NMTC 2018 Stage II - KAPREKAR (Class 7, 8) - Problems and Solutions

Problem 1

A lucky year is one in which at least one date, when written in the form day / month / year, has the following property. The product of the month times the day equals the last two digits of the year. For example, $1956$ is a lucky year because it has the date $7 / 8 / 56$ where $7 \times 8=56$, but 1962 is not a lucky year as $62=62 \times 1$ or $31 \times 2$, where $31 / 2 / 1962$ is not a valid date. From 1900 to 2018 how many years are not lucky (not including $1900$ and $2018$) ? Give proper explanation for your answer.

Problem 2

In the figure given, $\angle A, \angle B$ and $\angle C$ are right angles. If and $\angle A E B=40^{\circ}$ and $\angle B E D=\angle B D E$, then find $\angle \mathrm{CDE}$.

Problem 3

(a) $\quad \mathrm{ABCDEF}$ is a hexagon in which $\mathrm{AB}=\mathrm{BC}=\mathrm{CD}=\mathrm{DE}=2$ and $\mathrm{EF}=\mathrm{FA}=1$. Its interior angle $\mathrm{C}$ is between $90^{\circ}$ and $180^{\circ}$ and $\mathrm{F}$ is greater than $180^{\circ}$. The rest of the angles are $90^{\circ}$ each. What is its area?
(b) A convex polygon with ' $n$ ' sides has all angles equal to $150^{\circ}$, except one angle. List all possible values of $n$.

Problem 4

$a, b, c$ are distinct non-zero reals such that $$\frac{1+a^3}{a}=\frac{1+b^3}{b}=\frac{1+c^3}{c}.$$ Find all possible values of $a^3+b^3+c^3$

Problem 5

Find the smallest positive integer such that it has exactly $100$ different positive integer divisors including $1$ and the number itself.

Problem 6

(a) What is the sum of the digits of the smallest positive integer which is divisible by $99$ and has all of its digits equal to $2$ ?
(b) When $270$ is divided by the odd number $\mathrm{n}$, the quotient is a prime number and the remainder is $0$ . What is $n$ ?

Problem 7

Consider the sums
$$
\mathrm{A}=\frac{1}{1 \cdot 2}+\frac{1}{3 \cdot 4}+\ldots \ldots+\frac{1}{99 \cdot 100} \text { and } \mathrm{B}=\frac{1}{51 \cdot 100}+\frac{1}{52 \cdot 99}+\ldots \ldots+\frac{1}{100 \cdot 51}
$$

Express $\frac{\mathrm{A}}{\mathrm{B}}$ as an irreducible fraction.

Problem 8

Let $a, b, c$ be real numbers, not all of them are equal. Prove that if $a+b+c=0$, then $a^2+a b+b^2=b^2+b c+c^2=c^2+c a+a^2$.

Prove the converse, if $a^2+a b+b^2=b^2+b c+c^2=c^2=c a+a^2$, then $a+b+c=0$.