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NMTC 2018 Stage II - BHASKARA (Class 9, 10) - Problems and Solutions

Problem 1

is a right angled triangle with as hypotenuse. The medians drawn to and are perpendicular to each other. If has length , find the area of triangle .

Problem 2

(a) Find the smallest positive integer such that it has exactly different positive integer divisors including and the number itself.
(b) A rectangle can be divided into ' ' equal squares. The same rectangle can also be divided into equal squares. Find .

Problem 3

Prove that is divisible by for all odd .

Problem 4

Is it possible to have lines in a plane such that (1) no three lines have a common point and (2) they have exactly points of intersection. Validate.

Problem 5

In a trapezium with parallel to , the diagonals intersect at . The area of is area of is . Find the area of the trapezium.

Problem 6

Let be three positive integers. Prove that among any consecutive positive integers there exists three different numbers such that divides .

Problem 7

(a) Let be positive integers. If is the square of an integer, then prove that is not a product of two different prime numbers.
(b) are real numbers such that, . Prove .

Problem 8

is a quadrilateral in a circle whose diagonals intersect at right angles. Through the centre of the circle, and are drawn parallel to respectively, meeting in and , produced in . Prove are parallel to and respectively.

Problem 1

is a right angled triangle with as hypotenuse. The medians drawn to and are perpendicular to each other. If has length , find the area of triangle .

Problem 2

(a) Find the smallest positive integer such that it has exactly different positive integer divisors including and the number itself.
(b) A rectangle can be divided into ' ' equal squares. The same rectangle can also be divided into equal squares. Find .

Problem 3

Prove that is divisible by for all odd .

Problem 4

Is it possible to have lines in a plane such that (1) no three lines have a common point and (2) they have exactly points of intersection. Validate.

Problem 5

In a trapezium with parallel to , the diagonals intersect at . The area of is area of is . Find the area of the trapezium.

Problem 6

Let be three positive integers. Prove that among any consecutive positive integers there exists three different numbers such that divides .

Problem 7

(a) Let be positive integers. If is the square of an integer, then prove that is not a product of two different prime numbers.
(b) are real numbers such that, . Prove .

Problem 8

is a quadrilateral in a circle whose diagonals intersect at right angles. Through the centre of the circle, and are drawn parallel to respectively, meeting in and , produced in . Prove are parallel to and respectively.

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