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$\mathrm{n}, \mathrm{a}$ are natural numbers each greater than 1 . If $a+a+a+a+\ldots+a=2010$, and there are $n$ terms on the left hand side, then the number of ordered pairs $(a, n)$ is

Hint 1

Hint 2

Hint 3 with Final Answer

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

14

The number of times the digit occurs in the result of $1+11+111+\ldots .+111$

111 (100digits) is $\ldots \ldots .$

Hint 1

Hint 2

Hint 3 with Final Answer

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

11

Let us call a sum of integers a cool sum if the first and last terms are 1 and each term differs from its neighbours by at most. For example, $1+2+2+3+3+2+1$ and $1+2+3+4+3+2+1$ are cool sums. The minimum number of terms required to write 2019 as a cool sum is â€¦â€¦

Hint 1

Hint 2

Hint 3 with Final Answer

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

89

For each positive integer $n$ let $f(n)=n^{4}-3 n^{2}+9$. Then the sum of all $f(n)$ which are prime is

Hint 1

Hint 2

Hint 3 with Final Answer

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

20

The product of four positive integers $a, b, c$ and $d$ is 9 ! The number $a, b, c, d$ satisfy $a b+a+b=$ 1224, $b c+b+c=549$ and $c d+c+d=351$. The $a+b+c+d=\ldots \ldots$

Hint 1

Hint 2

Hint 3 with Final Answer

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

108

The number of 6 digit numbers of the form "ABCABC", which are divisible by 13 , where $A, B$ and $C$ are distinct digits, $A$ and $C$ being even digits is

Hint 1

Hint 2

Hint 3 with Final Answer

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

128

In the sequence $1,4,8,10,16,21,25,30$ and 43 , the number of blocks of consecutive terms whose sums are divisible by 11 is

Hint 1

Hint 2

Hint 3 with Final Answer

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

exactly four

In the subtraction below, what is the sum of the digits in the result ? $111 \ldots \ldots \ldots . .111$ (100 digits) $-222 \ldots . . .222$ (50 digits)

Hint 1

Hint 2

Hint 3 with Final Answer

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

450

The least odd prime factor of $2019^{8}+1$ is

Hint 1

Hint 2

Hint 3 with Final Answer

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

97

Let $a, b, c$ be positive integers each less than 50 , such that $a^{2}-b^{2}=100 c$. The number of such triples $(a, b, c)$ is

Hint 1

Hint 2

Hint 3 with Final Answer

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

25

If $4921 \times D=A B B B D$, then the sum of the digits of $A B B B D \times D$ is

Hint 1

Hint 2

Hint 3 with Final Answer

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

19

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

Hint 1

Hint 2

Hint 3 with Final Answer

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

8.

If $\mathrm{X}$ 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

Hint 1

Hint 2

Hint 3 with Final Answer

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

11

Let $x$ be the number $0.000$. 001 which has 2019 zeroes after the decimal point. Then which of the following numbers is the greatest?

Hint 1

Hint 2

Hint 3 with Final Answer

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

$\frac{1}{x^{2}}$

$A B C$

then the number of possible values for $A, B, C, D, E$ satisfying this equation where

If $\frac{C \mathrm{CBA}}{\mathrm{DEDD}}$ then the number of $\mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{D}$ and $\mathrm{E}$ are distinct digits is

Hint 1

Hint 2

Hint 3 with Final Answer

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

4

$\mathrm{R} A \mathrm{~T}$

If $+\mathrm{MAC}$ and each alphabet represents a different digit, what is the maximum possible value $+\frac{\mathrm{VAT}}{\mathrm{FLAT}}$

of FLAT?

Hint 1

Hint 2

Hint 3 with Final Answer

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

2450

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

Hint 1

Hint 2

Hint 3 with Final Answer

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

26

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

Hint 1

Hint 2

Hint 3 with Final Answer

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

47

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

Hint 1

Hint 2

Hint 3 with Final Answer

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

7

The number of natural numbers $n \leq 2019$ such that $\sqrt[3]{48 n}$ is an integer is

Hint 1

Hint 2

Hint 3 with Final Answer

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

3

The number of perfect cubes that lie between $2^{9}+1$ and $2^{18}+1$ is

Hint 1

Hint 2

Hint 3 with Final Answer

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

56

$\mathrm{n}, \mathrm{a}$ are natural numbers each greater than 1 . If $a+a+a+a+\ldots+a=2010$, and there are $n$ terms on the left hand side, then the number of ordered pairs $(a, n)$ is

Hint 1

Hint 2

Hint 3 with Final Answer

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

14

The number of times the digit occurs in the result of $1+11+111+\ldots .+111$

111 (100digits) is $\ldots \ldots .$

Hint 1

Hint 2

Hint 3 with Final Answer

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

11

Let us call a sum of integers a cool sum if the first and last terms are 1 and each term differs from its neighbours by at most. For example, $1+2+2+3+3+2+1$ and $1+2+3+4+3+2+1$ are cool sums. The minimum number of terms required to write 2019 as a cool sum is â€¦â€¦

Hint 1

Hint 2

Hint 3 with Final Answer

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

89

For each positive integer $n$ let $f(n)=n^{4}-3 n^{2}+9$. Then the sum of all $f(n)$ which are prime is

Hint 1

Hint 2

Hint 3 with Final Answer

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

20

The product of four positive integers $a, b, c$ and $d$ is 9 ! The number $a, b, c, d$ satisfy $a b+a+b=$ 1224, $b c+b+c=549$ and $c d+c+d=351$. The $a+b+c+d=\ldots \ldots$

Hint 1

Hint 2

Hint 3 with Final Answer

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

108

The number of 6 digit numbers of the form "ABCABC", which are divisible by 13 , where $A, B$ and $C$ are distinct digits, $A$ and $C$ being even digits is

Hint 1

Hint 2

Hint 3 with Final Answer

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

128

In the sequence $1,4,8,10,16,21,25,30$ and 43 , the number of blocks of consecutive terms whose sums are divisible by 11 is

Hint 1

Hint 2

Hint 3 with Final Answer

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

exactly four

In the subtraction below, what is the sum of the digits in the result ? $111 \ldots \ldots \ldots . .111$ (100 digits) $-222 \ldots . . .222$ (50 digits)

Hint 1

Hint 2

Hint 3 with Final Answer

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

450

The least odd prime factor of $2019^{8}+1$ is

Hint 1

Hint 2

Hint 3 with Final Answer

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

97

Let $a, b, c$ be positive integers each less than 50 , such that $a^{2}-b^{2}=100 c$. The number of such triples $(a, b, c)$ is

Hint 1

Hint 2

Hint 3 with Final Answer

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

25

If $4921 \times D=A B B B D$, then the sum of the digits of $A B B B D \times D$ is

Hint 1

Hint 2

Hint 3 with Final Answer

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

19

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

Hint 1

Hint 2

Hint 3 with Final Answer

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

8.

If $\mathrm{X}$ 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

Hint 1

Hint 2

Hint 3 with Final Answer

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

11

Let $x$ be the number $0.000$. 001 which has 2019 zeroes after the decimal point. Then which of the following numbers is the greatest?

Hint 1

Hint 2

Hint 3 with Final Answer

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

$\frac{1}{x^{2}}$

$A B C$

then the number of possible values for $A, B, C, D, E$ satisfying this equation where

If $\frac{C \mathrm{CBA}}{\mathrm{DEDD}}$ then the number of $\mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{D}$ and $\mathrm{E}$ are distinct digits is

Hint 1

Hint 2

Hint 3 with Final Answer

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

4

$\mathrm{R} A \mathrm{~T}$

If $+\mathrm{MAC}$ and each alphabet represents a different digit, what is the maximum possible value $+\frac{\mathrm{VAT}}{\mathrm{FLAT}}$

of FLAT?

Hint 1

Hint 2

Hint 3 with Final Answer

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

2450

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

Hint 1

Hint 2

Hint 3 with Final Answer

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

26

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

Hint 1

Hint 2

Hint 3 with Final Answer

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

47

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

Hint 1

Hint 2

Hint 3 with Final Answer

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

7

The number of natural numbers $n \leq 2019$ such that $\sqrt[3]{48 n}$ is an integer is

Hint 1

Hint 2

Hint 3 with Final Answer

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

3

The number of perfect cubes that lie between $2^{9}+1$ and $2^{18}+1$ is

Hint 1

Hint 2

Hint 3 with Final Answer

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

56

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