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I'm often curious where an interesting problem comes from; the only other place I find this one (besides Math Forums) is here: https://www.egmo.org/egmos/egmo13/paper-day1-English.pdf Interesting that it's dated yesterday. I don't know if that adds anything to our knowledge, except that it...

I would start by experimenting, to get a feel for what it means. Can you find a "peculiar number"? Can you find four of them? (It's conceivable that they are very rare and hard to find, so you might not be able to do this; but just trying should help you think about the problem better.) Let's...

No, that is a series whose sequence of terms converges to zero, and whose sum converges to 2. (Obviously it adds up to more than 1; I'm sure you didn't write what you meant.) But that's not related to folding a piece of paper, which you described here: The results of folding don't add up; why...

That makes sense.

Yes, that's my impression of "perpendicular". Can you confirm the meaning of "projection" in your context? (We usually talk about the projection of a point, not a line, which is part of the reason for my uncertainty.) Since the theorem of Apollonius (at least the one I know of) is not...

I wish I could tell what you're missing. You should be able to develop some level of intuition about this from enough exposure to examples, though anything involving infinity ultimately has to be left not fully explained, because, after all, we ourselves are finite. A big part of this is...

That was my first thought; but constructibility is linked to square roots, and this will probably involve cube roots, which is a different story. I am not pursuing this any deeper, because I don't have a lot of knowledge in this area.

The phrase is "regression to[ward] the mean"; neither of your phrases is standard, to my knowledge, so I don't know what you're trying to say. I recommend you read up on what the phrase means, perhaps starting with Wikipedia, where I often start (then taking links from there, or searching for...

The term is intentionally vague, I think, allowing for anything broadly analogous to the physical (3D Euclidean) space we are familiar with. This space is a set of points, related (structured) by coordinates or distances, and it's the model for what we mean by "space". So if we are thinking...

Does this help at all? https://en.wikipedia.org/wiki/Space_(mathematics) If not, what is lacking?

I would suppose that "perpendicular" in the context of a right triangle means the side perpendicular to the side being use as the base; and "projection" means the orthogonal (right-angle) projection of that side onto the base. Imagine the shadow when the sun is directly overhead. How long would...

Draw a picture? If A is a sweet that costs N10 for 2, and B is a sweet that costs N10 for 6, then I can't make a picture of 3 sweets with equal numbers of each type, but I can draw a jar of 60, with 30 of each type: AAAAA AAAAA AAAAA AAAAA AAAAA AAAAA =30 @ N10 for 2 = 15*N10 = N150 BBBBB...

No, 1 sweet does not correspond to N10, and 10+10+10+10 is not 4. Yes, this is one way to do it. Since N10 is 1/4 of N40, the number of sweets for N10 is 1/4 of the number you get for N40. But 10/40 is not 3! I suppose you meant \frac{10}{40}\times12=3, which is the correct answer. As I...

Not quite right. Give that another try! Maybe write a proportion. (You didn't use the 12.)

That's right, except that you haven't said that this is a total of 12 sweets, 6 of each type. Now what? Just to make sure we're clear, I'll rewrite what you said here a little more carefully: Six of the sweets sold at two for N10 cost \frac{10}{2} \times 6 = N 30. We have already been told...

That's a valid start (though not how I did it). What do you plan to do next? What are your ideas? You might, for example, now ask, how much will it cost for one of each? Then what is the unit cost of the mix? My own method avoided fractions, by asking how much 6 of each cost, rather than one...

Actually, they did the right division, since they got 46.66, not 0.4666. They just didn't write what they did. The "error" is in the rounding, not the dividing.

Let's keep it simple. Using a rounded decimal value changes the answer. Evaluating as a fraction instead, you get 7/0.15 = 700/15 = 140/3 = 46 2/3. The answer you wrote is an approximation of this, obtained by (inaccurately) rounding 46.666... to 46.66. This is a reason to avoid dividing by a...

Look it up: https://www.merriam-webster.com/dictionary/recourse Recourse means 1 a: a turning to someone or something for help or protection settled the matter without recourse to law b: a source of help or strength : RESORT had no recourse left 2 : the right to demand payment from the maker...

The suggestion of starting with a circle is good; assuming the cuts are perpendicular to an axis, the solution ends up being the same, as you can show by stretching. In that case, the area formula is found here: https://en.wikipedia.org/wiki/Circular_segment#Arc_length_and_area You'd use the...