Feb 2016

We are practicing Mathcounts 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.


Week of Feb 27-Mar 4. This week we will focus on lines and coordinates. Click GoogleForm to submit answers.

Answers: (3,3), 5, (-3,1), 45, 10, (c^2)/(2ab), 10

Feb 27.  Point A lies at the intersection of y = x and y = (− 2 / 3) x + 5. What are the coordinates of A? Express your answer as an ordered pair.

Feb 28.  What is the sum of the coordinates of the point at which y = x − 3 and y = −2x + 9 intersect?

Feb 29.  What are the coordinates of the point at which the segment with endpoints (2, 6) and (5, 9) intersects the segment with endpoints (−1, −1) and (5, −7)? Express your answer as an ordered pair.

Mar 1. A line segment has endpoints (−5, 10) and (a, b). If the midpoint of the segment is (13, −2), what is the absolute difference between a and b?

Mar 2. All points with coordinates (x, y) that are equidistant from the points (1, 3) and (7, 11) lie along a single line. When the equation of the line is written in the form y = mx + b, what is the value of b?

Mar 3. The line with equation ax + by = c, where a, b and c are positive, forms a right triangle with legs on the x- and y-axes. What is the area of the triangle? Express your answer as a common fraction in terms of a, b and c.

Mar 4. The point (8, k) in the first quadrant is the same distance from the point (0, 4) as it is from the x-axis. What is the value of k?


 

Week of Feb 20-26. This week we will focus on algebra and number theory. Click GoogleForm to submit.

Answers: 7/3, 11/6, 16, 1024, -5050, 5, 10

Feb 20.  For non-negative integers m and n, (m+n)/(m-n) = (25 /4) (m-n)/(m+n) and m > n. What is the value of m/n ?

Feb 21.  If 1/ n + 1/ (2n) + 1/ (3n) = k, what is the value of nk? Express your answer as a common fraction.

Feb 22.  If a and b are positive integers such that ab = 48 and a – b = 8, what is the value of a + b?

Feb 23. What is the value of 2 * 4 * 6 * 8 * …. *20 / 10!  ?

Feb 24. What is the value of 1^2 – 2^2 + 3^2 – 4^2 + … + 99^2 – 100^2?

Feb 25. What is the value of (20^2 – 15^2)/ (18^2 – 17^2) ?

Feb 26. If 6^12 = 6(6^n + 6^n + 6^n + 6^n + 6^n + 6^n ), what is the value of n?


Week of Feb 13-19.  This week we focus on evaluating functions and solving equations.  Click GoogleForm to submit.

Answers: 2, -7, 5, 4, 9, 0, 2

Feb 13.  The function f(x) is defined as x+4 when x <-1 and as x^2−6 when x ≥−1.  What is the value of f (f (2))?  (Notation: x^2 means x squared.)

Feb 14. Given that f (x) = 3x − 7 and g (x) = x^2 − 4, what is the value of  f(g(f(3)))?

Feb 15. If f (x) = sqrt(x + 4), for what value of x does f (x) = 3?  (Notation: sqrt is square root.)

Feb 16. If x + 2y + 3z = 6, 2x + 3y + z = 8 and 3x + y + 2z = 10, what is the value of x + y + z?

Feb 17.  If 5a − b − c = 36 and b = c = a/2, what is the value of a?

Feb 18. If (x + y)^2 = x^2 + y^2, what is the value of xy?

Feb 19.  If x and y are negative integers and x – y = 1, what is the least possible value for xy?


Week of Feb 6-12. This week we will focus on primes and factors. Click GoogleForm to submit answers. 

I realize that problems on Feb 6 and Feb 8 are hard if you don’t know the formula.  So let me offer some explanation here.
To find the sum of all factors of 144,
  • prime factorization:  144 = 2^4 * 3^2
  • sum of factors = (2^0 + 2^1 + 2^2 + 2^3 + 2^4) * (3^0+3^1+3^2) = 31*13  (If you expand the expression, do you see that all factors of 144 are included?)

To find the sum of all factors of 2016,

  • prime factorization: 2016 = 2^5 * 3^2 * 7^1
  • sum of factors = (2^0 + 2^1 + 2^2 + 2^3 + 2^4 + 2^5) * (3^0+3^1+3^2) * (7^0+7^1) = 63*13*8

 


 

Answers: 6552, 3456, 2015, 36, 41, 34, 76

Feb 6. What is the sum of the factors of 2016?

Feb 7. What is the product of the greatest common factor and least common multiple of 48 and 72? 

Feb 8. What is 5 times the sum of all the distinct positive factors of 144?

Feb 9. How many perfect squares are factors of 12!?

Feb 10. The product of three consecutive prime numbers is 2431. What is their sum? 

Feb 11. What is the sum of all positive integers less than 20 that cannot be written as the sum of two prime numbers?

Feb 12.  Becky tries an experiment. She writes some numbers on the blackboard and then applies the following rule: she picks any number on the board that is greater than 1, erases it and replaces it with the list of its proper divisors. For example, if the number 6 was on the board, she would apply the rule by erasing the 6 and replacing it with the numbers 1, 2 and 3. The experiment ends when there are only 1s left on the board. If Becky begins with just the number 72 on the board, how many 1s will be on the board when she is finished?


Week of Jan 30-Feb 5, 2016. This week we will focus on percentage calculations. Click GoogleForm to submit answers.

Answers: 58, 51, 14580, 29, 60, 61, 460

Jan 30. The peak of volcano Mauna Kea is 13,803 feet above sea level. When measured from its oceanic base, it measures 33,100 feet vertically to its peak. What percent of Mauna Kea’s altitude is below sea level? Express your answer to the nearest whole number.   (Calculator)

Jan 31. Kiera and Aubrey were the only candidates for SGA President. Kiera received 44% of the 7th graders’ votes, and Aubrey received 42% of the 8th graders’ votes. The student body consists of 325 7th graders and 350 8th graders, and every student voted for one of these two candidates. What percent of the students’ votes did the winner receive? Express your answer to the nearest whole number.(Calculator)

Feb 1. A car’s present value is $20,000, and its value decreases by the same percentage every year. At the end of one year, it will be worth $18,000. What will it be worth at the end of 3 years?

Feb 2. A path crosses a rectangular field on a diagonal. If someone travels across the field on the diagonal, instead of walking along the sides, what is the greatest possible percent reduction in total distance traveled? Express your answer to the nearest whole number. (Calculator)

Feb 3. The original price for a pair of shoes was increased by 150%, and then this new price was decreased by 75%. By what percent must the current price be increased to return to the original price?

Feb 4. Each year for the first five years of life, a baby elephant’s weight increases by 10%. By what percent of its birth weight does an elephant’s weight increase during these five years? Express your answer to the nearest whole number.

Feb 5. Bennie is ordering a new computer. A 6.25% sales tax will be added to the price of the computer, and then an $11 delivery charge will be added to that total. If Bennie has $500 to spend, what is the maximum price of a computer that he can afford? Express your answer to the nearest whole number. (Calculator)

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