2005 AMC 10A Problems

#1

While eating out, Mike and Joe each tipped their server $\$2$ . Mike tipped $10%$ (Error compiling LaTeX. Unknown error_msg) of his bll and Joe tipped $20%$ (Error compiling LaTeX. Unknown error_msg) of his bill. What was the difference, in dollars between their bills?

$\mathrm{(A) \ } 2\qquad \mathrm{(B) \ } 4\qquad \mathrm{(C) \ } 5\qquad \mathrm{(D) \ } 10\qquad \mathrm{(E) \ } 20$

#2

For each pair of real numbers $a$ $\neq$ $b$ , define the operation $\star$ as

$(a \star b) = \frac{a+b}{a-b}$ .

What is the value of $((1 \star 2) \star 3)$ ?

$\mathrm{(A) \ } -\frac{2}{3}\qquad \mathrm{(B) \ } -\frac{1}{5}\qquad \mathrm{(C) \ } 0\qquad \mathrm{(D) \ } \frac{1}{2}\qquad \mathrm{(E) \ } \textrm{This\, value\, is\, not\, defined.}$

#3

The equations $2x + 7 = 3$ and $bx - 10 = -2$ have the same solution $x$ . What is the value of $b$ ?

$\mathrm{(A) \ } -8\qquad \mathrm{(B) \ } -4\qquad \mathrm{(C) \ } -2\qquad \mathrm{(D) \ } 4\qquad \mathrm{(E) \ } 8$

#4

A rectangle with a diagonal of length $x$ is twice as long as it is wide. What is the area of the rectangle?

$\mathrm{(A) \ } \frac{1}{4}x^2\qquad \mathrm{(B) \ } \frac{2}{5}x^2\qquad \mathrm{(C) \ } \frac{1}{2}x^2\qquad \mathrm{(D) \ } x^2\qquad \mathrm{(E) \ } \frac{3}{2}x^2$

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2005 AMC 10A Problems