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# NMTC 2017 Stage II - BHASKARA (Class 9, 10) - Problems and Solutions

###### Problem 1

(a) Find all prime numbers such that and are also primes.
(b) Determine real numbers such that

###### Problem 2

If are distinct real numbers such that

find the numerical value of .

###### Problem 3

and are triangles such that and . If and , without using Trigonometry, prove that .

###### Problem 4

(a) a, b, c, d are positive real numbers such that . Prove that

(b) In a scalene triangle . The bisectors of the angles and meet the opposite sides in and respectively. Prove that the circle on as diameter passes through the point .

###### Problem 5

(a) Prove that cannot be expressed as a product of two polynomials of degree 2 with integer coefficients.
(b) segments are marked on a line. Each of these segments intersects at least other segments. Prove that one of these segments intersects all other segments.

###### Problem 6

If are positive real numbers such that and , find the value

###### Problem 1

(a) Find all prime numbers such that and are also primes.
(b) Determine real numbers such that

###### Problem 2

If are distinct real numbers such that

find the numerical value of .

###### Problem 3

and are triangles such that and . If and , without using Trigonometry, prove that .

###### Problem 4

(a) a, b, c, d are positive real numbers such that . Prove that

(b) In a scalene triangle . The bisectors of the angles and meet the opposite sides in and respectively. Prove that the circle on as diameter passes through the point .

###### Problem 5

(a) Prove that cannot be expressed as a product of two polynomials of degree 2 with integer coefficients.
(b) segments are marked on a line. Each of these segments intersects at least other segments. Prove that one of these segments intersects all other segments.

###### Problem 6

If are positive real numbers such that and , find the value

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