AMTI Practice corner !

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 .

AMTI, RMO, NMTC

Note by Aditya Narayan Sharma
2 years ago

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Problem 2 :

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

\(\text{Show that : } (n-1)(I_n+I_{n-2}) = 2sin(n-1)\theta\)

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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.

\(\mathbf{Clarification : } \text{ {.} denotes fractional part}\)

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Let \( f(x) = x^2 - 3x\{x\} \).

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 - 2 years ago

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Absolutely ! this is the standard approach. Let me just share another view which specifies the integers also.

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

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 \((p = [p] + 0.5 = 10k+1.5)\)

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@Aditya Narayan Sharma You may post the next problem.

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Great initiative! Do you know how to register for the exam and what are the list of test centers(I am from Bangalore)?

Brilliant Member - 2 years ago

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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 Bangalore

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thank you!

Brilliant Member - 2 years ago

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@Brilliant Member Oh bdw check my note of online junior mathematician search. It's an o online contest. Lasts til 10th may. Join it plz

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@Aditya Narayan Sharma I think that I already submitted the answers. I didn't get any message after the submission though.

Brilliant Member - 2 years ago

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@Brilliant Member Oh when diu submit? Was it shown that your response has been recorded?

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@Aditya Narayan Sharma Bdhoi amra pathaini ..otai bl6a

Sayandeep Ghosh - 2 years ago

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@Brilliant Member Yeah it must give u

Sayandeep Ghosh - 2 years ago

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