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Numbers - New Definition

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

Problem Num : 11

From : AMC10

Type:

Section:Numbers

Theme:
Adjustment# : 0

Difficulty: 1

'
Define $a@b = ab - b^{2}$ and $a#b = a + b - ab^{2}$ . What is $frac {6@2}{6#2}$ ?
$ext{(A)} - frac {1}{2}qquad ext{(B)} - frac {1}{4}qquad ext{(C)} frac {1}{8}qquad ext{(D)} frac {1}{4}qqu...$
'

Category New Definition

Analysis

Solution/Answer
$frac{6 @ 2}{6 # 2} = frac{(6) imes (2) - (2)^2}{(6) + (2) - (6) cdot (2)^2} = frac{8}{-16} = frac{-1}{2} Rightarrow ...$

Answer:

Problem Num : 12

From : AMC10

Type:

Section:Numbers

Theme:
Adjustment# : 0

Difficulty: 1

'
For the positive integer $n$ , let $<n>$ denote the sum of all the positive divisors of $n$ with the exception of $n$ itself. For example, $<4>=1+2=3$ and $<12>=1+2+3+4+6=16$ . What is $<<<6>>>$ ?
$mathrm{(A)} 6qquadmathrm{(B)} 12qquadmathrm{(C)} 24qquadmathrm{(D)} 32qquadmathrm{(E)} 36$
'

Category New Definition

Analysis

Solution/Answer
$<<<6>>> = <<6>> = <6> = 6quadLongrightarrowquadmathrm{(A)}$

Answer:

Problem Num : 13

From : AMC10B

Type:

Section:Numbers

Theme:
Adjustment# : 0

Difficulty: 1

'
For a real number $x$ , define $heartsuit(x)$ to be the average of $x$ and $x^2$ . What is $heartsuit(1)+heartsuit(2)+heartsuit(3)$ ?
$extbf{(A)} 3 qquad extbf{(B)} 6 qquad extbf{(C)} 10 qquad extbf{(D)} 12 qquad extbf{(E)} 20$
'

Category New Definition

Analysis

Solution/Answer
The average of two numbers, $a$ and $b$ , is defined as $frac{a+b}{2}$ . Thus the average of $x$ and $x^2$ would be $frac{x(x+1)}{2}$ . With that said, we need to find the sum when we plug, $1$ , $2$ and $3$ into that equation. So:

$frac{1(1+1)}{2} + frac{2(2+1)}{2} + frac{3(3+1)}{2} = frac{2}{2} + frac{6}{2} + frac{12}{2} = 1+3+6= oxed{ extbf{(C)...$

Answer:

Array ( [0] => 7844 [1] => 7864 [2] => 8128 ) 131 2