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## 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 $\rule{1cm}{0.15mm}$.

###### Problem 24

The sum of 8 positive integers is 22 and their LCM is 9. The number of integers among these that are less than 4 is $\rule{1cm}{0.15mm}$.

###### Problem 25

The number of natural numbers $n \leq 2019$ such that $\sqrt[3]{48 n}$ is an integer is $\rule{1cm}{0.15mm}$.