Jan 2016

From now until Feb 2016 we will practice Mathcounts problems. Mathcounts is a team competition.  To participate, your school must register a team.

I’ve chosen problems from the 2014-2015 exercises. Unlike AMC8, Mathcounts are short-answer questions, not multiple choices.  Some even allow using a calculator, and I’ll indicate those with “(calculator)” at the end of the problems.  Mathcounts and AMC8 test similar math skills, though I feel Mathcounts is somewhat harder.

Like before, I’ll group problems based on topics. I’ll post new problems and answers to old problems each Saturday, and I’ll give out prizes on a regular basis. Enjoy!

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.

The next round of prizes will be given out on January 24 at 5:15.  Due to the snow storm, I have to postpone the ceremony.  Sorry for the inconvenience.
Sunday Jan 31 at 5:30pm
36 Chatham Road, Summit
RSVP at http://evite.me/kSDrj9BaeP.

Week of Jan 23-29. This week we will focus on problems related to area. Click GoogleForm to submit answers.

Answers: 79, 27.7, 60, 400, 682, 30, 16

Jan 23. What is the area of a circle that circumscribes a 6-cm by 8-cm rectangle? Express your answer to the nearest whole number.   (Calculator)

Jan 24. A pizza of diameter 16 inches has an area large enough to serve 4 people. What is the diameter of a pizza with an area large enough to serve 12 people? Express your answer as a decimal to the nearest tenth. (Calculator)

Jan 25. What is the area of triangle ABC with vertices A(2, 3), B(17, 11) and C(17, 3) ?

Jan 26. A square residential lot is measured to be 100 feet on each side, with a possible measurement error of 1% in each of the length and width. What is the absolute difference between the largest and smallest possible measures of the area given this possible error? (Calculator)

Jan 27. Five rectangles are arranged in a row. Each rectangle is half as tall as the previous one. Also, each rectangle’s width is half its height. The first rectangle is 32 cm tall. What is the sum of the areas of all five rectangles?

Jan 28. The amount of paint required to cover a surface is directly proportional to the area of that surface. The amount of paint needed to cover five spheres of radius 10 inches is the same as the amount of paint required to cover a solid right circular cylinder with radius 20 inches. What is the height of the cylinder?

Jan 29. ABCD is a square with side length 4 units, and AEFC is a rectangle with point B on side EF. What is the area of AEFC?

Week of Jan 16-22. This week we will focus on problems related to volume. Click GoogleForm to submit answers.

Answers: 4, 3, 1207, 144w^3, 4576, 72sqrt(2), 1/27

Jan 16. Two cylinders are equal in volume. The radius of one is doubled, and the height of the other cylinder is increased to k times its original height. If the two new cylinders are equal in volume, what is the value of k?

Jan 17. The surface area of a sphere, in square meters, and its volume, in cubic meters, are numerically equal. What is the length of the radius of the sphere?

Jan 18. Boynton’s sheet cake measures 18 × 24 inches and has a height of 4 inches. However, these measurements include a 3/4-inch thick layer of frosting on the top and sides. What is the volume of Boynton’s cake excluding the frosting? Express your answer to the nearest whole number. (Calculator)

Jan 19. A certain box of width w has a length that is twice its width, and its height is three times its width. What is the total volume of 24 of these boxes? Express your answer in terms of w.

Jan 20. For a particular rectangular solid with integer dimensions, the sum of its length, width and height is 50 cm. What is the absolute difference between the greatest possible volume and the least possible volume of the solid?

Jan 21. A pyramid with a square base of side length 6 m has a height equal to the length of the diagonal of the base. What is the volume of the pyramid? Express your answer in simplest radical form.

Jan 22. Connecting the centers of the four faces of a regular tetrahedron creates a smaller regular tetrahedron. What is the ratio of the volume of the smaller tetrahedron to the volume of the original one? Express your answer as a common fraction.

Week of Jan 9-15. This week we will focus on problems related to speed. Click GoogleForm to submit answers.

Answers: 70, 14, 15, 60, 2900, 22.5, 8: 12

Jan 9. Kimba chewed a piece of gum 42 times in one minute. If she continued to chew at the same rate, how many times would she chew her gum in 100 seconds?

Jan 10. How many fewer minutes does it take to drive 35 miles at 30 mi/h than to drive the same distance at 25 mi/h?

Jan 11. Cara needs to drive to Greenville. She can drive 50 mi/h along a 200-mile highway, or she can take a different route, which requires her to drive 150 miles at 60 mi/h and then 50 miles at 40 mi/h. How many minutes would Cara save by taking the faster route?

Jan 12. When Mr. Tesla drives 60 mi/h, the commute from home to his office takes 30 minutes less than it does when he drives 40 mi/h. What is the distance of Mr. Tesla’s commute from home to his office?

Jan 13. At 1:00 p.m. an airplane left the local airport and flew due east at 300 mi/h. At 3:00 p.m. a second plane left the same airport and flew due north at 400 mi/h. Assuming the curvature of the earth is negligible, how many miles apart are the planes at 8:00 p.m.? (Calculator)

Jan 14. Normally, the hose in Elena’s garden will fill her small pool in 15 minutes. However, a leak in the hose allows 1/3 of the water flowing through the hose to spill into the flower bed. How many minutes will it take to fill the pool? Express your answer as a decimal to the nearest tenth. (Calculator.)

Jan 15. Ellie and Emma live 1.04 miles from each other. They decide to meet by walking toward each other, Ellie at 2.4 mi/h and Emma at 2.8 mi/h. If they both leave at 8:00 a.m., at what time in the morning will they meet? (Calculator.)

Week of Jan 2-8. This week we will focus on problems related to the new year! Click GoogleForm to submit answers by Friday, Jan 8.

Answers: 37, 40, 39, 89, 8, 757, 3

Jan 2. If 2015 = 101a + 19b, for positive integers a and b, what is the value of a + b?

Jan 3. How many even factors does 2016 have?

Jan 4. What is the sum of prime factors of 2015?

Jan 5. Let n represent the smallest positive integer such that 2016 + n is a perfect square. Let m represent the smallest positive integer such that 2016 − m is a perfect square. What is the value of n + m?

Jan 6. What is the units digit of the product 2^2016 × 7^2015?  (2 to the power of 2016, 7 to the power of 2015)

Jan 7.  If 2015 + a = b for positive integers a and b, both of which are palindromes, what is the smallest possible value of a?

Jan 8.  What is the 2016th digit after the decimal point in the decimal representation of 1 / 13 ?