# Difference between revisions of "1997 AHSME Problems"

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[[1997 AHSME Problems/Problem 29|Solution]] | [[1997 AHSME Problems/Problem 29|Solution]] | ||

− | ==Problem== | + | ==Problem 30== |

For positive integers <math>n</math>, denote <math>D(n)</math> by the number of pairs of different adjacent digits in the binary (base two) representation of <math>n</math>. For example, <math> D(3) = D(11_{2}) = 0 </math>, <math> D(21) = D(10101_{2}) = 4 </math>, and <math> D(97) = D(1100001_{2}) = 2 </math>. For how many positive integers less than or equal <math>97</math> to does <math>D(n) = 2</math>? | For positive integers <math>n</math>, denote <math>D(n)</math> by the number of pairs of different adjacent digits in the binary (base two) representation of <math>n</math>. For example, <math> D(3) = D(11_{2}) = 0 </math>, <math> D(21) = D(10101_{2}) = 4 </math>, and <math> D(97) = D(1100001_{2}) = 2 </math>. For how many positive integers less than or equal <math>97</math> to does <math>D(n) = 2</math>? |

## Revision as of 18:19, 23 August 2011

## 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

## Problem 1

If and are digits for which

$\begin{tabular}{ccc}& 2 & a\\ \times & b & 3\\ \hline & 6 & 9\\ 9 & 2\\ \hline 9 & 8 & 9\end{tabular}$ (Error compiling LaTeX. ! Missing $ inserted.)

then

## Problem 2

The adjacent sides of the decagon shown meet at right angles. What is its perimeter?

## Problem 3

If , , and are real numbers such that

then

## Problem 4

If is larger than , and is larger than , then is what percent larger than ?

## Problem 5

A rectangle with perimeter is divided into five congruent rectangles as shown in the diagram. What is the perimeter of one of the five congruent rectangles?

## Problem 6

Consider the sequence

whose th term is . What is the average of the first terms of the sequence?

## Problem 7

The sum of seven integers is . What is the maximum number of the seven integers that can be larger than ?

## Problem 8

Mientka Publishing Company prices its bestseller Where's Walter? as follows:

where is the number of books ordered, and is the cost in dollars of books. Notice that books cost less than books. For how many values of is it cheaper to buy more than books than to buy exactly books?

## Problem 9

In the figure, is a square, is the midpoint of , and is on . If is perpendicular to , then the area of quadrilateral is

## Problem 10

Two six-sided dice are fair in the sense that each face is equally likely to turn up. However, one of the dice has the replaced by and the other die has the replaced by . When these dice are rolled, what is the probability that the sum is an odd number?

## Problem 11

In the sixth, seventh, eighth, and ninth basketball games of the season, a player scored ,, , and points, respectively. Her points-per-game average was higher after nine games than it was after the first five games. If her average after ten games was greater than , what is the least number of points she could have scored in the tenth game?

## Problem 12

If and are real numbers and , then the line whose equation is *cannot* contain the point

## Problem 13

How many two-digit positive integers have the property that the sum of and the number obtained by reversing the order of the digits of is a perfect square?

## Problem 14

The number of geese in a flock increases so that the difference between the populations in year and year is directly proportional to the population in year . If the populations in the years , , and were , , and , respectively, then the population in was

## Problem 15

Medians and of triangle are perpendicular, , and . The area of triangle is

## Problem 16

The three row sums and the three column sums of the array

are the same. What is the least number of entries that must be altered to make all six sums different from one another?

## Problem 17

A line intersects the graph of and the graph of . The distance between the points of intersection is . Given that , where and are integers, what is ?

## Problem 18

A list of integers has mode and mean . The smallest number in the list is . The median of the list is a member of the list. If the list member were replaced by , the mean and median of the new list would be and , respectively. If were instead replaced by , the median of the new list would be . What is ?

## Problem 19

A circle with center is tangent to the coordinate axes and to the hypotenuse of the -- triangle as shown, where . To the nearest hundredth, what is the radius of the circle?

## Problem 20

Which one of the following integers can be expressed as the sum of consecutive positive integers?

## Problem 21

For any positive integer , let

What is ?

## Problem 22

Ashley, Betty, Carlos, Dick, and Elgin went shopping. Each had a whole number of dollars to spend, and together they had dollars. The absolute difference between the amounts Ashley and Betty had to spend was dollars. The absolute difference between the amounts Betty and Carlos had was dollars, between Carlos and Dick was dollars, between Dick and Elgin was dollars, and between Elgin and Ashley was dollars. How many dollars did Elgin have?

## Problem 23

In the figure, polygons , , and are isosceles right triangles; , , and are squares with sides of length ; and is an equilateral triangle. The figure can be folded along its edges to form a polyhedron having the polygons as faces. The volume of this polyhedron is

## Problem 24

A rising number, such as , is a positive integer each digit of which is larger than each of the digits to its left. There are five-digit rising numbers. When these numbers are arranged from smallest to largest, the number in the list does not contain the digit

## Problem 25

Let be a parallelogram and let , , , and be parallel rays in space on the same side of the plane determined by . If , , , and and and are the midpoints of and , respectively, then

## Problem 26

Triangle and point in the same plane are given. Point is equidistant from and , angle is twice angle , and intersects at point . If and , then

## Problem 27

Consider those functions that satisfy for all real . Any such function is periodic, and there is a least common positive period for all of them. Find .

## Problem 28

How many ordered triples of integers satisfy and ?

## Problem 29

Call a positive real number special if it has a decimal representation that consists entirely of digits and . For example, and are special numbers. What is the smallest such that can be written as a sum of special numbers?

## Problem 30

For positive integers , denote by the number of pairs of different adjacent digits in the binary (base two) representation of . For example, , , and . For how many positive integers less than or equal to does ?