I have heard about NMTC and AMTI , and this year I will be giving an olympiad for the first time . So let's have a AMTI practice corner, as the NMTC is 2 or 3 months from now.

Please contribute as many problems and their various solutions so as to help every other aspirant !!!

I am posting the first problem here .

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TopNewestProblem 2 :

\(I_n = \int \frac{cos(n\theta)}{cos\theta}\)

\(\text{Show that : } (n-1)(I_n+I_{n-2}) = 2sin(n-1)\theta\) – Aditya Sharma · 8 months ago

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Problem 1 :

Prove that there are infinitely many reals \(p\) such that \(p(p-3\text{{p}})\) is an integer where p is not an integer.

\(\mathbf{Clarification : } \text{ {.} denotes fractional part}\) – Aditya Sharma · 8 months ago

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Consider the interval \( [n, n + 1] \) for integer n.

\( f(n) = n^2 \)

\(f(n + 1) = (n + 1)^2 \)

For \( p \in [n, n + 1], f(p) = p^2 - 3pn \) is continuous, and hence takes every value between \( f(n) \) and \(f(n+1) \), by the intermediate value theorem.

But we can then find \( p \) such that \( f(p) = n^2 + 1 \) or whatever integer between \( n^2 \) and \( (n + 1)^2 \).

And since there are infinitely many \( n \), we are done. – Ameya Daigavane · 8 months ago

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\(f(p) = p(p-3\text{{p}}) = p([p] - 2\text{{p}})\)

Since f(p) is an integer and product of two factors wher p isn't a integer. So we must have ([p] - 2{p}) a integer as two non-integers never multiply to be a integer.

\([p] \in \mathbb{I} \implies \text{2{p}} \in \mathbb{I}\)

It's clear that {p} = 0.5 for 2{p} to be a integer .

2{p} = 1 .

So p = [p] + 0.5. we may write p as \(\frac{\bar{xyz....5}}{10} = p\) as it's non-integer part is 0.5.

so f(p) = \(\frac{\bar{xyz....5}}{10}([p]-1)\implies 10|([p]-1)\)

So \([p] = 10k+1\) \(k \in [0,\infty)\)

So we have infinitely many integers p of the form \((p = [p] + 0.5 = 10k+1.5)\) – Aditya Sharma · 8 months ago

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– Aditya Sharma · 8 months ago

You may post the next problem.Log in to reply

Great initiative! Do you know how to register for the exam and what are the list of test centers(I am from Bangalore)? – Svatejas Shivakumar · 8 months ago

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– Aditya Sharma · 8 months ago

Yeah. NMTC is the first step. Visit amtionline.com and check the toll free number there . the forms are available at the first week of June. It's an all India test and the centres will definitely include BangaloreLog in to reply

– Svatejas Shivakumar · 8 months ago

thank you!Log in to reply

– Aditya Sharma · 8 months ago

Oh bdw check my note of online junior mathematician search. It's an o online contest. Lasts til 10th may. Join it plzLog in to reply

– Svatejas Shivakumar · 8 months ago

I think that I already submitted the answers. I didn't get any message after the submission though.Log in to reply

– Aditya Sharma · 8 months ago

Oh when diu submit? Was it shown that your response has been recorded?Log in to reply

– Sayandeep Ghosh · 8 months ago

Bdhoi amra pathaini ..otai bl6aLog in to reply

– Sayandeep Ghosh · 8 months ago

Yeah it must give uLog in to reply