# Mar 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 Mar 26-Apr 1. Among all the problems posted in during the past 7 months, I feel counting and probability problems are the hardest for all. Therefore, I’ll finish this season with 7 probability problems. Click GoogleForm to submit answers.

Answers: 15/36, 1/24, 3/7, 1/7, 16/47, 1/5, 6/25

Mar 26. Will and Ian each randomly choose a positive divisor of 20. What is the probability that the least common multiple of their chosen numbers is 20? Express your answer as a common fraction.

Mar 27. When three fair dice are rolled, what is the probability that the product of the three numbers rolled is a prime number? Express your answer as a common fraction.

Mar 28. In a certain game of darts, a dart that lands on the bull’s-eye scores 11 points, and a dart that lands any other place on the board scores 7 points. A player throws three darts, and the probability that the player hits the bull’s-eye is 50% on each throw, independent of previous results. Given that each dart lands on the board, and at least one dart lands on the bull’s-eye, what is the probability that the total number of points scored will be prime? Express your answer as a common fraction.

Mar 29. Four husband-and-wife couples attend a show. From these four couples, two people are randomly selected when the performer asks for volunteers from the audience. What is the probability that the two who are selected are a married couple? Express your answer as a common fraction.

Mar 30. Seven contestants enter a drawing that begins with 100 balls numbered 1 through 100 in a box. Each contestant randomly selects a ball without replacement. The two contestants who select balls with the two highest numbers each will win a cash prize. The first six contestants select balls numbered 83, 5, 44, 67, 21 and 30. What is the probability that the last contestant will win a cash prize? Express your answer as a common fraction.

Mar 31. Two distinct numbers are selected at random from the set {1, 2, 3, 4, 5, 6}. What is the probability that their product is an odd number? Express your answer as a common fraction.

April 1.  The positive integers 1 through 50 are written on 50 cards with one integer on each card. If Matt draws one card at random, what is the probability that the number on the card is a multiple of 6 or 8? Express your answer as a common fraction. (Calculator)

Week of Mar 19-25. This week we will focus on algebra. Click GoogleForm to submit answers.

Answers: 2015, 331776, 6, 226, 12, 10, 40

Mar 19. When a > 5 and b <= 5, a @ b = (a + b)(a − b), and when a <= 5 and b > 5, a @ b = (b + a)(b − a). What is the value of 2 × ((2 @ 6) @ −4) − 1?

Mar 20. For positive integer n, n? = n! · (n − 1)! · … · 1! and n# = n? · (n − 1)? · … · 1?. What is the value of 4# · 3# · 2# · 1#?

Mar 21. If a  b = a · b + 3, what is the absolute difference between (10  11)  12 and 10  (11  12)?

Mar 22. If X # Y = (X/Y) + (XY), what is the value of 15 # (6 # 2)?

Mar 23. If a ▲ b = |a – b|, then what is the sum of all numbers x such that (3 ▲ x) ▲ 8 = 2?

Mar 24. If ▲+ ▲= + +  and  = 16 – ( ▲+ ▲ ), what is the value of ▲+  ?

Mar 25. The digits of the addends in the sum shown are represented by the letters A, B and G. What is the value of A × (B + G)?

G A B + B A G = 1090

Week of Mar 12-18. This week we will focus on miscellaneous puzzles. Click GoogleForm to submit answers.

Answers: 16, 1353, B and C, 15, 25, 2, 1

Mar 12.  Rob has 10 white, 8 red and 6 blue socks in his drawer. If he selects socks from the drawer randomly, without looking, what is the least number of socks Rob must select to guarantee that he has removed a pair of white socks?

Mar 13. How many people must be in a group to guarantee that at least 3 of them share the same first initial and last initial?

Mar 14. Which two of the following four statements, labeled A through D, are true statements?
A: Statement B is false, but statement C is true.
B: Statement C is true, but statement D is false.
C: Statement D is false, and statement A is false.
D: Statement A is true, and statement B is true.

Mar 15. In a survey, 30 people reported that they enjoy some combination of walking, hiking and jogging. The number who enjoy only walking, the number who enjoy only hiking and the number who enjoy only jogging are all equal. Likewise, the number who enjoy only walking and hiking, the number who enjoy only walking and jogging and the number who enjoy only hiking and jogging are equal. In addition, the survey showed that half as many people enjoy exactly two of these activities as those who enjoy only one activity. If three people enjoy all three activities, how many people enjoy jogging?

Mar 16. Given the following facts about the integers a, b, c, d, e and f, what is the value of a if 0 <a < 60?
a is odd.        b = (a −1)/ 2 is even.          c = b/2 is even.

d = c /2 is odd.        e = (d −1)/ 2 is odd.        f = (e −1)/ 2 is even.

Mar 17. The same digit A occupies both the thousands and tens places in the five-digit number 1A,2A2. For what value of A will 1A,2A2 be divisible by 9?

Mar 18. In Lewis Carroll’s Through the Looking-Glass, this conversation takes place between Tweedledee and Tweedledum. Tweedledum says, “The sum of your weight and twice mine is 361 pounds.” Tweedledee answers, “The sum of your weight and twice mine is 362 pounds.” What is the absolute difference in the weights of Tweedledee and Tweedledum?

Week of Mar 5-11. This week we will focus on arithmetic and geometric sequences. Click GoogleForm to submit answers.

Answers: 6, 9/8, 9, 12288, 12, 682, 10

March 5. A 63-inch long string is cut into pieces so that their lengths form a sequence. First, a 1-inch piece is cut from the string, and each successive piece that is cut is twice as long as the previous piece cut. Into how many pieces can the original length of string be cut in this way?

March 6. Three fractions are inserted between 1/ 4 and 1/ 2 so that the five fractions form an arithmetic sequence. What is the sum of these three new fractions? Express your answer as a common fraction.

March 7. If a1 = 13 and an = 77, for an arithmetic sequence of integers, a1, a2, a3, …, an , with n terms, what is the median of all possible values of n?

March 8. A sequence begins 1, 2, …, and each term after the second term is the sum of all preceding terms. What is the 15th term of this sequence?

March 9. In the sum 4 + 2 sqrt(2) + 2 + sqrt(2) + … each term is obtained by dividing the previous term by sqrt(2) . If the sum of the series, in simplest radical form, is m + n sqrt(2) , what is the value of m + n?  (sqrt(2) means square root of 2.)

March 10. 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?

March 11. A certain computer program will take 2000 years to run using current technology. Every year, advances in technology make it possible to run the program in half the time it would have taken starting in the previous year. However, once the program is started it cannot be interrupted to apply newer technology. Including the years spent waiting to start, what is the least number of years it will take to finish running the program? Express your answer to the nearest whole number.

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?