Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.3>Q 17

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For any non-empty finite set $A$ of real numbers, let $s (A)$ be the sum of the elements in $A$ . There are exactly $61 3$ -element subsets $A$ of $\{1, \dots, 23\}$ with $s (A) = 36 .$ The total number of $3$ -element subsets of $\{1, \dots, 23\}$ with $s (A) < 36$ is given by $N .$ Then evaluate $\frac{N + 45}{25}$

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Important Questions on Number System

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.3>Q 20

Let the number of distinct integers among the following numbers is $N .$ Find the sum of all digits of $N .$

$[\frac{1^{2}}{2007}], [\frac{2^{2}}{2007}], \dots, [\frac{2007^{2}}{2007}]$ , where, $[\cdot]$ is the greatest integer function.

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.3>Q 21

For each integer

n \geq 4,

let

a_{n}

denote the base-

n

number

0 . {\bar{133}}_{n} .

The product

a_{4} a_{5} \dots \dots \dots . a_{99}

can be expressed as

\frac{m}{n!}

, where

m

and

n

are positive integers and

n

is as small as possible. What is the value of sum of the digits of

m

?

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.3>Q 24

If

a, b,

and

c

are positive real numbers such that

a (b + c) = 152, b (c + a) = 162

, and

c (a + b) = 170,

then

\frac{a b c}{12}

is

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.3>Q 25

If

x, y,

and

z

are positive numbers satisfying

x + \frac{1}{y} = 4, y + \frac{1}{z} = 1

and

z + \frac{1}{x} = \frac{7}{3}

, then find

x y z ?

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.3>Q 29

How many odd positives

3

-digit integers are divisible by

3

but do not contain the digit

3 ?

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.3>Q 33

A five-digit palindrome is a positive integer with respective digits

a b c b a

, where

a

is non-zero. Let

S

be the sum of all five-digit palindromes. What is the sum of the digits of

S ?

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.3>Q 34

The products

(8) (888 \dots . 8)

, where the second factor has

k

digits is an integer whose digits have a sum of

1000

. What is the sum of digits of

k ?

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.3>Q 40

There are

88

numbers

a_{1}, a_{2}, a_{3}, \dots, a_{88}

and each of them is either equals to

- 3

or

- 1 .

Given that

a_{1}^{2} + a_{2}^{2} + \dots + a_{88}^{2} = 280

and

a_{1}^{4} + a_{2}^{4} + \dots + a_{88}^{4} = 10^{3} a + b .

Find the product of

a

and

b