Display Problems

Imagination is more important than knowledge... Albert Einstein

Guess is more important than calculation --- Knowhowacademy.com

Numbers - New Definition

For COMPETITION
Number of Total Problems: 13.
FOR PRINT ::: (Book)

Problem Num : 1

From : NCTM

Type: None

Section:Numbers

Theme:None
Adjustment# :

Difficulty: 1

Category New Definition

Analysis

Solution/Answer

Problem Num : 2

From : AMC10B

Type:

Section:Numbers

Theme:
Adjustment# : 0

Difficulty: 1

'
For the nonzero numbers a, b, and c, define
$(a,b,c)=frac{abc}{a+b+c}$
Find $(2,4,6)$ .
$mathrm{(A) } 1qquad mathrm{(B) } 2qquad mathrm{(C) } 4qquad mathrm{(D) } 6qquad mathrm{(E) } 24$
'

Category New Definition

Analysis

Solution/Answer
$frac{2cdot 4cdot 6}{2+4+6}=frac{48}{12}=4Longrightarrowmathrm{ (C) }$

Answer:

Problem Num : 3

From : AMC10B

Type:

Section:Numbers

Theme:
Adjustment# : 0

Difficulty: 1

'
The symbolism $lfloor x floor$ denotes the largest integer not exceeding $x$ . For example, $lfloor 3 floor = 3,$ and $lfloor 9/2 floor = 4$ . Compute $lfloor sqrt{1} floor + lfloor sqrt{2} floor + lfloor sqrt{3} floor + cdots + lfloor sqrt{16} floor.$
$extbf{(A) } 35 qquad extbf{(B) } 38 qquad extbf{(C) } 40 qquad extbf{(D) } 42 qquad extbf{(E) } 136$
'

Category New Definition

Analysis

Solution/Answer
The first three values in the sum are equal to $1,$ the next five equal to $2,$ the next seven equal to $3,$ and the last one equal to $4.$ For example, since $2^2=4$ any square root of a number less than $4$ must be less than $2.$ Sum them all together to get
$3cdot1 + 5cdot2 + 7cdot3 + 1cdot4 = 3+10+21+4 = oxed{ extbf{(B) } 38}$

Answer:

Problem Num : 4

From : AMC10

Type:

Section:Numbers

Theme:
Adjustment# : 0

Difficulty: 1

'
Given that a, b, and c are non-zero real numbers, define $(a, b, c) = frac{a}{b} + frac{b}{c} + frac{c}{a}$ , find $(2, 12, 9)$ .
$ext{(A)} 4 qquad ext{(B)} 5 qquad ext{(C)} 6 qquad ext{(D)} 7 qquad ext{(E)} 8$
'

Category New Definition

Analysis

Solution/Answer
$(2, 12, 9)=frac{2}{12}+frac{12}{9}+frac{9}{2}=frac{1}{6}+frac{4}{3}+frac{9}{2}=frac{1}{6}+frac{8}{6}+frac{27}{6}=fr...$ . Our answer is then $oxed{ ext{(C)} 6}$ .
Alternate solution for the lazy: Without computing the answer exactly, we see that $2/12= ext{a little}$ , $12/9= ext{more than }1$ , and $9/2=4.5$ . The sum is $4.5 + ( ext{more than }1) + ( ext{a little}) = ( ext{more than }5.5) + ( ext{a little})$ , and as all the options are integers, the correct one is obviously $6$ .

Answer:

Problem Num : 5

From : AMC10B

Type:

Section:Numbers

Theme:
Adjustment# : 0

Difficulty: 1

'
For real numbers $a$ and $b$ , define $a diamond b = sqrt{a^2 + b^2}$ . What is the value of
$(5 diamond 12) diamond ((-12) diamond (-5))$ ?
$mathrm{(A)} 0 qquad mathrm{(B)} frac{17}{2} qquad mathrm{(C)} 13 qquad mathrm{(D)} 13sqrt{2} qquad mathrm{(E)} 26$
'

Category New Definition

Analysis

Solution/Answer
$(5 diamond 12) diamond ((-12) diamond (-5))\ (sqrt{5^2+12^2}) diamond (sqrt{(-12)^2+(-5)^2})\ (sqrt{169})diamond(s...$

Answer:

Problem Num : 6

From : AMC10

Type:

Section:Numbers

Theme:
Adjustment# : 0

Difficulty: 1

'
For any three real numbers $a$ , $b$ , and $c$ , with $b eq c$ , the operation $otimes$ is defined by: $otimes(a,b,c)=frac{a}{b-c}$ What is $otimes<cmath>( otimes</cmath>(1,2,3),<cmath>otimes</cmath>(2,3,1),<cmath>otimes</cmath&g...$ ?
$mathrm{(A) } -frac{1}{2}qquad mathrm{(B) } -frac{1}{4} qquad mathrm{(C) } 0 qquad mathrm{(D) } frac{1}{4} ...$
'

Category New Definition

Analysis

Solution/Answer
$otimes<cmath> left(frac{1}{2-3}, frac{2}{3-1}, frac{3}{1-2} ight)=</cmath>otimes$ $(-1,1,-3)=frac{-1}{1+3}=-frac{1}{4}Longrightarrowmathrm{(B)}$

Answer:

Problem Num : 7

From : AMC10B

Type:

Section:Numbers

Theme:
Adjustment# : 0

Difficulty: 1

'
For real numbers $x$ and $y$ , define $x spadesuit y = (x+y)(x-y)$ . What is $3 spadesuit (4 spadesuit 5)$ ?
$mathrm{(A) } -72qquad mathrm{(B) } -27qquad mathrm{(C) } -24qquad mathrm{(D) } 24qquad mathrm{(E) } 72$
'

Category New Definition

Analysis

Solution/Answer
Since $x spadesuit y = (x+y)(x-y)$ :
$3 spadesuit (4 spadesuit 5) = 3 spadesuit((4+5)(4-5)) = 3 spadesuit (-9) = (3+(-9))(3-(-9)) = -72 Rightarrow A$

Answer:

Problem Num : 8

From : AMC10

Type:

Section:Numbers

Theme:
Adjustment# : 0

Difficulty: 1

'
For each pair of real numbers $a<cmath> eq</cmath>b$ , define the operation $star$ as
$(a star b) = frac{a+b}{a-b}$ .
What is the value of $((1 star 2) star 3)$ ?
$mathrm{(A) } -frac{2}{3}qquad mathrm{(B) } -frac{1}{5}qquad mathrm{(C) } 0qquad mathrm{(D) } frac{1}{2}qqu...$
'

Category New Definition

Analysis

Solution/Answer
$((1 star 2) star 3) = left(left(frac{1+2}{1-2} ight) star 3 ight) = (-3 star 3) = frac{-3+3}{-3-3} = 0 Longrightar...$

Answer:

Problem Num : 9

From : AMC10

Type:

Section:Numbers

Theme:
Adjustment# : 0

Difficulty: 1

'
Define $xotimes y=x^3-y$ . What is $hotimes (hotimes h)$ ?
$mathrm{(A)} -hqquadmathrm{(B)} 0qquadmathrm{(C)} hqquadmathrm{(D)} 2hqquadmathrm{(E)} h^3$
'

Category New Definition

Analysis

Solution/Answer
By the definition of $otimes$ , we have $hotimes h=h^{3}-h$ . Then $hotimes (hotimes h)=hotimes (h^{3}-h)=h^{3}-(h^{3}-h)=h$ . The answer is $mathrm{(C)}$ .

Answer:

Problem Num : 10

From : AMC10B

Type:

Section:Numbers

Theme:
Adjustment# : 0

Difficulty: 1

'
For real numbers $a$ and $b$ , define $a * b=(a-b)^2$ . What is $(x-y)^2*(y-x)^2$ ?
$mathrm{(A)} 0qquadmathrm{(B)} x^2+y^2qquadmathrm{(C)} 2x^2qquadmathrm{(D)} 2y^2qquadmathrm{(E)} 4xy$
'

Category New Definition

Analysis

Solution/Answer
Since $(-a)^2 = a^2$ , it follows that $(x-y)^2 = (y-x)^2$ , and $(x-y)^2 * (y-x)^2 = [(x-y)^2 - (y-x)^2]^2 = [(x-y)^2 - (x-y)^2]^2 = 0 mathrm{(A)}.$

Answer:

Array ( [0] => 5991 [1] => 7965 [2] => 7990 [3] => 7754 [4] => 8030 [5] => 7789 [6] => 8050 [7] => 7809 [8] => 7824 [9] => 8088 ) 131 2