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

#Algebra

Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

Markdown	Appears as
`italics` or `_italics_`	italics
`bold` or `__bold__`	bold
- bulleted - list	bulleted list
1. numbered 2. list	numbered list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1 paragraph 2	paragraph 1 paragraph 2
`[example link](https://brilliant.org)`	example link
`> This is a quote`	This is a quote
# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"	# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"

Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

Problem 2 :

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

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

Aditya Narayan Sharma - 5 years, 2 months 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.

$\mathbf{Clarification : } \text{ {.} denotes fractional part}$

Aditya Narayan Sharma - 5 years, 2 months 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.

Ameya Daigavane - 5 years, 2 months 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 $(p = [p] + 0.5 = 10k+1.5)$

Aditya Narayan Sharma - 5 years, 2 months ago

@Aditya Narayan Sharma – You may post the next problem.

Aditya Narayan Sharma - 5 years, 2 months ago

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

A Former Brilliant Member - 5 years, 2 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 Bangalore

Aditya Narayan Sharma - 5 years, 2 months ago

thank you!

A Former Brilliant Member - 5 years, 2 months ago

@A Former 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

Aditya Narayan Sharma - 5 years, 2 months ago

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

A Former Brilliant Member - 5 years, 2 months ago

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

Aditya Narayan Sharma - 5 years, 2 months ago

@Aditya Narayan Sharma – Bdhoi amra pathaini ..otai bl6a

Sayandeep Ghosh - 5 years, 2 months ago

@A Former Brilliant Member – Yeah it must give u

Sayandeep Ghosh - 5 years, 2 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$