Dec 2018 (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 Dec 9 – Dec 15, From the 2019 Mathcounts School Handboook.

Click GoogleForm to submit answers by Sat, Dec 15.

Dec 9: What is the probability that a sequence of five flips of a fair coin will not land heads up twice in a row? Express your answer as a common fraction.

Dec 10: Liz, Eva and Ace have played trivia as a team 33 times and won 24 times. What is the minimum number of games they must win to have an overall winning percentage of 80%?

Dec 11: A certain vending machine accepts nickels, dimes, quarters and dollar bills, and it provides change using nickels, dimes and quarters. If Sarah selects a healthful snack priced at $1.30 after inserting $2.00, in how many ways can the vending machine provide Sarah’s change?

Dec 12: What is the geometric mean of the median and mode of the set {23, 25, 3, 25, 20, 22, 21, 2, 1, 14, 12}? Express your answer in simplest radical form.

Dec 13: What is the mean of all three-digit positive integers whose digits are in the set {2, 0, 1, 9}?

Dec 14:  If we define the binary operation  as a  b = ab + b for all numbers a and b, what positive value of x satisfies the equation x  (4  x) = 550?

Dec 15: A circle with center P is inscribed in isosceles triangle ABC with apex angle A measuring 34 degrees. What is the degree measure of angle APC? Express your answer as a decimal to the nearest tenth.

 


Week of Dec 2 – Dec 8, Transformations (from the 2019 Mathcounts School Handboook)

Click GoogleForm to submit answers by Sat, Dec 8.

Dec 2: A point P(−3, 2) is translated right 4 units to its image P′. The point P′ is then translated up 3 units to its image P″. What is the distance from P to P″?

Answer: 5

Dec 3: A segment has endpoints A(0, 0) and B(−3, 4). Point C is the image of point B translated down 4 units and left 3 units. What is the perimeter of triangle ABC?

Answer: 16

Dec 4: A point Q(−3, 4) is reflected across the x-axis, and then the image Q′ is reflected across the line x = 2. What are the coordinates of the image Q″? Express your answer as an ordered pair.

Answer: (7,-4)

Dec 5: A point S(1, 6) is reflected across the line x − 2y = −6. What is the sum of the
coordinates of the image S′?

Answer: S'(3, 2). Sum of coordinates is 5.

Dec 6: What are the coordinates of the image of point D(−5, −3) when it is rotated 90 degrees clockwise about the origin? Express your answer as an ordered pair.

Answer: (-3, 5)

Dec 7: What are the coordinates of the image of the point E(3, −1) when it is rotated 90 degrees counterclockwise about the point F(5, 4)? Express your answer as an ordered pair.

Answer: (10,2)

Dec 8: A segment with endpoints G(−2, 3) and H(4, 7) is dilated by a scale factor of 2/3 with center of dilation (0, 0). What is the sum of all the coordinates of G′ and H′?

Answer: (-2+3+4+7)*2/3 = 8

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