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

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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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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 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 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} ?

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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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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 $R ?$

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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 30

Let

p

and

q

be positive integers such that

\frac{5}{9} < \frac{p}{q} < \frac{4}{7}

and

q

is as small as possible. What is

{(q - p)}^{2} ?

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.3>Q 31

A function

f

is defined recursively by

f (1) = f (2) = 1

and

f (n) = f (n - 1) - f (n - 2) + n

for all integers

n \geq 3 .

What is the sum of the digits of

f (2018) ?

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 2 - Number System>EXERCISE - 2.3>Q 32

Sunita has

30

thin rods, one each of every integer length from

1 cm

through

30 cm .

She places the rods with lengths

3 cm, 7 cm,

and

15 cm

on a table. She then wants to choose a fourth rod that she can put with these three to form a quadrilateral with positive area. How many of the remaining rods can she choose as the fourth rod?