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TL;DR:

Tell me which topics should i study the most, based on this three tests:

Mathematics (A): 2020 2019 2018

This question may sound a bit weird, since the natural answer would be "study whats already is on the test" but i'm wanna share a bit of context for this.

Context:

So i'm currently studying for the Japanese Scholarship known as MEXT in which every year the japanese embassy of each country gave the opportunity to enroll on a japanese university and go to study in that country. As an undergraduate student this is a big deal specially if there's a lack of opportunities in your current country.

I also want you to know that here in my country, there is a lack of proper mathematical education when it comes to high school, pretty much anyone who obtained a bachelor degree's are not well prepared to face one of this japanese test designed for undergraduate students since the test, present to you calculus, trigonometry, modular-arithmetic, number-theory, etc. problems which you don't see until you reach third semester of any university engineer-based career (at least here where i live)

Finally i want to tell that i've been studying calculus for a bit and i do know the basics and i don't have any problem with arithmetic or algebra.

So here is the question:

Could somebody look at those test and tell me which topics should i study the most? Below (in the comment section) you will find the three test that were used in past examinations and contains all the questions, i also will leave a link to the answers in case someone want to deep onto this:

Current approach and status:

In my current approach i tried solving most of the exercises presented and always fall in the same loop of no knowing, research, try solving, fail, research again, try solving again, failing, and finally making a question here. This of course may not be the best for a huge variety of topic since you have to make sure to understand what you are doing before going into any kind of examination but with so many themes its hard to pick one thing to study since you're pretty much dropping everything else in order to solve only one thing.

flow chart

The ideal:

When making this post i'm hoping someone could orient me through a path of things i should know and preferably master in order to be well prepared for the test.

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  • $\begingroup$ 2020 → Answers | 2019 → Answers | 2018 → Answers $\endgroup$
    – Carlos
    May 20, 2022 at 20:58
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    $\begingroup$ I don't know how to answer this. I think I'd have to see you trying out a problem to figure out where your weaknesses are. I am willing to do that with you once if you'd like. You may contact me at mathanthologyeditor on gmail. $\endgroup$
    – Sue VanHattum ♦
    May 21, 2022 at 1:01
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    $\begingroup$ To me, it is. It is asking 'how do I learn math more effectively'? And I think that's an important question for us to continue to address. Math education definitely includes our attempts to educate ourselves. $\endgroup$
    – Sue VanHattum ♦
    May 21, 2022 at 15:32
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    $\begingroup$ From your description, I don't think anyone can help much. Which topics you study won't matter nearly so much as studying more effectively. $\endgroup$
    – Sue VanHattum ♦
    May 21, 2022 at 21:51
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    $\begingroup$ I'm sympathetic to your concern, but the issue is significantly similar to asking for a detailed analysis of bipedal locomotion in application to walking down a stairs. :) Namely, if you think about it too much, it's just impossible. Very hard to make a robot do it. But, with practice, we can do it. Even small children. :) $\endgroup$
    – paul garrett
    May 22, 2022 at 4:15

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You have described your situation clearly. But you have asked the wrong question. I don't think your problem is lack of knowledge (which topics to study). It is a lack of understanding of how to play around (so how to study is actually a good question). Most of these problems need a creative exploration. (There were a few that I would throw my hands up at, but not many.)

I recommend that you check out Art of Problem Solving. They teach a number of math courses, but they also teach courses to prepare students for math competitions. It looks like this is the lowest level of those courses. If the $\$360$ pricetag is not reasonable to you, then you might consider The Art of Problem Solving, Volume 1, a textbook to help students get started on creative problem solving. That is $\$47$ for an online book only. Still too much money? They have free resources in Alcumus. If you follow some of the questions and answers there, you might learn something just from that. The people at AOPS are dedicated to helping students learn to think creatively in mathematics.

There is also a wonderful book by Paul Zeitz, The Art and Craft of Problem Solving. Here are some copies on a site that sells used books. Or, if you live in a big city, maybe you could find it in a library?

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    $\begingroup$ I have a technical question for SE experts. When I tried to type 360 dollars above with a dollar sign, it messed with the layout. Weirdly the $47 is fine. I tried to make some minor changes, but I have no idea what was causing that, and the only way I found to fix it was to ditch the dollar sign. (And it did it here in the comment too!) Who can help me? $\endgroup$
    – Sue VanHattum ♦
    May 22, 2022 at 0:49
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    $\begingroup$ \$ triggers LaTeX math, so it must be backslashed. I'm guessing there was interaction between the two \$-signs. $\endgroup$
    – Joseph O'Rourke
    May 22, 2022 at 12:23
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Carlos, Sue has a nice answer. To add to it, there are probably a lot of problems in regular algebra 2, trig, pre-calc, and calc books that would help you, if you concentrate on those that are the harder part of the review exercises. Usually those are with larger numbers. Most homework exercises start with plug and chug and move to those needing a big more trickery. Or sometimes they are "starred".

You seem to have very good English, so I think lots of math books in that language would be helpful. Look online or in libraries. Those that have answers in the back are more useful (can check your work, or sometimes seeing the answer will help you figure out how to do the problem. MSE is fine, but it is not efficient to go there for every hurdle.) And I'm not suggesting to look in the back until you've solved it (to check) or given it a solid try (do the whole section of problems before checking...often your subconsious will figure it out whjile working on other problems or even a nearby problem will have an insight to help you.)

By the way, that is a beautiful description of your issues, especially with the flow chart. But I think you lack some facility/familiarity with doing a lot of manipulations. Those problems you showed are not ball-buster hard. I looked up your MSE questions and found this one:

https://math.stackexchange.com/questions/4444621/help-with-frac-sin-theta-cos-theta-sin-theta-cos-theta

The key insight was sinsq + cossq =1. About the most common trig identity around! But you didn't try it, or mention trying it. When I see you missing that, I think you haven't done enough "trig identity playing around with" problems in general. Lots of the regular trig books will have a bunch of these style problems.

I'm far from a math jock, but some of those Japanese questions don't look THAT hard. Like 1-2 on the 2020 exam was one about minimum distance from a Cartesian coordinate point to a curve. Did you try just putting in the distance formula and minimizing? Take a derivative and set to zero to find extrema...basic min/max stuff! You probably can ignore the square root and just minimize the argument inside. If that's an issue, I wonder if you've done enough min/max problems in general. By the way, I just pulled down my AP Calculus book from the 80s (Thomas Finney) and a completely analogous problem is there (#17). I would think this is a pretty normal problem in calculus, really. And really the whole section of 40 problems are excellent. Lots of word problems (they are "harder" and make you think, also.) Not a single problem that is plug and chug.

So, net/net, I don't think you are that rock solid on algebra, trig, and calculus, at least as commonly taught. I think doing more problems in regular books will help you. Fine if you want to emphasize the "back of the section" ones more. But I wouldn't be surprised if some drill with the middle of the section or even the beginning of the section (at least to get more so these things are familiar) wouldn't hurt either.

P.s. I think there's a general issue of students in general thinking that "I get it" is sufficient. The human mind is not a perfect computer, not a rules based system. "I get it" is just the start. You need to build familiarity by doing large volumes of problems. sinsq plus cossq should be as familiar to you as the roof of your mouth is to your tongue. Or like if you were learning the hip throw in judo...I get it is the start...we need to do a lot of them and develop comfort/familiarity.

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