Algebra Problems, Geometry Problems, Math Problems, Number Theory Problems, Probability Problems

Daily Quiz #2: March 23, 2014

As of writing this post, it looks like many of you have not seen the contest yet. I’m hoping those of you that do know about it will tell the others, and I’ll give you a day to get points for the questions you answer on Quiz #1. Remember, I’m preparing you for your competition this weekend!

6. Willy Wonka’s famous ice cream shop has three types of ice cream cones: the Rocky Road Regular, the Tiger Stripe Tall, and the White Chocolate Wide. Yum. They all have the same dimensions, except the tall’s height is three times the regular’s, and the wide’s diameter is twice the regular’s. Does the tall cone or the wide cone hold more ice cream, and how much more? Express your answer as a common fraction (ratio of the bigger cone to the smaller one). (10 pts.)

7. Given a standard, fair coin, what is the probability of flipping the coin three times and getting exactly two heads? (10 pts.)

8. If the sum of 10 consecutive numbers is 395, what is the sum of the odd numbers among them? (10 pts.)

9. Bonus. Joh can mow half an acre of lawn in 1 hour. His good friend Num can mow a full acre in one and a half hours. If they start mowing a 10.5-acre lawn at 10:00 AM together, at what time will they finish? (20 pts.)

10. Bonus. The height of a falling object in feet as time passes is given by h = -16t2vts, where v is the initial velocity of the object and s is the initial height. If a projectile is launched off a 24-foot building at an initial velocity of 40 feet per second, how many seconds later will the projectile hit the ground? (20 pts.)

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Algebra Problems, Geometry Problems, Math Problems, Number Theory Problems, Probability Problems

Daily Quiz #1: March 22, 2014

The state competition is coming up! We’re already making great progress towards doing well at state – now we just need to keep up the skills until next weekend. So, here’s what I have in mind: every day from now until Thursday, I’ll post five problems on this blog. You, the Mathlete, should solve everything you can and post your answers (along with your name) in the comments. The next day, the answers will be posted so you can check your work.

Here’s the game, though: you will receive points for each question you answer correctly. Every day, I’ll post each person’s rankings along with the answers. Who will be the best Mathlete of all? We shall see….

1. I bought a one-year membership to the Abscissa Nature Park the other day for $99. The membership lets me visit the park for only $5, while non-members must pay $18. How many visits do I have to make this year to make buying the membership worth my money? (10 pts.)

2. Srikar is looking for books at the Broadmoor Library, and there are 70 fiction books and 80 nonfiction books on the nearest shelf. If no two books are the same, what is the probability that he will randomly choose one book of each kind without replacement? (10 pts.)

3. Sunjay is thinking of three positive integers. He tells his friend Robert that the sum of the first and second numbers is 8 less than the sum of the first and third, and 14 less than the sum of the second and third. If the second number is three times the first number, what is the sum of all three numbers? (10 pts.)

4. Bonus. What is the sum of 17268 and 3246? Express your answer in base 10. (20 pts.)

5. Bonus. The plot of land for Caroline’s new house is a trapezoid with two right angles, two sides of length 300 m, and a diagonal of length 500 m. What is the length of the other diagonal? Express your answer as a decimal to the nearest tenth of a meter. (20 pts.)?

Geometry Problems, Math Problems

Problem of the Day: 3/13/13

In the figure below showing three right triangles, segment PR measures 9 cm. What is the measure of segment US? Express your answer in simplest radical form.

20130313-072539.jpg

Solution to yesterday’s problem:

Let’s use the blanks method to solve this problem. We have 10 people to choose from and 10 seats.

__10__ __9__ __8__ __7__ __6__ __5__ __4__ __3__ __2__ __1__

However, we have to divide by 10 because each arrangement could start at any of the 10 seats around the table. Multiplying the blanks and dividing by 10, we find a total of 362,880 ways.

Geometry Problems, Math Problems

Problem of the Day: 3/4/13

Two sides of a triangle measure 10 cm and 16 cm. What is the sum of the possible values of the other side?

Solution to yesterday’s problem:

Since the triangle has all integer lengths, it must be a Pythagorean triple. The side of the triangle that measures 24 inches could be the short side or the long side of the triangle. Let’s cover the cases where this side is the short side.

Pythagorean triples that could be candidates for this triangle are 3-4-5, 5-12-13, and 8-15-17. For 3-4-5, we know that if the short side is 8 times 3, then the long side would be 8 times 4, or 32. The triangle could not be 5-12-13 because 24 does not divide into 5. For 8-15-17, if the short side is 3 times 8, then the long side would be 3 times 25, or 45.

Moving on to the cases where the long side is 24 inches. Candidates for the triangle are again 3-4-5, 5-12-13, and 8-15-17. For 3-4-5, if the long side is 6 times 4, then the short side is 6 times 3, or 18. For 5-12-13, if the long side is 2 times 12, then the short side is 2 times 5, or 10.

Adding up all these values gives a final answer of 105.

Geometry Problems, Math Problems

Problem of the Day: 3/3/13

A certain right triangle has all integer lengths. One of the triangle’s legs is 24 inches. What is the sum of all possible lengths of the other leg?

Solution to yesterday’s problem:

Let’s use our favorite method to solve this problem: the blanks method. There will be 10 blanks for 10 digits:

____ ____ ____ ____ ____ ____ ____ ____ ____ ____

The first digit can be any number. The second digit can be any number except the one used in the first digit, and the same principle goes for the third digit.

__10__ __9__ __8__ ____ ____ ____ ____ ____ ____ ____

The fourth and fifth digits can be any number from 0-9.

__10__ __9__ __8__ __10__ __10__ ____ ____ ____ ____ ____

The last five digits, however, can only be one digit: the digit used in its corresponding spot of the palindrome. For instance, the last digit must be the same as the first digit.

__10__ __9__ __8__ __10__ __10__ __1__ __1__ __1__ __1__ __1__

We don’t need to divide by anything because in phone numbers, order matters! Multiplying the numbers together, we find that there are 72,000 phone numbers that fit the bill.

Geometry Problems, Math Problems

Problem of the Day: 2/17/13

In the figure below, PQSR is a square with side length 3 inches, and RT is 5 inches. What is the length of the lower segment of QS?

20130217-104810.jpg

Solution to yesterday’s problem:

We need to find the two parts of the probability: the total number of possible occurrences and the number of occurrences that satisfy our conditions. We can find the first number using the blanks method. There are two dice, so there are two blanks:

____ ____

There are six possibilities for both the first and second dice:

__6__ __6__

So there are 36 possibilities. However, we need to divide by 2 because there are two possibilities listed for each desired possibility; for instance, 2 and 4 is listed as 2 and 4 and 4 and 2. Finally the first number in the probability is 18.

For the second number, we count the number of combinations whose difference is 2. This is quite simply 1 and 3, 2 and 4, 3 and 5, and 4 and 6. So the second number is 4. Simplifying the probability, we find an answer of 2/9.