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# 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 Sharma
5 months, 3 weeks ago

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Problem 2 :

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

$$\text{Show that : } (n-1)(I_n+I_{n-2}) = 2sin(n-1)\theta$$ · 5 months, 3 weeks ago

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.

$$\mathbf{Clarification : } \text{ {.} denotes fractional part}$$ · 5 months, 3 weeks ago

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. · 5 months, 3 weeks ago

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 $$(p = [p] + 0.5 = 10k+1.5)$$ · 5 months, 3 weeks ago

You may post the next problem. · 5 months, 3 weeks ago

Great initiative! Do you know how to register for the exam and what are the list of test centers(I am from Bangalore)? · 5 months, 3 weeks 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 Bangalore · 5 months, 3 weeks ago

thank you! · 5 months, 3 weeks ago

Oh bdw check my note of online junior mathematician search. It's an o online contest. Lasts til 10th may. Join it plz · 5 months, 3 weeks ago

I think that I already submitted the answers. I didn't get any message after the submission though. · 5 months, 3 weeks ago

Oh when diu submit? Was it shown that your response has been recorded? · 5 months, 3 weeks ago

Bdhoi amra pathaini ..otai bl6a · 5 months, 3 weeks ago