# Difference between revisions of "1982 AHSME Problems"

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If the operation <math>x \aster y</math> is defined by <math>x \aster y = (x+1)(y+1) - 1</math>, then which one of the following is FALSE? | If the operation <math>x \aster y</math> is defined by <math>x \aster y = (x+1)(y+1) - 1</math>, then which one of the following is FALSE? | ||

− | \text{(A)} \ x \aster y = y\aster x \text{ for all real } x,y. | + | <math>\text{(A)} \ x \aster y = y\aster x \text{ for all real } x,y. |

\text{(B)} \ x \aster (y + z) = ( x \aster y ) + (x \aster z) \text{ for all real } x,y, \text{ and } z. | \text{(B)} \ x \aster (y + z) = ( x \aster y ) + (x \aster z) \text{ for all real } x,y, \text{ and } z. | ||

\text{(C)} \ (x-1) \aster (x+1) = (x \aster x) - 1 \text{ for all real } x. | \text{(C)} \ (x-1) \aster (x+1) = (x \aster x) - 1 \text{ for all real } x. | ||

\text{(D)} \ x \aster 0 = x \text{ for all real } x. | \text{(D)} \ x \aster 0 = x \text{ for all real } x. | ||

− | \text{(E)} \ x \aster (y \aster z) = (x \aster y) \aster z \text{ for all real } x,y, \text{ and } z. <math> | + | \text{(E)} \ x \aster (y \aster z) = (x \aster y) \aster z \text{ for all real } x,y, \text{ and } z. </math> |

[[1982 AHSME Problems/Problem 7|Solution]] | [[1982 AHSME Problems/Problem 7|Solution]] | ||

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== Problem 8 == | == Problem 8 == | ||

− | By definition, < | + | By definition, <math>r! = r(r - 1) \cdots 1</math> and <math>\binom{j}{k} = \frac {j!}{k!(j - k)!}</math>, where <math>r,j,k</math> are positive integers and <math>k < j</math>. |

− | If < | + | If <math>\binom{n}{1}, \binom{n}{2}, \binom{n}{3}</math> form an arithmetic progression with <math>n > 3</math>, then <math>n</math> equals |

− | < | + | <math>\textbf{(A)}\ 5\qquad |

\textbf{(B)}\ 7\qquad | \textbf{(B)}\ 7\qquad | ||

\textbf{(C)}\ 9\qquad | \textbf{(C)}\ 9\qquad | ||

\textbf{(D)}\ 11\qquad | \textbf{(D)}\ 11\qquad | ||

− | \textbf{(E)}\ 12<math> | + | \textbf{(E)}\ 12</math> |

[[1982 AHSME Problems/Problem 8|Solution]] | [[1982 AHSME Problems/Problem 8|Solution]] | ||

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== Problem 9 == | == Problem 9 == | ||

− | A vertical line divides the triangle with vertices < | + | A vertical line divides the triangle with vertices <math>(0,0), (1,1)</math>, and <math>(9,1)</math> in the <math>xy\text{-plane}</math> into two regions of equal area. |

− | The equation of the line is < | + | The equation of the line is <math>x=</math> |

− | < | + | <math>\text {(A)} 2.5 \qquad |

\text {(B)} 3.0 \qquad | \text {(B)} 3.0 \qquad | ||

\text {(C)} 3.5 \qquad | \text {(C)} 3.5 \qquad | ||

\text {(D)} 4.0\qquad | \text {(D)} 4.0\qquad | ||

− | \text {(E)} 4.5 <math> | + | \text {(E)} 4.5 </math> |

[[1982 AHSME Problems/Problem 9|Solution]] | [[1982 AHSME Problems/Problem 9|Solution]] | ||

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== Problem 10 == | == Problem 10 == | ||

− | In the adjoining diagram, < | + | In the adjoining diagram, <math>BO</math> bisects <math>\angle CBA</math>, <math>CO</math> bisects <math>\angle ACB</math>, and <math>MN</math> is parallel to <math>BC</math>. |

− | If < | + | If <math>AB=12, BC=24</math>, and <math>AC=18</math>, then the perimeter of <math>\triangle AMN</math> is |

<asy> | <asy> | ||

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[[1982 AHSME Problems/Problem 11|Solution]] | [[1982 AHSME Problems/Problem 11|Solution]] | ||

− | + | ||

== Problem 11 == | == Problem 11 == | ||

## Revision as of 14:06, 3 October 2014

## Contents

- 1 Problem 1
- 2 Problem 2
- 3 Problem 3
- 4 Problem 4
- 5 Problem 5
- 6 Problem 6
- 7 Problem 7
- 8 Problem 8
- 9 Problem 9
- 10 Problem 10
- 11 Problem 11
- 12 Problem 12
- 13 Problem 13
- 14 Problem 14
- 15 Problem 15
- 16 Problem 16
- 17 Problem 17
- 18 Problem 18
- 19 Problem 19
- 20 Problem 20
- 21 Problem 21
- 22 Problem 22
- 23 Problem 23
- 24 Problem 24
- 25 Problem 25
- 26 Problem 26
- 27 Problem 27
- 28 Problem 28
- 29 Problem 29
- 30 Problem 30
- 31 See also

## Problem 1

When the polynomial is divided by the polynomial , the remainder is

## Problem 2

If a number eight times as large as is increased by two, then one fourth of the result equals

## Problem 3

Evaluate at .

## Problem 4

The perimeter of a semicircular region, measured in centimeters, is numerically equal to its area, measured in square centimeters. The radius of the semicircle, measured in centimeters, is

## Problem 5

Two positive numbers and are in the ratio where . If , then the smaller of and is

## Problem 6

The sum of all but one of the interior angles of a convex polygon equals . The remaining angle is

## Problem 7

If the operation $x \aster y$ (Error compiling LaTeX. ! Undefined control sequence.) is defined by $x \aster y = (x+1)(y+1) - 1$ (Error compiling LaTeX. ! Undefined control sequence.), then which one of the following is FALSE?

$\text{(A)} \ x \aster y = y\aster x \text{ for all real } x,y. \text{(B)} \ x \aster (y + z) = ( x \aster y ) + (x \aster z) \text{ for all real } x,y, \text{ and } z. \text{(C)} \ (x-1) \aster (x+1) = (x \aster x) - 1 \text{ for all real } x. \text{(D)} \ x \aster 0 = x \text{ for all real } x. \text{(E)} \ x \aster (y \aster z) = (x \aster y) \aster z \text{ for all real } x,y, \text{ and } z.$ (Error compiling LaTeX. ! Undefined control sequence.)

## Problem 8

By definition, and , where are positive integers and . If form an arithmetic progression with , then equals

## Problem 9

A vertical line divides the triangle with vertices , and in the into two regions of equal area. The equation of the line is

## Problem 10

In the adjoining diagram, bisects , bisects , and is parallel to . If , and , then the perimeter of is

## Problem 11

How many integers with four different digits are there between and such that the absolute value of the difference between the first digit and the last digit is ?

## Problem 12

Let , where and are constants. If , the equals

## Problem 13

If , and , then equals

## Problem 14

In the adjoining figure, points and lie on line segment , and , and are diameters of circle , and , respectively. Circles , and all have radius and the line is tangent to circle at . If intersects circle at points and , then chord has length

## Problem 15

Let [z] denote the greatest integer not exceeding z. Let x and y satisfy the simultaneous equations

\[\begin{array}{1} y=2[x]+3 \\ y=3[x-2]+5. \end{array}\] (Error compiling LaTeX. ! LaTeX Error: Illegal character in array arg.)

If is not an integer, then is

## Problem 16

A wooden cube has edges of length meters. Square holes, of side one meter, centered in each face are cut through to the opposite face. The edges of the holes are parallel to the edges of the cube. The entire surface area including the inside, in square meters, is

## Problem 17

How many real numbers satisfy the equation ?

## Problem 18

In the adjoining figure of a rectangular solid, and . Find the cosine of .

## Problem 19

Let for . The sum of the largest and smallest values of is

## Problem 20

The number of pairs of positive integers which satisfy the equation is

## Problem 21

In the adjoining figure, the triangle is a right triangle with . Median is perpendicular to median , and side . The length of is

## Problem 22

In a narrow alley of width a ladder of length a is placed with its foot at point P between the walls. Resting against one wall at , the distance k above the ground makes a angle with the ground. Resting against the other wall at , a distance h above the ground, the ladder makes a angle with the ground. The width is equal to

## Problem 23

The lengths of the sides of a triangle are consecutive integers, and the largest angle is twice the smallest angle. The cosine of the smallest angle is

## Problem 24

In the adjoining figure, the circle meets the sides of an equilateral triangle at six points. If , and , then equals

## Problem 25

The adjacent map is part of a city: the small rectangles are rocks, and the paths in between are streets. Each morning, a student walks from intersection to intersection , always walking along streets shown, and always going east or south. For variety, at each intersection where he has a choice, he chooses with probability whether to go east or south. Find the probability that through any given morning, he goes through .

## Problem 26

If the base representation of a perfect square is , where , then equals

## Problem 27

Suppose is a solution of the polynomial equation , where , and are real constants and . Which of the following must also be a solution?

## Problem 28

A set of consecutive positive integers beginning with is written on a blackboard. One number is erased. The average (arithmetic mean) of the remaining numbers is . What number was erased?

## Problem 29

Let , and be three positive real numbers whose sum is . If no one of these numbers is more than twice any other, then the minimum possible value of the product is

## Problem 30

Find the units digit of the decimal expansion of

.

## See also

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.