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

NMTC 2023 Stage II - Kaprekar (Grade 7 & 8) - Problems and Solutions

Join Trial or Access Free Resources
Problem 1

If $b\left(a^2-b c\right)(1-a c)=a\left(b^2-c a\right)(1-b c)$ where $a \neq b$ and $a b c \neq 0$, prove that $a+b+c=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$

Problem 2

$a, b, c$ are three distinct positive integers. Show that among the numbers $a^5 b-a b^5, b^5 c-b c^5, c^5 a-c a^5$ there must be one which is divisible by 8 .

Problem 3

There are four points $P, Q, R, S$ on a plane such that no three of them are collinear. Can the triangles $P Q R, P Q S, P R S$ and $Q R S$ be such that at least one has an interior angle less than or equal to $45^{\circ}$ ? If so, how? If not, why?

Problem 4

A straight line $\ell$ is drawn through the vertex $\mathrm{C}$ of an equilateral triangle $A B C$, wholly lying outside the triangle. $\mathrm{AL}, \mathrm{BM}$ are drawn perpendiculars to the straight line $\ell$. If $N$ is the midpoint of $A B$, prove that $\triangle L M N$ is an equilateral triangle.

Problem 5

$A B C D$ is a parallelogram. Through $C$, a straight line is drawn outside the parallelogram. $A P, B Q$ and $D R$ are drawn perpendicular to this line Show that $A P=B Q+D R$. If the line through $C$ cuts one side internally, then will the same result hold? If so prove it. If not, what is the corresponding result? Justify your answer.

Problem 6

$m, n$ are non-negative real numbers whose sum is 1 . Prove that the maximum and minimum values of $\frac{m^3+n^3}{m^2+n^2}$ are respectively 1 and $1 / 2$.

Problem 7

(a) Solve for $x: \frac{x+5}{2018}+\frac{x+4}{2019}+\frac{x+3}{2020}+\frac{x+2}{2021}+\frac{x+1}{2022}+\frac{x}{2023}=-6$

(b) If $\frac{a^2+b^2}{725}=\frac{b^2+c^2}{149}=\frac{c^2+a^2}{674}$ and $a-c=18$, find the value of $(a+b+c)$.

Problem 8

If $a+b+c+d=0$, prove that $a^3+b^3+c^3+d^3=3(a b c+b c d+c d a+d a b)$

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Your email address will not be published.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

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.
JOIN TRIAL
support@cheenta.com
Trial
Math Olympiad Program
magic-wandrockethighlight