2010 Camp Selection Problems – due 27th August 2010
Sunday, August 1, 2010 22:15With one IMO concluded – and so successfully for New Zealand! – it’s time to start thinking about the next.
Our first step in choosing the team to represent New Zealand at the 2011 IMO in Amsterdam is to choose 24 students to attend a week long training camp in Christchurch in January. These students will be chosen using the 2010 Camp Selection Problems (pdf, 105kb). Students submitting solutions to the Camp Selection Problems must intend to be in school in 2011; must have been born on or after 20 July 1991; and must be NZ citizens or hold NZ Resident status.
Those selected for the camp will do 1-2 assignments as preparation for sitting Round 1 of the British Mathematical Olympiad in December. At the camp in January we will choose a squad of 10-12 students for further training, and to sit the Australian and Asia-Pacific Mathematical Olympiads and Round 2 of the BMO. The final team of six will be chosen based on the results of these competions, and perhaps some additional selection tests, if needed.
We look forward to receiving your solutions!
- Chris Tuffley
Leader, 2010 NZIMO team
Important update, 11th August: At last January’s camp we decided that only year 12 students and members of the squad would be required to do just the senior problems; students in year 11 or below, who have been to a January camp but not been in the squad, can do both the junior and senior problems. The instructions for the problems have been updated to reflect this.
Files (all pdf):
- the problems only
- the instructions and registration form only
- Both in one.
George Han says:
August 4th, 2010 at 6:50 pm
I unfortunately do not understand question four in the junior division.
Can you please explain what ‘lcm’ and ‘gcd’ mean please.
Thank you
Chris says:
August 5th, 2010 at 11:32 am
Dear George,
“lcm” is short for “lowest common multiple”, and “gcd” is short for “greatest common divisor”.
In case you haven’t met these before, the lowest common multiple of two numbers a and b is the smallest positive number that is divisible by both a and b, and the greatest common divisor is the largest number that divides both a and b. For example, lcm(18,30) is 90, and gcd(18,30) is 6.
Best wishes,
Chris
Eddy says:
August 5th, 2010 at 10:39 pm
Good luck guys! Have fun with these
Tzu-chien Yeh says:
August 6th, 2010 at 6:07 pm
i need to know what i got! im pisssed
Emily says:
August 7th, 2010 at 7:59 pm
for the question with this in do you mean a= 2007 and b=2010 or do you mean it is n?
(a) n = 2007;
(b) n = 2010.
Franklin He says:
August 8th, 2010 at 12:39 pm
Can someone explain problem S1 for me? I have no idea what they are talking about.
And what’s convex?
Ian Seong says:
August 10th, 2010 at 11:28 pm
Hello~ Emily: I think what you wrote on there is correct (n=2007)
Ian Seong says:
August 10th, 2010 at 11:30 pm
Franklin: problem S1 is basically:
In a series of numbers, the n+1th term is 4 x (bigger number between nth term and 4) divided by n-1th term
Ian Seong says:
August 10th, 2010 at 11:31 pm
Convex can be defined in two ways:
1. All the internal angles are less than 180 degrees
2. If you draw a line, the line intersect the sides of the polygon at most twice
Chris says:
August 11th, 2010 at 6:54 pm
Emily – Problem J4 has two parts; in part (a) you’re to find all solutions (a,b) for n=2007, and in part (b) you’re to find all solutions (a,b) for n=2010.
The numbering of the parts was rather unfortunate, given the question! So I have changed it to (i) and (ii) rather than (a) and (b).
Franklin – max{x,y} means the maximum of x and y, i.e. whichever of x and y is the larger – so for example max{2,3}=3. So, given x_0 and x_1, to find x_2 you compare 4 and x_1, take whichever is larger, multiply that number by 4, and divide the result by x_0. You repeat this process with x_1 and x_2 to get x_3, and so on.
As Ian says, a convex polygon is one in which all the internal angles are less than 180 degrees. More generally, a region in the plane is convex if, given any two points inside the region, the line segment that joins them lies entirely within the region.
Tzu-Chien: do you actually have a question for us? I’ve no idea what you might be referring to.
Raj says:
August 13th, 2010 at 3:24 pm
Can you explain what J1 means? Does it mean that each row contains the numbers 1-8 and each column contains the numbers 1-8? and how do we know how much rice to put on each square?
Chris says:
August 13th, 2010 at 6:03 pm
Dear Raj,
the rows are numbered from 1 to 8, and so are the columns, and the row and column numbers of each square tell you how many grains of rice are on it – for example, the square at row 3, column 4 has 3×4=12 grains of rice on it.
Best wishes,
Chris
Natalia says:
August 14th, 2010 at 7:56 pm
For S6, can one number be in two pairs? e.g. {5,9} and {5,1}
What do you mean by “not necessarily disjoint”?
Matt says:
August 15th, 2010 at 9:58 am
For J6, if X knows Y, does Y know X?
Chris says:
August 16th, 2010 at 10:39 am
Natalia: Yes, one number can be in two pairs, just as in your example. This is what the “not necessarily disjoint” bit means. The pairs (5,9) and (1,5) are different, but they’re not disjoint, because they share 5; saying that the pairs need not be disjoint means that this is allowed.
Matt: Yes, in this problem we assume that if X knows Y, then Y knows X.
Best wishes,
Chris
Matthew says:
August 17th, 2010 at 5:11 pm
For question S4, does the circumcircle fit inside triangle ABC or have tangents on the vertices?
Chris says:
August 17th, 2010 at 6:03 pm
Dear Matthew,
the circumcircle of a triangle is the circle that passes through the vertices; the incircle of a triangle is the circle that is tangent to all three sides.
Best wishes,
Chris
Ian Seong says:
August 17th, 2010 at 11:11 pm
Just a warning
the solutions are due in 10 days!!!
Ian Seong says:
August 18th, 2010 at 9:54 pm
I have a question, do you want the solutions double-sided or one-sided?
Chris says:
August 19th, 2010 at 12:10 pm
Dear Ian,
it really doesn’t matter, either one- or two-sided is fine.
Best wishes,
Chris
Matthew says:
August 19th, 2010 at 5:47 pm
Hi there, once again.
For question S3, is it possible for (x,y) to be the same integer? And if not, if you invert the numbers, is it a different answer? E.g x=3 and y=4, would x=4 and y=3 be a different answer?
Chris says:
August 19th, 2010 at 8:36 pm
Dear Matthew,
yes, in a problem like this it’s possible to have x=y. In addition, if x=3, y=4 is a solution, and so is x=4, y=3, then these are considered to be different solutions. When we write (x,y) we typically mean an ordered pair. So x=3,y=4 would be (3,4), and x=4,y=3 would be (4,3), and we consider (3,4) and (4,3) to be different.
For this particular problem, the equation is symmetric in x and y, so if (x,y)=(a,b) is a solution, then (x,y)=(b,a) will automatically be one too. However, not every equation is symmetric like this, so in general we need to keep track of which number we’re substituting for which variable. In this situation, treating a solution as an ordered pair (x,y) is very helpful.
Best wishes,
Chris
hannah says:
August 22nd, 2010 at 9:09 pm
Hello
What does it mean when it says distinct integers ? (in S6)
And can you use the same number twice in a set ?
eg. have i = 1 and j = 1
Thanks
hannah says:
August 22nd, 2010 at 9:13 pm
actually don’t worry, i just understood the question (!)
Matthew says:
August 23rd, 2010 at 5:18 pm
It seems I need a lot of help here… I do not understand Question S6, could you please explain?
Emma says:
August 23rd, 2010 at 11:02 pm
Hi just asking, if i send the answers from chch to auckland, how long will it take to arrive??
Ian Seong says:
August 24th, 2010 at 7:25 pm
Emma:
It actually depends.
Last year when I sent it, it arrived in a day, because I sent it as fast post
If you send it with other posts, it might take 3 days to weeks (?)
Teddy says:
August 25th, 2010 at 10:13 pm
Hello!
Sorry to ask, how would we know if our solution papers had arrived?
Ilya Chevyrev says:
September 1st, 2010 at 12:17 am
Hi Teddy,
Sorry for late reply, we have all been away recently. Generally speaking, due to the large number of solutions, we do not individually let everyone know that we received their work until after we marked them with the results already, which should be done in a few weeks, maybe earlier. In the past I do not think we have had any issues with not receiving work from students so I wouldn’t worry about it. If after a while you are worried then it may be possible for you contact either myself or Ivan and we may be able to check if your work has arrived.
won says:
September 5th, 2010 at 10:36 pm
i have a question about the last division when i was solving before the dead line. at the last question on senior, when i calculated combinations between the integers. actually there are quiet much, which i cannot distinguish for each one. so how that is possible?
P.S could you check my one( won or wonjae.kim) arrived safely?
Ian Seong says:
September 13th, 2010 at 7:00 pm
Hello
About when do we received the mail about the result?
Hojeong says:
October 4th, 2010 at 2:39 pm
I still havn’t recieved the mail about the result… should I be worried or is it to soon?
Hojeong says:
October 4th, 2010 at 2:40 pm
I still havn’t recieved the mail about the result… should I be worried or is it too soon?
Ian Seong says:
October 4th, 2010 at 8:22 pm
I don’t think you have to be worried. The result for mine is not sent yet.