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.

We are taking a break. Happy Holidays to all! I’ll be back Jan 2019.

Week of Dec 16 – Dec 22, From the 2019 Mathcounts School Handbook.

Dec 16: What is the degree measure of the complement to an angle that is a supplement to an angle of measure 163 degrees?

Answer: 90 – (180-163) = 73

Dec 17: Every morning, the Sharetrain arrives in Mountain View at 9:19 a.m. It takes Miranda between 17 and 21 minutes to walk to the train station from home. If she wants to guarantee that she will arrive at the station with at least 5 minutes to spare, what is the latest time she can leave home?

Answer: She needs to leave 26 minutes before 9:19am. So it’ll be 8:53am.

Dec 18: Ryan picks two different numbers from the set {2, 3, 5, 7} and multiplies them. What is the absolute difference between the greatest and the least products that Ryan can get?

Dec 19: Zu’s zoo offers a promotional deal: get a free \$3 cotton candy and a free \$2 soda with the purchase of five \$12 admission tickets. A group of 20 students will each purchase an admission ticket. If half of them want cotton candy and one-fourth of them want soda, how many dollars would they save by using the promotional deal?

Answer: Group tickets for 20 students will offer them 4 free cotton candies and 4 free sodas. This will save them \$12+\$8 = \$20.

Dec 20: Preston has four potatoes, each of which weighs a whole number of ounces. The median weight of his potatoes is 11 ounces, and the mean weight of his potatoes is 12 ounces. What is the greatest possible difference between the weight of the heaviest and lightest of his potatoes?

Answer: The total weight of 4 potatoes is 48 ounces. The total weight of the two potatoes that are neither heaviest or lightest is 22 ounces. So the total weight of the heaviest and lightest is 26 ounces. The heaviest one is 25 ounces and lightest 1 ounce. The difference is 24 ounces.

Dec 21: What is the measure of an interior angle of a regular polygon with 90 sides?

Answer: An exterior angle is 360/90 = 4. An interior angle is 180-4=176.

Dec 22: What is the value of 1 × 12 + 2 × 11 + 3 ×10 + 4 × 9 + 5 × 8 + 6 × 7 + 7 × 6 + 8 × 5 + 9 × 4 + 10 × 3 + 11 × 2 + 12 × 1?

Week of Dec 9 – Dec 15, From the 2019 Mathcounts School Handbook.

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.

Answer: There is 1 way to arrange 5 tails and 0 heads, 5 ways to arrange 4 tails and 1 head,  6 ways to arrange 3 tails and 2 heads, and 1 way to arrange 2 tails and 3 heads. The probability is 13/32.

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%?

Answer: (33+x)*80% >= 24+x. Therefore, x=12.

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?

Answer: For 70 cents, 2Q2D, 2Q1D2N, 2Q4N, 1Q4D1N, 1Q3D3N, 1Q2D5N, 1Q1D7N, 1Q9N, 7N, 6N2D, 5N4D, 4N6D, … A total of 16 ways.

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.

Answer: Sort the numbers: 1, 2, 3, 12, 14, 20, 21, 22, 23, 25, 25. Median=20, Mode=25. Geometric mean=sqrt(500)=10sqrt(5).

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

Answer: 2 appears in the 100s position 16 times, so that contributes 200*16 to the sum. 2 appears in the 10s position 12 times, and appears in the 1s position 12 times. The sum equals 200*16+900*16+100*16+20*12+90*12+10*12+2*12+9*12+1*12 = 20784. The number of 3 digit numbers is 3*4*4=48. So the mean is 20784/48=433.

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.

Answer: Angle C=73. Angle APC = 180-(34+73)/2=126.5.

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

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″?

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?

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.

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.