Kaprekar Contest (NMTC Sub-Junior 2019 - VII and VIII Grades) - Stage I- Problems and Solution

Part A

Problem 1

If $4921 \times D=A B B B D$ , then the sum of the digits of $A B B B D \times D$ is
(A) 19
(B) 20
(C) 25
(D) 26

Problem 2

What is the $2019^{th}$ digit to the right of the decimal point, in the decimal representation of $\frac{5}{28}$ ?

(A) 2
(B) 4
(C) 8
(D) 7

Problem 3

If $X$ is a 1000 digit number, $Y$ is the sum of its digits, $Z$ the sum of the digits of $Y$ and $W$ the sum of the digits of $Z$ , then the maximum possible value of $W$ is
(A) 10
(B) 11
(C) 12
(D) 22

Problem 4

Let $x$ be the number $0.000 \ldots . . .001$ which has 2019 zeroes after the decimal point. Then which of the following numbers is the greatest?
(A) $10000+x$
(B) $10000 \cdot x$
(C) $\frac{10000}{x}$
(D) $\frac{1}{x^2}$

Problem 5

Where A, B, C, D, E are distinct digits satisfying this addition fact, then E is

(A) 3
(B) 5
(C) 2
(D) 4

Problem 6

In a $5 \times 5$ grid having 25 cells, Janani has to enter 0 or 1 in each cell such that each sub square grid of size $2 \times 2$ has exactly three equal numbers. What is the maximum possible sum of the numbers in all the 25 cells put together?
(A) 23
(B) 21
(C) 19
(D) 18

Problem 7

$A B C D$ is a square. $E$ is one fourth of the way from $A$ to $B$ and $F$ is one fourth of the way from $B$ to C. $X$ is the centre of the square. Side of the square is $8 \mathrm{~cm}$ . Then the area of the shaded region in the figure in $\mathrm{cm}^2$ is

(A) 14
(B) 16
(C) 18
(D) 20

Problem 8

$A B C D$ is a rectangle with $E$ and $F$ are midpoints of $C D$ and $A B$ respectively and $G$ is the mid-point of $\mathrm{AF}$ . The ratio of the area of $\mathrm{ABCD}$ to area of $\mathrm{AECG}$ is

(A) $4: 3$
(B) $3: 2$
(C) $6: 5$
(D) $8: 3$

Problem 9

each alphabet represents a different digit, what is the maximum possible value
of FLAT?

(A) 2450
(B) 2405
(C) 2305
(D) 2350

Problem 10

How many positive integers smaller than 400 can you get as a sum of eleven consecutive positive integers?
(A) 37
(B) 35
(C) 33
(D) 31

Problem 11

Let $x, y$ and $z$ be positive real numbers and let $x \geq y \geq z$ so that $x+y+z=20.1$ . Which of the following statements is true?
(A) Always $x y<99$
(B) Always $x y>1$
(C) Always $x y \neq 75$
(D) Always $yz \neq 49$

Problem 12

A sequence $\left[a_n\right]$ is generated by the rule, $a_n=a_{n-1}-a_{n-2}$ for $n \geq 3$ Given $a_1=2$ and $a_2=4$ , then sum of the first 2019 terms of the sequence is given by
(A) 8
(B) 2692
(C) -2692
(D) -8

Problem 13

There are exactly 5 prime numbers between 2000 and 2030 . Note: $2021=43 \times 47$ is not a prime number. The difference between the largest and the smallest among these is
(A) 16
(B) 20
(C) 24
(D) 26

Problem 14

Which of the following geometric figures is possible to construct?

(A) A pentagon with 4 right angled vertices
(B) An octagon with all 8 sides equal and 4 angles each of measure $60^{\circ}$ and other four angles of measure $210^{\circ}$
(C) A parallelogram with 3 vertices of obtuse angle measures.
(D) $A$ hexagon with 4 reflex angles.

Problem 15

If $y^{10}=2019$ , then
(A) $2<y<3$
(B) $1<y<2$
(C) $4<y<5$
(D) $3<y<4$

Part B

Problem 16

A sequence of all natural numbers whose second digit (from left to right) is 1 , is written in strictly increasing order without repetition as follows: $11,21,31,41,51,61,71,81,91,110,111, \ldots$ Note that the first term of the sequence is 11 . The third term is 31 , eighth term is 81 and tenth term is 110. The 100th term of the sequence will be $\rule{1cm}{0.15mm}$

Problem 17

In $\triangle \mathrm{ABC}, \mathrm{AB}=6 \mathrm{\textrm {cm }}, \mathrm{AC}=8 \mathrm{\textrm {cm }}$ , median $A D=5 \mathrm{~cm}$ . Then, the area of $\triangle \mathrm{ABC}$ in $\mathrm{cm}^2$ is $\rule{1cm}{0.15mm}$ .

Problem 18

Given $a, b, c$ are real numbers such that $9 a+b+8 c=12$ and $8 a-12 b-9 c=1$ . Then $a^2-b^2+c^2=\rule{1cm}{0.15mm}$

Problem 19

In the given figure, $\triangle A B C$ is a right angled triangle with $\angle A B C=90^{\circ} . D, E, F$ are points on $A B, A C$ , $\mathrm{BC}$ respectively such that $\mathrm{AD}=\mathrm{AE}$ and $\mathrm{CE}=\mathrm{CF}$ . Then, $\angle \mathrm{DEF}= \rule{1cm}{0.15mm}$ (in degree).

Problem 20

Numbers of 5-digit multiples of 13 is $\rule{1cm}{0.15mm}$ .

Problem 21

The area of a sector and the length of the arc of the sector are equal in numerical value. Then the radius of the circle is $\rule{1cm}{0.15mm}$ .

Problem 22

If $a, b, c, d$ are positive integers such that $a+\frac{1}{b+\frac{1}{c+\frac{1}{d}}}=\frac{43}{30}$ , then $d$ is $\rule{1cm}{0.15mm}$ .

Problem 23

A teacher asks 10 of her students to guess her age. They guessed it as $34,38,40,42,46,48,51$ , 54,57 and 59. Teacher said "At least half of you guessed it too low and two of you are off by one. Also my age is a prime number". The teacher's age is $\rule{1cm}{0.15mm}$ .