美国数学题2

美国数学题2
1、Jump Like a flea
A flea can jump 350 times its body length. If humans could jump like fleas, how far could you jump?
Solution
Answers will vary, depending on the height of the person. The average height of an 11–14-year-old male is 60 inches. The average height of a female of the same age is 62 inches, according to the National Institute of Health. A student who is 5 feet tall could jump about 1750 feet, or 583 yards. For varying answers, multiply the height by 350 and approximate that distance with a known landmark.
From Menu of Problems, April 2001
2、Product of Primes
2006 is the product of three primes, 2 × 17 × 59. Find the first year that occurs after 2006, which is the product of three consecutive primes.
Solution
2431, which is the product of 11 × 13 × 17.
From Menu of Problems, April 2006
Adding Digits
If you add the digits in a number, how many numbers between 0 and 1000 will have a sum of 15?
Solution
73. Only two ways are possible to add two digits to get a sum of 15: 6 + 9 and 7 + 8. Each combination results in six numbers: 69, 96, 609, 690, 906, and 960. Eight ways exist to combine three distinct digits for a sum of 15: 1 + 5 + 9, 1 + 6 + 8, 2 + 4 + 9, 2 + 5 + 8, 2 + 6 + 7, 3 + 4 + 8, 3 + 5 + 7, and 4 + 5 + 6. Again, each combination results in six numbers. Four other ways to get 15 are 1 + 7 + 7, 3 + 3 + 9, 3 + 6 + 6, and 4 + 4 + 7. Each combination results in three numbers. Finally, 555 fits the bill. All told, we have 12 + 48 + 12 + 1 = 73.
From April's Menu of Problems, April 1999
3、Twins
The product of the ages of twins and their younger brother is 36. How old are the children?
Solution
1, 6, and 6. The possible ages of the children are 1, 1, 36; 2, 2, 9; 3, 3, 4; and 1, 6, 6. The only choice that has the twins as the oldest children is 1, 6, 6.
From May's Menu of Problems, May 1999
4、5, 7, and 11 Minute Glasses
How can you measure 13 minutes exactly with a 5-minute, 7-minute, and 11-minute hourglass?
Solution
Start the 5- and 11-minute timers at the same time. When the 5-minute runs out, start timing for the 13 minutes. When the 6 minutes remaining on the 11-minute timer are finished, start the 7-minute timer.
From Menu of Problems, April 2002
5、Make the Statement True
Add one straight line segment to make the mathematical sentence 5 + 5 + 5 = 550 a true statement.
Solution
Change one of the plus signs to a number 4 by connecting the left and top endpoints. There might be other possibilities as well.
From April's Menu of Problems, April 2007
6、Smallest Odd Number
What is the smallest odd number you can obtain from the product of four different prime numbers?
Solution
1155. Remember that 1 is not a prime number. Two is the first prime number, but if it is one of the prime numbers used, the product will be even. Therefore, to obtain the smallest value, you would use 3, 5, 7, and 11, the product of which is 1155. From April's Menu of Problems, April 2007
7、Multiples of 6
Find six consecutive multiples of 6 whose sum is the least common multiple (LCM) of 13 and 18.
Solution
24, 30, 36, 42, 48, and 54. The LCM of 13 and 18 is 13 × 18 = 234, since the numbers are relatively prime. Since we are looking for 6 numbers, we can divide 234 by 6 to get 39. Therefore, we know that 39 is greater than 3 of the numbers and less than 3 of the numbers; 24, 30, and 36 are the multiples of 6 less than 39, and 42, 48, and 54 are the multiples of 6 greater than 39. The sum of 24, 30, 36, 42, 48, and 54 is 234. From March's Menu of Problems, March 2007
8、Drawing Rectangles
How many different rectangles can be drawn using the hour marks on a clock's face as vertices?
Solution
15.
The rectangles that can be drawn are shown below.
From February's Menu of Problems, February 2004 MTMS.
9、Paper Money
In how many different ways can you receive $20 from your bank if you ask for paper money only?
Solution
Ten, excluding the use of $2 bills.
Students can solve this problem by making an organized list.
Using $2 bills creates another thirty-one choices.
From November's Menu of Problems, November 1999 MTMS.
Contributed by Cynthia Barb, Kent State University--Stark Campus, Canton, Ohio; and Anne Larson Quinn, Edinboro University of Pennsylvania, Edinboro, Pennsylvania. 10、Half of a Half of a Half
What would you get if you divided a half of a half of a half by a half of a fourth of a half?
Solution
2. A half of a half of a half is 1/2×1/2×1/2 = 1/8, and a half of a fourth of a half is
1/2×1/4×1/2 = 1/16; 1/8 divided by 1/16 is 2.
From November's Menu of Problems, November 2006
11、Prime Numbers
Find the sum of the least and greatest two-digit prime numbers whose digits are also prime.
Solution
96.
Consider prime numbers such that each digit is 2, 3, 5, or 7. The smallest two-digit number that can be formed with these digits is 22, but it is not prime. Therefore, 23 is the least value.
The greatest two-digit numbers that can be formed with these digits are 77 and 75, but they are not prime. The number with the greatest value that meets the conditions is 73; therefore, 23 + 73 = 96.
From February's Menu of Problems, February 2005 MTMS.
Courtesy of David Rock at the University of Mississippi.
12、Multiple of 12
Find the greatest ten-digit positive multiple of 12 using each digit once and only once.
Solution
9,876,543,120. For a number to be a multiple of 12, it must be divisible by both 3 and 4. The sum of the ten digits 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 + 0 = 45, which is divisible by 3. To be divisible by 4, the last two digits must be divisible by 4. The largest number obtained using all ten digits once and only once is 9,876,543,210, but 10, the last two digits, is not divisible by 4. The next greatest value is 9,876,543,201, but 01 is not divisible by 4. The next greatest value is 9,876,543,120; since 20 is divisible by 4, the number is therefore both divisible by 3 and 4 and 12.
From March's Menu of Problems, March 2007
13、Positive Primes
What is the last digit (ones) of the product of the positive prime numbers less than 50?
Solution
0. Prime numbers less than 50 end in 1, 3, 7, or 9, with the exception of 2 and 5. Since 2 and 5 will have to be factors and 2×5 equals 10, the product must has a ones digit of 0.
From May's Menu of Problems, May 2007
14、Hands of a Clock
Determine the smaller angle between the hands of a clock at 8:20, 12:20, and 1:30. Solution
12:20, 110°; 8:20, 130°; 1:30, 135°.
The hour hand moves 360°/12 = 30° each hour, or 10° each 20 minutes and 5° each 10 minutes. The minute hand moves 30° every 5 minutes.
At 12:20, the angle between the hour and minute hands is 4(30°) - 10° = 110°, or
3(30°) + 20°.
At 8:20, the hour hand is 10° past the 8. The angle between the 4 and the 8 at 20 minutes past 8 is 4(30°) - 120°, so the angle between the hour and minute hands at 8:20 is 130°.
At 1:30, the angle between hands is 6(30°) - 45° = 135°, or 4(30°) + 15°.
Students should share other ways that they use to calculate these angles.
From January's Menu of Problems, January 2003 MTMS.
15、4-digit Even Numbers
How many four-digit numbers consist of only even digits?
Solution
500. The even digits are 0, 2, 4, 6, and 8. The first digit cannot be a 0, because the number would not be a four-digit number, so you may choose only 2, 4, 6, or 8 for the first digit. Five digits can be chosen for the remaining columns. Therefore, there are 4 × 5 × 5 × 5, or 500, ways to choose the digits. (Note that if the second digit were also 0, it would not be a three-digit number.)
From October's Menu of Problems, October 2006
16、Squares on a Board
How many squares are on a traditional 8 × 8 checkerboard? (By the way, the answer is not 64.)
Solution
204 squares. You have one 8 × 8 square, four 7 × 7 squares, nine 6 × 6 squares, sixteen 5 × 5 squares, twenty-five 4 × 4 squares, thirty-six 3 × 3 squares, forty-nine 2 × 2 squares, and sixty-four 1 × 1 squares, or 1 + 4 + 9 + 16 + 25 + 36 + 49 + 64 = 204. By constructing a table, you can see a pattern. Refer to print version to view table. From Menu of Problems, September 2006
17、Remainder
What is the remainder when the product (1492)(1776)(1812)(1999) is divided by 5? Solution
1. When considering the remainder when dividing by 5, we need to be concerned with only the units digit of each number. The product of the units digits is 2 × 6 × 2 × 9, or 216, which leaves a remainder of 1 when divided by 5.
From Menu of Problems, September 1999
18、Prime Factors
Find the difference between the least and greatest prime factors of 33,660. Solution
15. The prime factorization of 33,660 is 2 × 2 × 3 × 3 × 5 × 11 × 17. The difference between 17 and 2 is 15.
From January's Menu of Problems, January 2007
19、How Many Minutes
How many minutes represent 10 percent of one full week?
Solution
1,008 minutes; 7 days × 24 hours × 60 minutes = 10,080 minutes in a week; 10 percent is 1,008 minutes.
From April's Menu of Problems, April 2007
20、Candy Corn
Each Halloween, 20 million pounds of candy corn are sold. If each piece weighs about 1 gram, how many pieces are bought?
Solution
Roughly 9 billion. Each pound of candy corn is 453 grams, so 20 million x 453 grams ≅9 billion pieces of candy corn.
From October's Menu of Problems, October 2002 MTMS.。