hello everyone this is part one of my review of my Smo Junior mock paper I did not actually have a fixed number of questions that I want to discuss on the contrary I would like to just do a quick intro first and I'll keep this video length to somewhere about 20 to 25 minutes um that's because my general take is that for the Smo it is okay after about 25 minutes to just give yourself a quick breather to catch your breath little Lally as well as mentally so I do not want to make this video too long either as well as the subsequent parts so we'll see how many questions we get to but the first thing to just do a quick outline of is what sort of targets you may have okay so if you haven't already tried the junior mock paper please look in the description below feel free to try it first before coming back to this and the subsequent review videos the targets for the Smo as a whole are not fixed both in terms of score as well as what you personally want to aim for but of course goal depends on round one plus round two so we'll talk about that another time the round two cut off is usually around 15 and then below that is silver which is around 12 to 13 you have got bronze which is usually around uh 9 to 11 and you have an honorable mention which is somewhere about 7 to 9 now this literally changes every time I write it so some of you may look at my old videos and say uh didn't you say something different and the answer is yes because in my mind I'm kind of just averaging what I know of the last two or three years and it does move it's a moving average in a way so don't worry too much about the exact score except to know that for this paper I think it is also a relative tricky paper so I wouldn't expect it to be in a hypothetical situation that the cut off be much higher than the lower ends of the spectrum now what this means is that for example if your aim is that I'm new to the Smo this the first time I'm happy with let's say a bronze will be great now that you could get relatively safely you something about 10 questions and therefore the amount of time you have is quite plentiful therefore whatever Pace we go in these videos is just for me to talk a bit about the questions as well as talk a bit about the Smo as a whole but the solutions themselves are pretty short you have quite a lot of time to find them so in your experience during the Smo you'll spend more time thinking about what to do than actually doing it okay so this is just a quick word on how the Smo would be especially since this is the Smo Junior video and many of you are taking the Smo Junior for the first time all right so let's oops B uh let's go straight to the first question uh exactly one of the following is a prime number uh which one is it multiple choice questions in the Smo allow you to have a little bit of of chances to do guess work elimination intuition it's not guessing in terms of random guessing but it's that it's multiple choice for a reason so for this one prime numbers are very hard to verify to be prime but it is very easy to verify that something is not prime so just do it by elimination uh Parts options B and D are even they're not prime option A is a ending in five that is certainly not Prime and for c and e the next easiest divisibility test to apply is three and the test is of course adding up the digits and seeing whether that total is divisible by three this is 22 and this is 30 so this is divisible by three this is not divisible by three and therefore you can eliminate option e 4567 should be primed by elimination now like many problems actually a question Setter like myself will be doing it differently from you so I would just need to check that 45567 is really Prime in order for the question to be accurate you do not need to track that as long as you know that the rest of them are not prime that's good enough for you second question 150 students set for a mock Smo Junior paper um all scores are between 0 and 25 inclusive like a an actual Smo Junior they total score is 2024 what is the maximum number of students who could have obtained the same score if you have immediate intuition for a question like this good for you if you do not have immediate intuition for a question like this look at options and you see it goes to 148 now obviously 150 would not work because 2024 is not a multiple of 50 but let's try and understand why 149 will not work now if 149 students obtain the same score what I want to do is to just divide this by 149 and I would get 13 remainder 87 for calculations like this I'm just shortcutting it in my explanation but obviously obvously you would take more than 5 Seconds to write it down right I set the question so I know the values don't worry about calculating too fast anyway this explains why it's not possible because if there were 149 students who scored 13 the last guy can't possibly score 87 if you get 87 for the Smo Junior it just means that the Smo system has messed up so it is the same logic if I change this to one 148 can I get it well if I divide by 148 I should still get 13 13 * 148 is going to have the remainder go up by 13 so goes up to 100 now this still doesn't work because I would not be able to have the other two students collectively score 100 points same if I adjust it up certainly three students cannot get a total of 113 four students cannot get a total of 126 and five students cannot get a total of 139 but when you go to 144 what happens is that it's not 13 remainder 152 it overflows to 14 remainder 2 so 14 sorry not remainder two remainder eight 152 - 144 is 8 but uh doesn't matter really the idea is that the other six students obviously can get a total score of eight it's not very impressive but it is mathematically possible so therefore our answer is a question three now I put question three here apart from the content of it it is also a meta message that I'm trying to convey please do not assume that the multiple choice questions are the five easiest questions of a Smo paper no one promise that this is not like let's say uh psle where the first 10 questions in your multiple are supposed to be one Mark questions for example because they are easiest and they multiple choice now these ones are multiple choice but everything's wor one Mark they may be easy they may not be easy and this is certainly not easy now in terms of the question itself what I want to say is that equations that you don't really have any clear understanding about just try to rearrange them to be nicer and nicer until you can figure out what to do so the first thing I'll do is add the two fractions common denominator and then I will cross multiply and I see XY is in three of the options there but looking at this I have no way of telling whether it is positive or negative rational or irrational since I have no way of telling whether it is any of these I shall continue trying to solve it but how exactly do I continue to solve it the idea comes from looking at x over y you see this is a quadratic equation where you have got y^2 x s and X Y everything is power two X and Y also have two things but I can make it look like X over y by dividing by Y2 and this would give you a quadratic equation in terms of X over y now take note that this is a quadratic equation in terms of X over y that you cannot factorize and that would already imply option e anytime when you cannot factorize a quadratic it means that it doesn't have any rational roots and since the question tells X Y are real numbers we do not need to worry about what if there's actually no real Roots at all it does exist because the question tells me so so this is enough for those of you who know the quadratic formula you would be able to compute that the value of x over y would be Min -1 plus or minus 5/2 which certainly is irrational it also tells you that depending on your choice of X and Y uh it is possible for this to be rational or irrational it is possible for this to be positive or negative not an easy question problem number four if you're looking at the diagram looks like a right angle triangle is it a right angle triangle it turns out it doesn't matter it could be it could not it may be it may not be but it doesn't help you at all so whether it is or it isn't is largely irrelevant to the problem instead what is more relevant apart from the lengths that we have so this is 13 and this is five apart from the lengths we have got a pair of parallel lines now during my classes I often will tell my students about similar triangles always come from parallel lines in two patterns pattern number one is a so-called pyramid pattern if you have got parallel lines like this then the small triangle is going to be similar to the whole thing pattern number two is this So-Cal hourglass pattern where the top half is going to be similar to the bottom half similar triangles mean same shape different size and same shape means same angles so in this case because of corresponding angles and in the second case because of alternate angles we do know that they are similar the use of similar triangles is that you can move ratios around and that's what this question is all about the first pattern of the pyramid appears in ad and ABC now if AE to AC is 5 is to 13 then the 5 is 13 will also apply to well I guess I should say 5 is to 8 for this one because it's a bit difficult to Mark out but for de to BC that is also 5 is to 13 now they are not the lengths being 5 and 13 It's just in terms of ratio and that's enough for me to use my second pattern the second pattern of course being the for mentioned hourglass pattern The Hourglass pattern here means that the 5 is to 13 carries to even more places namely DF to FC and EF to FB that's quite a lot now all of this being implemented we have no more information we need to find we can just go straight for the area of ACF start with ABC and First Slice it into ADC ADC is 5 over 13 of the triangle because the base is 5 over 13 of the whole triangle so 5 over3 13 of 39 is 15 that's ADC and ACF is going to be 13 over 18 of what we have just gotten value once you comput it is 65 over 6 that's option C question five now in the Smo one of the general trends that distinguishes it from the primary school Olympiad is simply quite a lot of algebra now all the questions that use algebra will require the same few identities that could be geometry questions counting questions number Theory questions but there will be problems like this that are particularly challenging your algebra fluency I suspect that working with the cube roots is not going to be very enjoyable and also quite difficult to do accurately I also suspect that some of you would try to guess X and Y and you are not going to find nice values of X and Y and therefore we should do something rather different but if you want to do all of the algebraic manipulations with cube roots why not get rid of them first instead of cubing this substitute it away now no one ever um has forbidden you from changing the names of your variables as long as you do it consistently so a minus B is 2 a cubed minus B Cubed is 20 and x² is the square of a cub so it's a the 6 + b the 6 is what we want to find I can almost get this by squaring if I Square this I have a to the 6 and B to the 6 as well as a junk term in between that tells me I need to find AB so to find a I use the first two equations firstly by factorizing using the difference of Cubes identity to conclude that a 2 + a + b 2 = to 20 / 2 which is 10 at the same time when I square a minus b a 2 - 2 a b + b 2 is = 4 so I subtract the two equations 3 a b = 6 AB = to 2 and therefore this is equal to 416 now with that we are able to proceed on to the short answer questions and the short answer questions do not necessarily progress on in difficulty from the multiple choice right usually not always but usually question six kind of resets to a quote unquote easy question which I may have failed in slightly because based on the statistics I got from the submitted answer so far um a lot of you found question six to be difficult so that allows me to bring in one of the editors that always feel is important for the asmo now we always talk about how the is a difficult contest which is true because a contest where 50% is a very good score is certainly a difficult one but that doesn't mean you should go into the Smo thinking about each question in isolation being very difficult so you have heard of people running a marathon and just breaking it down into let's say one minute intervals or 100 m intervals or 1 kilm intervals like something that they consider manageable and they tell the oh m is easy I just need to run that a lot of times I'm not sure whether running 4 and 2200 m is going to be easy but you don't think that you think of 100 m at a time so likewise when you see a question ask yourself why is this question easy assuming you are stuck ask yourself why is it easy in this case it is easy because 26 is just really big it's a big digit sum because it's such a big digit sum it's quite unlikely and because it's quite unlikely it is possible to list listing is not wrong it's only bad if there is a lot to list you see if there are three digits two digits you can forget about three digits just means that your maximum is 9 + 9 + 9 is 27 this is 26 so you get to drop off one and it would be 998 9989 899 for fourth digits the first digit has to be one it can't be two because if the first digit is two second digit is zero that's not enough so you have got it starts with one and now these have to sum to 25 which is still very close to 27 and so you have to take away two from the 999 that would either be 7 9 and 9 or 88 and 9 that's all answer is there are nine of them all right let's do one more question and we will conclude this section of the video so we have got number seven which looks pretty scary and I'll just give a quick suggestion as to what you can do now the first thing is that the substitute directly and then calculate X the power of 4 is not explicitly wrong uh if you know the binomial theorem it can work however it is a little bit scary now I noticed that 256 is actually going to be 4 the^ of 4 so this is x 4 + 4x the^ 4 now what I'm curious about is what is 4X that might be easier to compute and so I am going to go for that the way we simplify this is using the conjugate the conjugate of a combination of square roots is that you flip the sign of one of them and this works well because of your difference of squares identity at the bottom you'll be left with 24- 20 which is four which cancels off that's a very sweet sound every time we say it cancels off we know we on the right track because it means that X is this 4X is this and so I can start from x + 4 x is TW is 2 > 24 which is 4un 6 and then I shall Square this you get x2 + 8 + 16 / x^2 is equal to 96 move this over and then Square again now this is how you would avoid the binomial theorem it is of course not wrong to use the binomial theorem it's only contingent on whether you know what it says okay and so your answer is 7712 the Smo answers are up to five digit whole numbers for the short answer section so 0 to 99999 as I indicated in the answer form so this is um about the limit on the size of computations which you will need to do on your own just remember that there's no calculators and so if you tend to make a bit more careless mistakes while Computing just do it twice to make sure that you don't mess up all right and this is where I'll conclude the first part of the video thanks for watching and see you soon for the second part of our review of this paper