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

Problem 1

$ABC$ is a right angled triangle with $BC$ as hypotenuse. The medians drawn to $BC$ and $AC$ are perpendicular to each other. If $AB$ has length $1 \mathrm{~cm}$ , find the area of triangle $ABC$ .

Problem 2

(a) Find the smallest positive integer such that it has exactly $100$ different positive integer divisors including $1$ and the number itself.
(b) A rectangle can be divided into ' $n$ ' equal squares. The same rectangle can also be divided into $(n+76)$ equal squares. Find $\mathbf{n}$ .

Problem 3

Prove that $1^n+2^n+3^n+\ldots \ldots \ldots +15^n$ is divisible by $480$ for all odd $n \geq 5$ .

Problem 4

Is it possible to have $19$ lines in a plane such that (1) no three lines have a common point and (2) they have exactly $95$ points of intersection. Validate.

Problem 5

In a trapezium $ABCD$ with $AB$ parallel to $CD$ , the diagonals intersect at $P$ . The area of $\triangle ABP$ is $72 \mathrm{~cm}^2$ area of $\triangle CDP$ is $50 \mathrm{~cm}^2$ . Find the area of the trapezium.

Problem 6

Let $\mathrm{a}<\mathrm{b}<\mathrm{c}$ be three positive integers. Prove that among any $2 \mathrm{c}$ consecutive positive integers there exists three different numbers $x, y, z$ such that $abc$ divides $xyz$ .

Problem 7

(a) Let $m, n$ be positive integers. If $m^3+n^3$ is the square of an integer, then prove that $(m+n)$ is not a product of two different prime numbers.
(b) $a, b, c$ are real numbers such that, $ab +bc+ca=-1$ . Prove $a^2+5b^2+8c^2 \geq 4$ .

Problem 8

$ABCD$ is a quadrilateral in a circle whose diagonals intersect at right angles. Through $O$ the centre of the circle, $GOG^{\prime}$ and $HOH^{\prime}$ are drawn parallel to $\mathrm{AC}, \mathrm{BD}$ respectively, meeting $\mathrm{AB}, \mathrm{CD}$ in $\mathrm{G}, \mathrm{H}$ and $\mathrm{DC}$ , $A B$ produced in $\mathrm{G}^{\prime}, \mathrm{H}^{\prime}$ . Prove $\mathrm{GH}, \mathrm{G}^{\prime} \mathrm{H}^{\prime}$ are parallel to $B C$ and $A D$ respectively.

Problem 1

$ABC$ is a right angled triangle with $BC$ as hypotenuse. The medians drawn to $BC$ and $AC$ are perpendicular to each other. If $AB$ has length $1 \mathrm{~cm}$ , find the area of triangle $ABC$ .

Problem 2

(a) Find the smallest positive integer such that it has exactly $100$ different positive integer divisors including $1$ and the number itself.
(b) A rectangle can be divided into ' $n$ ' equal squares. The same rectangle can also be divided into $(n+76)$ equal squares. Find $\mathbf{n}$ .

Problem 3

Prove that $1^n+2^n+3^n+\ldots \ldots \ldots +15^n$ is divisible by $480$ for all odd $n \geq 5$ .

Problem 4

Is it possible to have $19$ lines in a plane such that (1) no three lines have a common point and (2) they have exactly $95$ points of intersection. Validate.

Problem 5

In a trapezium $ABCD$ with $AB$ parallel to $CD$ , the diagonals intersect at $P$ . The area of $\triangle ABP$ is $72 \mathrm{~cm}^2$ area of $\triangle CDP$ is $50 \mathrm{~cm}^2$ . Find the area of the trapezium.

Problem 6

Let $\mathrm{a}<\mathrm{b}<\mathrm{c}$ be three positive integers. Prove that among any $2 \mathrm{c}$ consecutive positive integers there exists three different numbers $x, y, z$ such that $abc$ divides $xyz$ .

Problem 7

(a) Let $m, n$ be positive integers. If $m^3+n^3$ is the square of an integer, then prove that $(m+n)$ is not a product of two different prime numbers.
(b) $a, b, c$ are real numbers such that, $ab +bc+ca=-1$ . Prove $a^2+5b^2+8c^2 \geq 4$ .

Problem 8

$ABCD$ is a quadrilateral in a circle whose diagonals intersect at right angles. Through $O$ the centre of the circle, $GOG^{\prime}$ and $HOH^{\prime}$ are drawn parallel to $\mathrm{AC}, \mathrm{BD}$ respectively, meeting $\mathrm{AB}, \mathrm{CD}$ in $\mathrm{G}, \mathrm{H}$ and $\mathrm{DC}$ , $A B$ produced in $\mathrm{G}^{\prime}, \mathrm{H}^{\prime}$ . Prove $\mathrm{GH}, \mathrm{G}^{\prime} \mathrm{H}^{\prime}$ are parallel to $B C$ and $A D$ respectively.

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