1997 AJHSME Problems
1997 AJHSME (Answer Key) Printable versions: • AoPS Resources • PDF | ||
Instructions
| ||
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 |
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 See also
Problem 1
Problem 2
Ahn chooses a two-digit integer, subtracts it from 200, and doubles the result. What is the largest number Ahn can get?
Problem 3
Which of the following numbers is the largest?
Problem 4
Julie is preparing a speech for her class. Her speech must last between one-half hour and three-quarters of an hour. The ideal rate of speech is 150 words per minute. If Julie speaks at the ideal rate, which of the following number of words would be an appropriate length for her speech?
Problem 5
There are many two-digit multiples of 7, but only two of the multiples have a digit sum of 10. The sum of these two multiples of 7 is
Problem 6
In the number the value of the place occupied by the digit 9 is how many times as great as the value of the place occupied by the digit 3?
Problem 7
The area of the smallest square that will contain a circle of radius 4 is
Problem 8
Walter gets up at 6:30 a.m., catches the school bus at 7:30 a.m., has 6 classes that last 50 minutes each, has 30 minutes for lunch, and has 2 hours additional time at school. He takes the bus home and arrives at 4:00 p.m. How many minutes has he spent on the bus?
Problem 9
Three students, with different names, line up in single file. What is the probability that they are in alphabetical order from front-to-back?
Problem 10
What fraction of this square region is shaded? Stripes are equal in width, and the figure is drawn to scale.
Problem 11
Define as the number of whole number divisors of . For example, because 3 has two divisors, 1 and 3. Find the value of
Problem 12
Find
Problem 13
Three bags of jelly beans contain 26, 28, and 30 beans. The ratios of yellow beans to all beans in each of these bags are , , and , respectively. All three bags of candy are dumped into one bowl. Which of the following is closest to the ratio of yellow jelly beans to all beans in the bowl?
Problem 14
There is a set of five positive integers whose average (mean) is 5, whose median is 5, and whose only mode is 8. What is the difference between the largest and smallest integers in the set?
Problem 15
Each side of the large square in the figure is trisected (divided into three equal parts). The corners of an inscribed square are at these trisection points, as shown. The ratio of the area of the inscribed square to the area of the large square is
Problem 16
Penni Precisely buys $100 worth of stock in each of three companies: Alabama Almonds, Boston Beans, and California Cauliflower. After one year, AA was up 20%, BB was down 25%, and CC was unchanged. For the second year, AA was down 20% from the previous year, BB was up 25% from the previous year, and CC was unchanged. If A, B, and C are the final values of the stock, then
Problem 17
A cube has eight vertices (corners) and twelve edges. A segment, such as , which joins two vertices not joined by an edge is called a diagonal. Segment is also a diagonal. How many diagonals does a cube have?
Problem 18
At the grocery store last week, small boxes of facial tissue were priced at 4 boxes for $5. This week they are on sale at 5 boxes for $4. The percent decrease in the price per box during the sale was closest to
Problem 19
If the product , what is the sum of and ?
Problem 20
A pair of 8-sided dice have sides numbered 1 through 8. Each side has the same probability (chance) of landing face up. The probability that the product of the two numbers that land face-up exceeds 36 is
Problem 21
Each corner cube is removed from this cube. The surface area of the remaining figure is
Problem 22
A two-inch cube of silver weighs 3 pounds and is worth $200. How much is a three-inch cube of silver worth?
Problem 23
There are positive integers that have these properties:
- the sum of the squares of their digits is 50, and
- each digit is larger than the one to its left.
The product of the digits of the largest integer with both properties is
Problem 24
Diameter is divided at in the ratio . The two semicircles, and , divide the circular region into an upper (shaded) region and a lower region. The ratio of the area of the upper region to that of the lower region is
Problem 25
All of the even numbers from 2 to 98 inclusive, excluding those ending in 0, are multiplied together. What is the rightmost digit (the units digit) of the product?
See also
1997 AJHSME (Problems • Answer Key • Resources) | ||
Preceded by 1996 AJHSME |
Followed by 1998 AJHSME | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AJHSME/AMC 8 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.