2006 AMC 10B

What is $(-1)^1 + (-1)^2 + \cdots + (-1)^{2006}?$

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

A football game was played between two teams, the Cougars and the Panthers. The two teams scored a total of $34$ points, and the Cougars won by a margin of $14$ points. How many points did the Panthers score?

Circles of diameter $1$ inch and $3$ inches have the same center. The smaller circle is painted red, and the portion outside the smaller circle and inside the larger circle is painted blue. What is the ratio of the blue-painted area to the red-painted area?

A $2 \times 3$ rectangle and a $3 \times 4$ rectangle are contained within a square without overlapping at any interior point, and the sides of the square are parallel to the sides of the two given rectangles. What is the smallest possible area of the square?

A region is bounded by semicircular arcs constructed on the sides of a square whose sides measure $\tfrac{2}{\pi},$ as shown. What is the perimeter of this region?

Which of the following is equivalent to

$$
\sqrt{\dfrac{x}{1-\dfrac{x-1}{x}}}
$$

when $x \lt 0?$

A square of area $40$ is inscribed in a semicircle as shown. What is the area of the semicircle?

Francesca uses $100$ grams of lemon juice, $100$ grams of sugar, and $400$ grams of water to make lemonade. There are $25$ calories in $100$ grams of lemon juice and $386$ calories in $100$ grams of sugar. Water contains no calories. How many calories are in $200$ grams of her lemonade?

In a triangle with integer side lengths, one side is three times as long as a second side, and the length of the third side is $15.$ What is the greatest possible perimeter of the triangle?

What is the tens digit in the sum $7! + 8! + 9! + \cdots + 2006!\,?$

The lines $x=\tfrac14 y+a$ and $y=\tfrac14 x+b$ intersect at the point $(1,2).$ What is $a+b?$

Joe and JoAnn each bought $12$ ounces of coffee in a $16$-ounce cup. Joe drank $2$ ounces of his coffee and then added $2$ ounces of cream. JoAnn added $2$ ounces of cream, stirred the coffee well, and then drank $2$ ounces. What is the resulting ratio of the amount of cream in Joe’s coffee to that in JoAnn’s coffee?

Let $a$ and $b$ be the roots of the equation $x^2-mx+2=0.$ Suppose that $a+\tfrac1b$ and $b+\tfrac1a$ are the roots of the equation $x^2-px+q=0.$ What is $q?$

Rhombus $ABCD$ is similar to rhombus $BFDE.$ The area of rhombus $ABCD$ is $24,$ and $\angle BAD=60^\circ.$ What is the area of rhombus $BFDE?$

Leap Day, February $29,$ $2004,$ occurred on a Sunday. On what day of the week will Leap Day, February $29,$ $2020,$ occur?

Bob and Alice each have a bag that contains one ball of each of the colors blue, green, orange, red, and violet. Alice randomly selects one ball from her bag and puts it into Bob’s bag. Bob then randomly selects one ball from his bag and puts it into Alice’s bag. What is the probability that after this process the contents of the two bags are the same?

Let $a_1,$ $a_2,$ $\ldots$ be a sequence for which $a_1=2,$ $a_2=3,$ and $a_n=\dfrac{a_{n-1}}{a_{n-2}}$ for each positive integer $n\ge 3.$ What is $a_{2006}?$

A circle of radius $2$ is centered at $O.$ Square $OABC$ has side length $1.$ Sides $\overline{AB}$ and $\overline{CB}$ are extended past $B$ to meet the circle at $D$ and $E,$ respectively. What is the area of the shaded region in the figure, which is bounded by $\overline{BD},$ $\overline{BE},$ and the minor arc connecting $D$ and $E?$

In rectangle $ABCD,$ we have $A=(6,-22),$ $B=(2006,178),$ and $D=(8,y)$ for some integer $y.$ What is the area of rectangle $ABCD?$