from: category_eng

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

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

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

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

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 $(2, 12, 9)$ .

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

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

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

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

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

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

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

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

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

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

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

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

10.

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

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

11.

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

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

12.

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 $<<<6>>>$ ?

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

13.

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

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

1.

2.

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

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

$3cdot1 + 5cdot2 + 7cdot3 + 1cdot4 = 3+10+21+4 = oxed{ extbf{(B) } 38}$

$(2, 12, 9)=frac{2}{12}+frac{12}{9}+frac{9}{2}=frac{1}{6}+frac{4}{3}+frac{9}{2}=frac{1}{6}+frac{8}{6}+frac{27}{6}=fr...$ . 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 $6$ .

5.

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

6.

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

Since $x spadesuit y = (x+y)(x-y)$ :

$3 spadesuit (4 spadesuit 5) = 3 spadesuit((4+5)(4-5)) = 3 spadesuit (-9) = (3+(-9))(3-(-9)) = -72 Rightarrow A$

8.

$((1 star 2) star 3) = left(left(frac{1+2}{1-2}<br /> ight) star 3<br /> ight) = (-3 star 3) = frac{-3+3}{-3-3} = 0 Longrightar...$

9.

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 $mathrm{(C)}$ .

10.

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

11.

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

12.

$<<<6>>> = <<6>> = <6> = 6quadLongrightarrowquadmathrm{(A)}$

13.

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:

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