How Cheenta works to ensure student success?
Explore the Back-Story

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

Join Trial or Access Free Resources
Problem 1

(a) Find all prime numbers $p$ such that $4 p^2+1$ and $6 p^2+1$ are also primes.
(b) Determine real numbers $x, y, z, u$ such that
& x y z+x y+y z+z x+x+y+z=7 \\
& y z u+y z+z u+u y+y+z+u=9 \\
& z u x+z u+u x+x z+z+u+x=9 \\
& u x y+u x+x y+y u+u+x+y=9

Problem 2

If $x, y, z, p, q, r$ are distinct real numbers such that
& \frac{1}{x+p}+\frac{1}{y+p}+\frac{1}{z+p}=\frac{1}{p} \\
& \frac{1}{x+q}+\frac{1}{y+q}+\frac{1}{z+q}=\frac{1}{q} \\
& \frac{1}{x+r}+\frac{1}{y+r}+\frac{1}{z+r}=\frac{1}{r}
find the numerical value of $\left(\frac{1}{p}+\frac{1}{q}+\frac{1}{r}\right)$.

Problem 3

$\mathrm{ADC}$ and $\mathrm{ABC}$ are triangles such that $\mathrm{AD}=\mathrm{DC}$ and $\mathrm{CA}=\mathrm{AB}$. If $\angle \mathrm{CAB}=20^{\circ}$ and $\angle \mathrm{ADC}=100^{\circ}$, without using Trigonometry, prove that $\mathrm{AB}=\mathrm{BC}+\mathrm{CD}$.

Problem 4

(a) a, b, c, d are positive real numbers such that $a b c d=1$. Prove that $$\frac{1+a b}{1+a}+\frac{1+b c}{1+b}+\frac{1+c d}{1+c}+\frac{1+d a}{1+d} \geq 4.$$
(b) In a scalene triangle $\mathrm{ABC}, \angle \mathrm{BAC}=120^{\circ}$. The bisectors of the angles $\mathrm{A}, \mathrm{B}$ and $\mathrm{C}$ meet the opposite sides in $\mathrm{P}, \mathrm{Q}$ and $\mathrm{R}$ respectively. Prove that the circle on $\mathrm{QR}$ as diameter passes through the point $P$.

Problem 5

(a) Prove that $x^4+3 x^3+6 x^2+9 x+12$ cannot be expressed as a product of two polynomials of degree 2 with integer coefficients.
(b) $2 n+1$ segments are marked on a line. Each of these segments intersects at least $n$ other segments. Prove that one of these segments intersects all other segments.

Problem 6

If $a, b, c, d$ are positive real numbers such that $a^2+b^2=c^2+d^2$ and $a^2+d^2-a d=b^2+c^2+bc$, find the value $$\frac{a b+c d}{a d+b c}$$

Leave a Reply

Your email address will not be published. Required fields are marked *

This site uses Akismet to reduce spam. Learn how your comment data is processed.

Knowledge Partner

Cheenta is a knowledge partner of Aditya Birla Education Academy

Cheenta Academy

Aditya Birla Education Academy

Aditya Birla Education Academy

Cheenta. Passion for Mathematics

Advanced Mathematical Science. Taught by olympians, researchers and true masters of the subject.
Math Olympiad Program