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IOQM - PRMO and RMO

IMPORTANT

Earn 100

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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Important Questions on Number System

HARD

IOQM - PRMO and RMO

IMPORTANT

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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IOQM - PRMO and RMO

IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.3>Q 22

The square is a multiplicative magic square. That is, the product of the numbers in each row, column, and diagonal is the same. If all the entries are positive integers, what is the sum of the possible values of $g$ ?

$50$	$b$	$c$
$d$	$e$	$f$
$g$	$h$	$2$

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.3>Q 23

How many four-digit numbers

N

have the property that the three-digit number obtained by removing the leftmost digit is one ninth of

N

EASY

IOQM - PRMO and RMO

IMPORTANT

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

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}

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

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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IOQM - PRMO and RMO

IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.3>Q 26

There are

x

integer values of

n

(not necessarily positive) so that value of

4000 \cdot {(\frac{2}{5})}^{n}

is an integer. What is

x^{2} ?

HARD

IOQM - PRMO and RMO

IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.3>Q 27

Let

S

be a set of

6

integers taken from

\{1, 2, \dots . . 12\}

with the property that if

a

and

b

are elements of

S

with

a < b

, then

b

is not a multiple of

a

. What is the square of the least possible value of an element in

S ?

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IOQM - PRMO and RMO

IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.3>Q 28

$R$ non-negative integers can be written in the form $a_{7} \cdot 3^{7} + a_{6} \cdot 3^{6} + a_{5} \cdot 3^{5} + a_{4} \cdot 3^{4} + a_{3} \cdot 3^{3} + a_{2} \cdot 3^{2} + a_{1} \cdot 3^{1} + a_{0} 3^{0}$

where $a_{i} \in \{- 1, 0, 1\}$ for $0 \leq i \leq 7 .$ What is the last two digits of $R ?$

Let the number of distinct integers among the following numbers is N. Find the sum of all digits of N. 122007,222007,…,200722007, where, · is the greatest integer function.

Important Questions on Number System

Learn and Prepare for any exam you want !

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.