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Background

I am a class 10 student who is fond of maths. I like making math questions.

Question

I do not know of any platform where I can post these questions for others to practice and learn. I really want not to let them go unnoticed. Is there any well-known website or any other platform for this?

My questions require some critical thinking and are different from the ones which are found the textbooks that follow the syllabus.

Example Questions

Q - 1 Let the function $S(n)$ denote the sum of the first $n$ terms of Arithematic Progression. Given that the degree of $S(n)$ is $1$ and the first term of the AP is 10, find the $10^2$th term of the AP.

Q - 2 Let there be 2 squares, whose area is given by $[p(x)]^2$ and $[g(x)]^2$ for $x\geq0$, where $p(x)$ and $g(x)$ are quadratic polynomials.

It is given that $p(x) > g(x) \quad \forall x\geq0$

When the smaller square is placed on the larger square, such that it overlaps completely with the larger square then the area not overlapped by the smaller square is given by the polynomial $r(x) = 3x^4 + 20x^3 + 46x^2 + 44x + 15$

If the difference in side lengths of both squares is $3,8,15$ for values of $x$ as $0,1,2$, then find how many times $p(x)$ and $g(x)$ intersect on the graph. Also evaluate $\frac{p(12) - 1}{g(10)}$

Q - 3 Prove that for any natural number $x$, $1101$ cannot be the sum of $x$ and the number obtained by reversing the digits of $x$.

Q - 4 Prove that for natural numbers $a,b$ and $n$, $HCF(a^n, b^n) = [HCF(a,b)]^n$

Edit

Edit 1 - My problems are relatively straightforward for students who have taken specific courses that cover the required problem-solving techniques. My original intent was to look for an audience that had just learnt the basics and for whom these questions would require critical thinking, but now I realize I should leave that idea and post my problems for a general maths audience.

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  • $\begingroup$ Lots of math enthousiasts are looking for questions, while you want to deliver questions. Looks like a great match! What about contacting a moderator of this site? Maybe you can put it a list of questions here on the site (or on another math related stackexcchange site, or, who knows, you can create your own stackexchange site)? $\endgroup$
    – Dominique
    Commented Feb 15 at 8:04
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    $\begingroup$ @Dominique I am only a beginner, but who knows one day I may come back with something greater. $\endgroup$ Commented Feb 15 at 8:52

1 Answer 1

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Perhaps reddit.com/r/mathriddles. The problems in the tag "easy" seem somewhat similar to the type of problems you posted, and they do get upvoted, which indicates that the community is receptive to them.

And of course nothing is stopping you from putting up a blog and posting the problems there, and/or making some YouTube videos about them.


Something to aspire to: Math.SE has some tags where people often post problems that they believe are interesting & non-trivial in some way.

I think the problems you provided here are too straightforward for Math.SE -- Q1 and Q4 look like standard textbook exercises, and I'll elaborate on Q2 and Q3 below. But perhaps you could try to create some more challenging problems to post there as well. It could be a good source of inspiration to continue developing your math chops! :)

  • Q2 seems straightforward if you know how solve systems of arbitrarily many linear equations (which is covered at the beginning of Linear Algebra, and sometimes the end of Precalculus). You have $6$ unknowns ($3$ coefficients for each of $p$ and $g$) and $8$ linear equations ($5$ equations from equating the coefficients in $p(x)^2 - g(x)^2 = r(x),$ and $3$ more equations from the differences in side lengths for $x=0,1,2$), so you can just solve the system to find $p$ and $g$ exactly, and then the problem is essentially solved.

  • Q3 looks a little bit more interesting but it's still straightforward for people who have taken an Introduction to Proofs course -- they'll naturally think of this number as having digits $abc,$ and arrive at \begin{align*} 1101 &= (a + 10b + 100c) + (100a + 10b + c) \\ &= 101(a+c) + 20b, \end{align*} and notice a contradiction: $a+c$ must end in $1,$ yet also $a+c=10$ (because $\frac{1101-(9)(20)}{101} \leq a+c \leq \frac{1101}{101}$).

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  • $\begingroup$ Thanks for informing me about Reddit and the specific tags of Math.SE as a platform. Actually, I had a different solution for Q-2 in mind, which requires relatively less effort in my opinion. We can first find the polynomial p(x)-q(x) by solving the system of 3 linear equations for coefficients of p(x)-g(x), then use polynomial division on r(x) to find p(x)+g(x). Now we can find p(x) and g(x) by adding/subtracting {p(x)+g(x)} and {p(x)-g(x)} and dividing by 2, in a way similar to how we would solve a pair linear equation in two variable of form a+b=k1 and a-b=k2. $\endgroup$ Commented Feb 15 at 9:04
  • $\begingroup$ When p(x)-q(x) is known, we can find how many times p(x) and q(x) intersect on graph by equating p(x)-q(x) to 0. To evaluate [p(12)-1]/g(10) I believe we need to find the exact polynomials p(x) and g(x) $\endgroup$ Commented Feb 15 at 9:09
  • $\begingroup$ @GameTimeWithAryan that's an interesting and clever solution! Maybe you can try to tweak the problem a bit to remove information so that this solution of yours still works, but mine doesn't. If you're able to do that, then the problem might be suitable for the contest-math tag on Math.SE. $\endgroup$ Commented Feb 15 at 15:06
  • $\begingroup$ I forgot to mention that, for Q-3, it is not sufficient to only check for 3 digit numbers, we also need to verify the 4 digit case for complete proof. This is because 1000 + 0001 = 1001 < 1101 and 1100 + 0011 = 1111 > 1101 $\endgroup$ Commented Feb 15 at 16:30
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    $\begingroup$ That is a great suggestion! I will try to tweak my problem. If I am successful then I will definitely post its link of Math SE in this post $\endgroup$ Commented Feb 17 at 6:45

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