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 .

No vote yet

1 vote

×

Problem Loading...

Note Loading...

Set Loading...

## Comments

Sort by:

TopNewestProblem 2 :

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

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

Log in to reply

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.

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

Log in to reply

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.

Log in to reply

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

Log in to reply

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)?

Log in to reply

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

Log in to reply

thank you!

Log in to reply

Log in to reply

Log in to reply

Log in to reply

Log in to reply

Log in to reply