Numbers - New Definition

SOURCE:COMPETITION
Number of Problems: 13. : (Book)

1.

2.

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For the nonzero numbers a, b, and c, define

$(a,b,c)=frac{abc}{a+b+c}$

Find $(2,4,6)$ .

$mathrm{(A) } 1qquad mathrm{(B) } 2qquad mathrm{(C) } 4qquad mathrm{(D) } 6qquad mathrm{(E) } 24$

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3.

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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 $lfloor sqrt{1} floor + lfloor sqrt{2} floor + lfloor sqrt{3} floor + cdots + lfloor sqrt{16} floor.$

$extbf{(A) } 35 qquad extbf{(B) } 38 qquad extbf{(C) } 40 qquad extbf{(D) } 42 qquad extbf{(E) } 136$

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4.

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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 $(2, 12, 9)$ .

$ext{(A)} 4 qquad ext{(B)} 5 qquad ext{(C)} 6 qquad ext{(D)} 7 qquad ext{(E)} 8$

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5.

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For real numbers $a$ and $b$ , define $a diamond b = sqrt{a^2 + b^2}$ . What is the value of

$(5 diamond 12) diamond ((-12) diamond (-5))$ ?

$mathrm{(A)} 0 qquad mathrm{(B)} frac{17}{2} qquad mathrm{(C)} 13 qquad mathrm{(D)} 13sqrt{2} qquad mathrm{(E)} 26$

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6.

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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 $otimes<cmath>( otimes</cmath>(1,2,3),<cmath>otimes</cmath>(2,3,1),<cmath>otimes</cmath&g...$ ?

$mathrm{(A) } -frac{1}{2}qquad mathrm{(B) } -frac{1}{4} qquad mathrm{(C) } 0 qquad mathrm{(D) } frac{1}{4} ...$

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7.

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For real numbers $x$ and $y$ , define $x spadesuit y = (x+y)(x-y)$ . What is $3 spadesuit (4 spadesuit 5)$ ?

$mathrm{(A) } -72qquad mathrm{(B) } -27qquad mathrm{(C) } -24qquad mathrm{(D) } 24qquad mathrm{(E) } 72$

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8.

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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 $((1 star 2) star 3)$ ?

$mathrm{(A) } -frac{2}{3}qquad mathrm{(B) } -frac{1}{5}qquad mathrm{(C) } 0qquad mathrm{(D) } frac{1}{2}qqu...$

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9.

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Define $xotimes y=x^3-y$ . What is $hotimes (hotimes h)$ ?

$mathrm{(A)} -hqquadmathrm{(B)} 0qquadmathrm{(C)} hqquadmathrm{(D)} 2hqquadmathrm{(E)} h^3$

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10.

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For real numbers $a$ and $b$ , define $a * b=(a-b)^2$ . What is $(x-y)^2*(y-x)^2$ ?

$mathrm{(A)} 0qquadmathrm{(B)} x^2+y^2qquadmathrm{(C)} 2x^2qquadmathrm{(D)} 2y^2qquadmathrm{(E)} 4xy$

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11.

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Define $a@b = ab - b^{2}$ and $a#b = a + b - ab^{2}$ . What is $frac {6@2}{6#2}$ ?

$ext{(A)} - frac {1}{2}qquad ext{(B)} - frac {1}{4}qquad ext{(C)} frac {1}{8}qquad ext{(D)} frac {1}{4}qqu...$

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12.

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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 $<<<6>>>$ ?

$mathrm{(A)} 6qquadmathrm{(B)} 12qquadmathrm{(C)} 24qquadmathrm{(D)} 32qquadmathrm{(E)} 36$

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13.

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For a real number $x$ , define $heartsuit(x)$ to be the average of $x$ and $x^2$ . What is $heartsuit(1)+heartsuit(2)+heartsuit(3)$ ?

$extbf{(A)} 3 qquad extbf{(B)} 6 qquad extbf{(C)} 10 qquad extbf{(D)} 12 qquad extbf{(E)} 20$

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