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    Hey!

    This year I am hoping to be selected as part of a team participating in the UKMT Senior Team Maths Challenge (TMC) - I am currently in S6 in Scotland, studying Advanced Higher Maths.

    I was wondering if anyone here had participated in the challenge before, and if they had any tips on important things to know going into the challenge?

    Currently our maths teacher is providing us with practice questions via Facebook which we are diligently working out, but if anyone had any more general ideas of what types of questions are likely to appear, and important things to learn to help us deal with them? (learning prime numbers, fibonacci sequences, etc.)

    Any advice or help would be much appreciated!
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    Our team learned the squares up to 1000 (go up to 2016 if you can but that might be a bit much), cubes up to 2016, Fibonacci sequence up to 2016, prime factorisation of 2016, nearest few primes either side of 2016, how to write 2016 in base 2 (maybe learn base 8/16 as well?). There might have been some other things too, but I can't remember them. A few useful websites can be found here and here.

    Since you're in pairs for everything it could be helpful to split the memorising between you and the person you're with - for example, one person in each pair learns squares/cubes and the other person learns everything else.
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    That's a good idea, they do a lot with the year of the challenge - I know that I've seen a lot of questions on things like sequences, primes etc - anyone have any good links for any learning on that we would need to know?

    (thanks for the links too, very helpful!)
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    (Original post by Sensei_Stig)
    That's a good idea, they do a lot with the year of the challenge - I know that I've seen a lot of questions on things like sequences, primes etc - anyone have any good links for any learning on that we would need to know?

    (thanks for the links too, very helpful!)
    Ooh yes, learning Primes up to 100 will be useful too. You probably know most of them already. Learn divisibility tests for multiples of 2, 3, 5, 7, 11 and 13 to help with numbers above that.
    Spoiler:
    Show
    2: ends in 0, 2, 4, 6, 8

    3: sum of digits is multiple of 3

    5: ends in 0 or 5

    7: Subtract 2 times the last digit from the rest. If the result is divisible by 7, then the original number is divisible by 7.
    e.g. to test if 483 is divisible:
    48 - (3*2) = 42, which is divisible by 7. So 483 is also divisible by 7.

    11: Find the alternating sum of the digits. If the result is divisible by 11, then the original number is divisible by 11.
    e.g. to test if 918082 is divisible:
    9 - 1 + 8 - 0 + 8 - 2 = 22, so 918082 is divisible by 11.

    13: Add 4 times the last digit to the rest. If the result is divisible by 13, then the original number is divisible by 13.
    e.g. to test if 637 is divisible:
    63 + (7*4) = 91, which is divisible by 13. So 637 is also divisible by 13.
    For sequences, try to look at the first 5-10 terms and look for a pattern: often it'll go in a loop, or it'll be a sequence you're familiar with already. I can't think of anything specific you need to learn/practise for sequences. You're probably already familiar with infinite geometric series (S_\infty = \dfrac{a}{1-r}), which would probably be the hardest thing they can ask you. Remember nobody is expected to know much more than the very basics past GCSE.


    I decided to write up some more techniques (not related to primes/sequences) in case you're interested. The following are probably a better use of your time to learn than memorising lists of numbers:



    To square a number ending in .5, for example 3.5, multiply the number below it (3) by the number above it (4) and add a quarter (12.25). You can do a similar thing to calculate squares of 15, 25, 35 etc.

    To get from n^2 to (n+1)^2, add n and then add n+1. You can do the same thing to get to a consecutive square below one you already know.

    An example using both of these: What's 46^2?
    Spoiler:
    Show
    45^2 = 40 \times 50 + 25 = 2025

    46^2 = (2025 + 45) + 46 = 2070 + 46 = 2116
    Other tricks:
    Multiplying 2-digit numbers (and above) by 11:
    Spoiler:
    Show
    Two digits:
    Add the digits together and put it in the middle. So 54*11 = 594 and 23*11 = 253. If they add to more than 9, carry the 1 over to the hundreds column. e.g. 68*11 = 748.

    More digits:
    Add adjacent digits and put them inbetween where they were before, just like you were making Pascal's triangle. Again, carry any 1s if necessary. For example; 1238*11 = 13618.
    Circle theorems:
    Learn the most common ones - tangents meet at right angles, opposite angles in cyclic quadrilateral, alternate segment theorem etc. You'll find them helpful in geometry questions.

    Sine rule:
    I can't remember this being useful, but just in case:
    \frac{a}{\sin A} = \frac{b}{\sin B} =  \frac{c}{\sin C} = d
    Where d is the diameter of the circumcircle of the triangle.

    Pythagoras' theorem:
    This is really useful to find unknown lengths. Question 8 in the national final group round 2013/2014 is a great example of this.
    Spoiler:
    Show

    Try it, then take a look at my solution (a bit messy, sorry!):
    Spoiler:
    Show


    Not much working need for a 16-minute question!

    You could also use the sine rule here:
    \sin C = \frac{8}{10}

    d = 10 \div \frac{8}{10} = 25/2

    R = \frac{d}{2} = \frac{25}{4}

    I'll to think of more if I can; it was a while since I did this. Let me know if you have any other questions!
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    (Original post by TLDM)
    Our team learned the squares up to 1000 (go up to 2016 if you can but that might be a bit much), cubes up to 2016, Fibonacci sequence up to 2016, prime factorisation of 2016, nearest few primes either side of 2016, how to write 2016 in base 2 (maybe learn base 8/16 as well?). There might have been some other things too, but I can't remember them. A few useful websites can be found here and here.

    Since you're in pairs for everything it could be helpful to split the memorising between you and the person you're with - for example, one person in each pair learns squares/cubes and the other person learns everything else.
    Lol.
    The divisibility tests would be enough I think.


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    (Original post by physicsmaths)
    Lol.
    The divisibility tests would be enough I think.
    Yeah, you're probably right. I think some of the things about finding squares I talked about can be good though, because you can use those two to help find a lot of larger square numbers quite quickly, which means you don't need to memorise anything! I should have suggested doing that really. My mistake.

    I think the nearest 2-3 primes as well as 2016's prime factors might come in useful as well, it's likely they'll ask something to do with the properties of 2016.
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    Wow, this is amazing, thank you so much for your help! I'll be sure to pass all this on to the team
 
 
 
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