Problem of the Week

4/26/19

Dog Show

Of the animals entered in a dog show, the number of poodles is at least one-fifth of the number of beagles and at most one-sixth the number of collies. The number of dogs that are poodles or beagles is at least 23. Find the minimum number of collies entered in the show.

4/12/19

Less than 2005°

The sum of the interior angles of a polygon is less than 2005°. What is the largest possible number of sides of the polygon?

Solution

13. The sum of the interior angles of a polygon = 180°(n– 2) < 2005° →180°n< 2365° →n< 13.139. Since nmust be an integer, the largest value for nis 13.

4/1/19

One or Ten?

Given that they are made of the same material, which is heavier: a ball with a radius of 10 inches or 10 balls each with a radius of 1 inch? For credit, show your answer and PROVE WHY it is correct.

Solution

The ball with a radius of 10 inches is heavier. Since the volume of a sphere is (4/3)πr³, one ball with a radius of 10 inches would have a volume of (4/3)π⋅10³= (4000/3)π in³. Thus 10 balls of radius 1 inch would have an accumulated volume of only

10⋅[(4/3)π⋅1³] = (40/3)π in³.

3/25/19

Tamika > Carlos?

Tamika selects two different numbers at random from the set {8, 9, 10} and adds them. Carlos takes two different numbers at random from the set {3, 5, 6} and multiplies them. What is the probability that Tamika's result is greater than Carlos's result?

Solution

4/9. Tamika can get the numbers 8 + 9 = 17, 8 + 10 = 18, or 9 + 10 = 19. Carlos can get 3 × 5 = 15, 3 × 6 = 18, or 5 × 6 = 30. The possible ways to pair these numbers are (17, 15), (17, 18), (17, 30), (18, 15), (18, 18), (18, 30), (19, 15), (19, 18), and (19, 30). Four of these nine pairs show Tamika with a higher result, so the probability is 4/9.

3/18/19

A New Mathematical Operation

If a*b= a^b– b, find (2*3)*4

Solution

621. (2*3)*4 = (2³– 3)*4 = (8 – 3)*4 = 5*4 = 5⁴– 4 = 625 – 4 = 621.

3/11/19

A Quarter Pounder

A quarter-pound hamburger contains approximately 80 calories per ounce of meat, an average french-fry contains about 14 calories, a cola contains about 10 calories per ounce, and a bun contains 200 calories. Suppose you have a quarter-pound hamburger with a bun and six ounces of cola. How many french-fries can you eat and still keep your meal below 800 calories?

Solution

15. We have 80h+ 14f+ 10c+ 200 < 800. Using the information in the problem we can write 80(4) + 14f+ 10(6) + 200 < 800. This implies that 14f< 220 and f< 16. Therefore, you can eat 15 french fries.

3/4/19

Perfect Squares and Cubes...NOT

How many numbers from 1 to 1 million, inclusive, are not perfect squares or perfect cubes?

Solution

998,910. There are 1000 perfect squares between 1 and 1 million; these are the squares of the first 1000 integers. Similarly, there are 100 perfect cubes—the cubes of the numbers from 1 to 100. Subtract the squares and the cubes from 1 million to get 998,900. However, every number that is a perfect sixth power has been subtracted twice (the largest of these is 10⁶= 1,000,000). Adding these back in gives 998,910.

2/25/19

999 Coins

Starting with a single pile of 999 coins, a person does the following in a series of steps: In step one, he splits the pile into two nonempty piles. Thereafter, at each step, he chooses a pile with 3 or more coins and splits this pile into two piles. What is the largest number of steps that is possible?

Solution

997. The number of steps is one less than the number of piles, and 998 is the largest number of piles, 997 with 1 coin and 1 with two coins.

2/11/19

House Prices

The average house price in Boomtown rose 30 percent each year for the last five years. If the average house price is currently $250,000, what was the average house price five years ago?

Solution

Approximately $67,332. If P was the average house price five years ago, then the current average price is 1.3^5P or 3.71293P. Thus, 250,000 = 3.71293P, so P=250,000/3.71293, which equals $67,332.27

2/4/19

Upright Integers

An integer is defined as upright if the sum of its first two digits equals its third digit. For example, 145 is an upright integer since 1 + 4 = 5. How many positive three-digit integers are upright?

Solution

45. From the definition, the first and second digits of an upright integer automatically determine the third digit, which is the sum of the first two digits. Consider those upright integers beginning with 1: 101, 112, 123, 134, 145, 156, 167, 178, and 189; there is a total of 9 such numbers. (Note that the second digit may not be 9; otherwise, the last “digit” would be 1 + 9 = 10.) Beginning with 2, the upright integers are 202, 213, 224, 235, 246, 257, 268, and 279; there is a total of 8 such numbers. We may continue this pattern of analysis to show that the numbers of upright integers beginning with a digit of 3, 4, 5, 6, 7, 8, or 9 are 7, 6, 5, 4, 3, 2, and 1, respectively. Therefore, there is a total of 9+8+7+6+5+4+3+2+1 = 45 three-digit upright integers.

1/28/19

Investments

A man has $10,000 to invest. He invests $4,000 at 5 percent and $3,500 at 4 percent. To have a yearly income of $500 from the investment, at what rate must he invest the remainder of the money?

Solution

About 65 minutes (to the nearest minute). The losing rates are $1000/hr., $500/hr., and $333.33/hr. The combined losing rate is $1833.33/hr. Thus,


1/14/19

What Is the Price?

Matthew and Matilda want to buy a set of DVDs. Matthew has $47 less than the purchase price, and Matilda has $2 less than the purchase price. If they pool their money, they still do not have enough to buy the DVDs. If the set costs a whole number of dollars, what is its price?

Solution

$48.

If Pis the price of the DVDs, Matthew has P- 47 dollars and Matilda has P- 2 dollars.

We know that (P- 47) + (P- 2) < P, since the amount of their combined savings is still less than the price of the DVDs.

Solving for P, we have P< 49; but P> 47, since Matthew had some money. Therefore, Pis a whole number such that 47 < P< 49, so the price of the DVDs is $48.

1/7/19

Tom and Bill

Tom is standing in a hole that is 4 feet deep. Bill asks him how much deeper he is going to dig the hole. Tom replies that he will dig 4 feet 2 inches deeper and that the top of his head will then be the same distance below ground level that it is now above ground level. How tall is Tom?

Solution

6 feet 1 inch. The top of his head will go down 4 feet 2 inches with the additional digging. Half that distance was above the hole before the additional digging, so he is 4 feet + 2 feet 1 inch, or 6 feet 1 inch, tall.

12/17/18

Wire Length

In the diagram, the rectangular wire grid contains 15 identical squares. The length of the grid is 10. What is the length of wire needed to construct the grid?

Solution

76. Since the length of the rectangular grid is 10, the side of each square in the grid is 10 ÷ 5 = 2. The height of the grid is therefore 6 (3 squares). There are four horizontal wires, each of length 10, and six vertical wires, each of length 6, for a total length of wire of 4 • 10 + 6 • 6 = 40 + 36 = 76.