NMTC Final Inter 2016

Q1) A, B, C are three points on a circle. The distance of C from the tangents at A and B to the circle are a and b respectively. If the distance of C from the chord AB is c, show that c is the geometric mean of a and b.

Q2)Find all integer solutions to the equation x³+(x+4)²=y².

Q3)Two right angled triangles are such that the incircle of one triangle is equal in size to the circumcircle of the other. If P is the area of the first triangle and Q is the area of the second triangle, then show that P/Q≥3+2√2.

Q4) (a)Find the maximum value k for which one can choose k integers from 1,2,.....,2n so that none of the chosen integers is divisible by any other chosen integer.

(b) F(x) is a polynomial of degree 2016 such that all the coefficients are non negative and non exceed F(0).Show that the coefficient of x^2017 in (F(x))² is at most F(1)²/2.

Q5) (a) n>=3 and a1, a2, a3,....., an are different positive integers. Given that, except the first and the last, each one is a harmonic mean of its immediate neighbors. Show that none of the given integers is less than n-1. (b)Show that the shortest side of a cyclic quadrilateral with circumradius 1 is at most √2.

Q6)C1,C2,C3 are circles with radii 1,2,3 respectively, touching each other as shown in Figure. Two circles can be drawn touching all these circles. Find the radii of these two circles.

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2 \times 3

2 \times 3

2^{34}

2^{34}

a_{i-1}

a_{i-1}

\frac{2}{3}

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\sqrt{2}

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\sum_{i=1}^3

\sum_{i=1}^3

\sin \theta

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\boxed{123}

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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

@Svatejas Shivakumar @Harsh Shrivastava @Abhay Kumar @Nihar Mahajan @Alan Joel @easha manideep d @Adarsh Kumar @Swapnil Das @Mehul Arora @Dev Sharma @Akshat Sharda

naitik sanghavi - 4 years, 7 months ago

Thanks for mentioning. Do you want me post the solutions, or you were just kind enough to help me have a glimpse of the paper?

Swapnil Das - 4 years, 7 months ago

As you wish! 😊

For question 6, use Descartes' Theorem.

Akshat Sharda - 4 years, 7 months ago

I have posted Junior Level Paper here.

A Former Brilliant Member - 4 years, 7 months ago

You have not mentioned question 5 b

Ayush G Rai - 4 years, 7 months ago

I have some questions about NMTC, please care to answer:

Is NMTC objective or subjective?
Is it IITJEE or Olympiad Mathematics?

Well NMTC screening test is objective and NMTC final test is subjective.It is Olympiad Mathematics

Done!BTW I am getting only two solutions for (Q2)!Am I correct?

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}`	$\boxed{123}$