# Feb 2019 (Mathcounts)

I will post seven new problems every weekend, one for each day of the week. The idea is that you work on your own during the week and I will post answers the following Saturday. For those of you who practiced with me last year, the process is similar.

For instructions on submitting weekly answers, click on the “Logistics” tab.
For previously posted problems, click on the “Archive” tab.
For previous prizes, click on the “News” tab.

Week of Feb 10 – Feb 16.

Click GoogleForm to submit answers by Sat, Feb 16.

Feb 10: On a certain math exam, 10% of the students got 70 points, 25% got 80 points, 20% got 85 points, 15% got 90 points, and the rest got 95 points. What is the difference between the mean and the median score of this exam?

Feb 11: Every high school in the city of Euclid sent a team of 3students to a math contest. Each participant in the contest received a different score. Andrea’s score was the median among all students, and hers was the highest score on her team. Andrea’s teammates Beth and Carla placed 37th and 64th, respectively. How many schools are in the city?

Feb 12: In 1995 the population of a town was a perfect square. Ten years later, after an increase of 150 people, the population was 9 more than a perfect square. Now, in 2015, with an increase of another 150 people, the population is once again a perfect square. Which of the following is closest to the percent growth of the town’s population during this twenty-year period?

Feb 13: Find the number of ordered pairs of integers $(x, y)$ that satisfy $1/x+1/y=1/7$.

Feb 14: All even numbers from 2 to 98 inclusive, except those ending in 0, are multiplied together. What is the rightmost (units digit) of the product?

Feb 15: A charity sells 140 benefit tickets for a total of \$2001. Some tickets sell for full price (a whole dollar amount), and the rest sell for half price. How much money is raised by the full-price tickets?

Feb 16: How many positive cubes divide 3!⋅5!⋅7!?

Week of Feb 3 – Feb 9. Happy Lunar New Year!

Click GoogleForm to submit answers by Sat, Feb 9.

Feb 3: Two tour guides are leading six tourists. The guides decide to split up. Each tourist must choose one of the guides, but with the stipulation that each guide must take at least one tourist. How many different groupings of guides and tourists are possible?

Answer: $2^6 - 2 = 62$.

Feb 4: How many three-digit numbers satisfy the property that the middle digit is the average of the first and the last digits?

Answer: The first and last digits are both odd numbers: 5*5; the first and last numbers are both even numbers: 4*5. So the total is 45.

Feb 5: How many line segments have both their endpoints located at the vertices of a given cube?

Answer: ${8 \choose 2} = 28$.

Feb 6: Three tiles are marked X and two other tiles are marked O. The five tiles are randomly arranged in a row. What is the probability that the arrangement reads XOXOX?

Answer: 3! 2! / 5! = 1/10.

Feb 7: Pat wants to buy four donuts from an ample supply of three types of donuts: glazed, chocolate, and powdered. How many different selections are possible?

Answer: Four donuts have the same flavor: 3. Three donuts have the same flavor: 3*2=6. Two donuts have one flavor and the other two have the same flavor: 3. Two donuts have the same flavor and the other two have different flavors: 3. So the total is 3+6+3+3=15.

Feb 8: How many four-digit positive integers have at least one digit that is a 2 or a 3?

Answer: The number of 4-digit integers is 9*10*10*10. Among those 7*8*8*8 have no digit that is 2 or 3. So the number that has at least one digit that is 2 or 3 is 9*10*10*10-7*8*8*8=5416.

Feb 9: A student must choose a program of four courses from a menu of courses consisting of English, Algebra, Geometry, History, Art, and Latin. This program must contain English and at least one mathematics course. In how many ways can this program be chosen?