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

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 $a^3+b^3+c^3+d^3=3(a b c+b c d+c d a+d a b)$

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 $a^3+b^3+c^3+d^3=3(a b c+b c d+c d a+d a b)$

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