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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.

Note by Naitik Sanghavi
1 month, 1 week ago

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

  • Is NMTC objective or subjective?
  • Is it IITJEE or Olympiad Mathematics?
Swapnil Das · 1 month, 1 week ago

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@Swapnil Das Well NMTC screening test is objective and NMTC final test is subjective.It is Olympiad Mathematics Ayush Rai · 1 month, 1 week ago

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@Ayush Rai Done!BTW I am getting only two solutions for (Q2)!Am I correct? Naitik Sanghavi · 1 month, 1 week ago

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You have not mentioned question 5 b Ayush Rai · 1 month, 1 week ago

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I have posted Junior Level Paper here. Svatejas Shivakumar · 1 month, 1 week ago

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For question 6, use Descartes' Theorem. Akshat Sharda · 1 month, 1 week ago

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@Naitik Sanghavi 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 · 1 month, 1 week ago

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@Swapnil Das As you wish! 😊 Naitik Sanghavi · 1 month, 1 week ago

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