AMC 10 2001 — AMC // 10

// PROBLEM 1

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The median of the list $n, n + 3, n + 4, n + 5, n + 6, n + 8, n + 10, n + 12, n + 15$ is $10$ . What is the mean?

// PROBLEM 2

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A number $x$ is $2$ more than the product of its reciprocal and its additive inverse. In which interval does the number lie?

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The sum of two numbers is $S$ . Suppose $3$ is added to each number and then each of the resulting numbers is doubled. What is the sum of the final two numbers?

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What is the maximum number of possible points of intersection of a circle and a triangle?

// PROBLEM 5

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The twelve pentominoes are the twelve distinct shapes made by joining five unit squares edge-to-edge. How many of the twelve pentominoes have at least one line of reflectional symmetry?

// PROBLEM 6

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Let $P(n)$ and $S(n)$ denote the product and the sum, respectively, of the digits of the integer $n$ . For example, $P(23) = 6$ and $S(23) = 5$ . Suppose $N$ is a two-digit number such that $N = P(N) + S(N)$ . What is the units digit of $N$ ?

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When the decimal point of a certain positive decimal number is moved four places to the right, the new number is four times the reciprocal of the original number. What is the original number?

// PROBLEM 8

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Wanda, Darren, Beatrice, and Chi are tutors in the school math lab. Their schedule is as follows: Darren works every third school day, Wanda works every fourth school day, Beatrice works every sixth school day, and Chi works every seventh school day. Today they are all working in the math lab. In how many school days from today will they next be together tutoring in the lab?

// PROBLEM 9

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The state income tax where Kristin lives is levied at the rate of $p\%$ of the first $\textdollar 28000$ of annual income plus $(p + 2)\%$ of any amount above $\textdollar 28000$ . Kristin noticed that the state income tax she paid amounted to $(p + 0.25)\%$ of her annual income. What was her annual income?

// PROBLEM 10

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If $x$ , $y$ , and $z$ are positive with $xy = 24$ , $xz = 48$ , and $yz = 72$ , then $x + y + z$ is

// PROBLEM 11

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Consider a dark center square surrounded by rings of unit squares. The first ring around the center square contains $8$ unit squares. The second ring contains $16$ unit squares. If we continue this process, the number of unit squares in the $100$ th ring is

// PROBLEM 12

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Suppose that $n$ is the product of three consecutive integers and that $n$ is divisible by $7$ . Which of the following is not necessarily a divisor of $n$ ?

// PROBLEM 13

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A telephone number has the form $ABC\text{-}DEF\text{-}GHIJ$ , where each letter represents a different digit. The digits in each part of the number are in decreasing order; that is, $A > B > C$ , $D > E > F$ , and $G > H > I > J$ . Furthermore, $D$ , $E$ , and $F$ are consecutive even digits; $G$ , $H$ , $I$ , and $J$ are consecutive odd digits; and $A + B + C = 9$ . Find $A$ .

// PROBLEM 14

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A charity sells $140$ benefit tickets for a total of $\$ 2001$. Some tickets sell for full price (a whole dollar amount), and the rest sell for half price. How much money is raised by the full-price tickets?

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A street has parallel curbs $40$ feet apart. A crosswalk bounded by two parallel stripes crosses the street at an angle. The length of the curb between the stripes is $15$ feet and each stripe is $50$ feet long. Find the distance, in feet, between the stripes.

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The mean of three numbers is $10$ more than the least of the numbers and $15$ less than the greatest. The median of the three numbers is $5$ . What is their sum?

// PROBLEM 17

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A $252^\circ$ sector of a circle of radius $10$ is rolled up by aligning the two straight sides to form a cone. Which cone is formed?

// PROBLEM 18

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The plane is tiled by congruent squares and congruent pentagons as indicated below. The percent of the plane that is enclosed by the pentagons is closest to

In the repeating $3\times 3$ block shown, the four shaded $1\times 1$ regions are the small squares, and the rest of the block is divided into four congruent pentagons.

// PROBLEM 19

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Pat wants to buy four donuts from an ample supply of three types of donuts: glazed, chocolate, and powdered. How many different selections are possible?

// PROBLEM 20

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A regular octagon is formed by cutting an isosceles right triangle from each of the four corners of a square with sides of length $2000$ . What is the length of each side of the octagon?

// PROBLEM 21

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A right circular cylinder with its diameter equal to its height is inscribed in a right circular cone. The cone has diameter $10$ and altitude $12$ , and the axes of the cylinder and cone coincide. Find the radius of the cylinder.

// PROBLEM 22

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In the magic square shown, the sums of the numbers in each row, column, and diagonal are the same. Five of these numbers are represented by $v$ , $w$ , $x$ , $y$ , and $z$ . Find $y + z$ .

The $3 \times 3$ magic square has the following entries (row by row, top to bottom):

Row 3 (top): $v$ , $24$ , $w$
Row 2 (middle): $18$ , $x$ , $y$
Row 1 (bottom): $25$ , $z$ , $21$

// PROBLEM 23

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A box contains exactly five chips, three red and two white. Chips are randomly removed one at a time without replacement until all the red chips are drawn or all the white chips are drawn. What is the probability that the last chip drawn is white?

// PROBLEM 24

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In trapezoid $ABCD$ , $\overline{AB}$ and $\overline{CD}$ are perpendicular to $\overline{AD}$ , with $AB + CD = BC$ , $AB < CD$ , and $AD = 7$ . What is $AB \cdot CD$ ?

// PROBLEM 25

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How many positive integers not exceeding $2001$ are multiples of $3$ or $4$ but not $5$ ?