2010 AMC 10B

What is the value of the following expression? $$
100(100-3)-(100 \cdot 100-3)
$$

Makayla attended two meetings during her $9$-hour work day. The first meeting took $45$ minutes and the second meeting took twice as long. What percent of her work day was spent attending meetings?

A drawer contains red, green, blue, and white socks with at least $2$ of each color. What is the minimum number of socks that must be pulled from the drawer to guarantee a matching pair?

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

A month with $31$ days has the same number of Mondays and Wednesdays. How many of the seven days of the week could be the first day of this month?

A circle is centered at $O,$ $\overline{AB}$ is a diameter and $C$ is a point on the circle with $\angle COB = 50^\circ.$ What is the degree measure of $\angle CAB?$

A triangle has side lengths $10,$ $10,$ and $12.$ A rectangle has width $4$ and area equal to the area of the triangle. What is the perimeter of this rectangle?

A ticket to a school play costs $x$ dollars, where $x$ is a whole number. A group of $9$th graders buys tickets costing a total of $\$48,$ and a group of $10$th graders buys tickets costing a total of $\$64.$ How many values for $x$ are possible?

Lucky Larry’s teacher asked him to substitute numbers for $a,$ $b,$ $c,$ $d,$ and $e$ in the expression $$
a-(b-(c-(d+e)))
$$ and evaluate the result. Larry ignored the parentheses but added and subtracted correctly and obtained the correct result by coincidence. The numbers Larry substituted for $a,$ $b,$ $c,$ and $d$ were $1,$ $2,$ $3,$ and $4,$ respectively. What number did Larry substitute for $e?$

Shelby drives her scooter at a speed of $30$ miles per hour if it is not raining, and $20$ miles per hour if it is raining. Today she drove in the sun in the morning and in the rain in the evening, for a total of $16$ miles in $40$ minutes. How many minutes did she drive in the rain?

A shopper plans to purchase an item that has a listed price greater than $\$100$ and can use any one of the three coupons. Coupon A gives $15\%$ off the listed price, Coupon B gives $\$30$ off the listed price, and Coupon C gives $25\%$ off the amount by which the listed price exceeds $\$100.$

Let $x$ and $y$ be the smallest and largest prices, respectively, for which Coupon A saves at least as many dollars as Coupon B or C. What is $y - x?$

At the beginning of the school year, $50\%$ of all students in Mr. Wells’ math class answered “Yes” to the question “Do you love math”, and $50\%$ answered “No.” At the end of the school year, $70\%$ answered “Yes” and $30\%$ answered “No.” Altogether, $x\%$ of the students gave a different answer at the beginning and end of the school year. What is the difference between the maximum and the minimum possible values of $x?$

What is the sum of all the solutions of the equation below? $$
x = |2x-|60-2x||
$$

The average of the numbers $1,$ $2,$ $3,$ $\cdots,$ $98,$ $99,$ and $x$ is $100x.$ What is $x?$

On a $50$-question multiple choice math contest, students receive $4$ points for a correct answer, $0$ points for an answer left blank, and $-1$ point for an incorrect answer. Jesse’s total score on the contest was $99.$ What is the maximum number of questions that Jesse could have answered correctly?

A square of side length $1$ and a circle of radius $\dfrac{\sqrt{3}}{3}$ share the same center. What is the area inside the circle, but outside the square?

Every high school in the city of Euclid sent a team of $3$ students to a math contest. Each participant in the contest received a different score. Andrea’s score was the median among all students, and hers was the highest score on her team. Andrea’s teammates Beth and Carla placed $37$th and $64$th, respectively. How many schools are in the city?

Positive integers $a,$ $b,$ and $c$ are randomly and independently selected with replacement from the set $\{1, 2, 3,\dots, 2010\}.$

What is the probability that $abc + ab + a$ is divisible by $3?$

A circle with center $O$ has area $156\pi.$ Triangle $ABC$ is equilateral, $\overline{BC}$ is a chord on the circle, $OA = 4\sqrt{3},$ and point $O$ is outside $\triangle ABC.$ What is the side length of $\triangle ABC?$

Two circles lie outside regular hexagon $ABCDEF.$ The first is tangent to $\overline{AB},$ and the second is tangent to $\overline{DE}.$ Both are tangent to lines $BC$ and $FA.$ What is the ratio of the area of the second circle to that of the first circle?