible. What is the rightmost digit of your answer? (For example, if you think the answer is 48 then you should answer 8.)

It is the year 2050. Once again the question of Australia's becoming a republic has arisen and is being put to a referendum. The prime minister (who has been in office for several decades) has devised the following rule to determine the outcome of the referendum.

The republicans have devised an advertising campaign which they believe will persuade the majority of people in a state to vote in favour. Unfortunately it is quite expensive, costing $1 for each voter in the state. They wish to advertise in enough states to pass the referendum, but they do not wish to spend any more money than they need to.

Your task is to determine the smallest amount of money that the republicans can spend and still pass the referendum. You are presented with three scenarios below. For each scenario you must give a single answer between 0 and 999, measured in millions of dollars.

As an example, consider scenario (i). In this scenario the total population of Australia is 291 million. By advertising in Qld, NSW, WA and the ACT, the republicans can win 54+38+50+36 = 178 million votes counted in favour of the referendum. The referendum is therefore passed (since 178 million is at least half of 291 million), with a total cost of $178 million. If you believe this is the smallest cost possible then you would write 178 as your answer (though in reality you can do better than this).

(i)	State		QLD	NSW	VIC	TAS	SA	WA	ACT	NT
	Population (millions)		54	38	32	17	27	50	36	37
(ii)	State		QLD	NSW	VIC	TAS	SA	WA	ACT	NT
	Population (millions)		10	9	29	60	18	27	12	14
(iii)	State		QLD	NSW	VIC	TAS	SA	WA	ACT	NT
	Population (millions)		8	18	7	17	10	50	11	12