Jan 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 going to have a Probability Stretch this week.  For this week, calculator is allowed.

Click GoogleForm to submit answers by Sat, Feb 3.

Week of Jan 28 – Feb 3

Jan 28:  Petra randomly selects a card from a standard deck of 52 playing cards. What is the percent probability that the card shows a red number greater than 6? Express your answer to the nearest hundredth.

Answer: 8/52 =  15% (7, 8, 9, 10 hearts and diamonds)

Jan 29: Max has eight identical cups. Each cup contains a different combination of nickels, dimes and quarters, each totaling 45 cents. Max randomly selects a cup. What is the probability that the cup he selects contains at least three dimes? Express your answer as a common fraction.

Answer:  2/8 = 1/4 (The 8 cups are 9N; 7N1D; 5N2D; 4N1Q; 3N3D; 2N1D1Q; 1N4D; 2D1Q)

Jan 30: A bag contains five chips numbered 2 through 6. Danya draws chips from the bag one at a time and sets them aside. After each draw, she totals the numbers on all the chips she has already drawn. What is the probability that at any point in this process her total will equal 10? Express your answer as a decimal to the nearest tenth.

Answer: 1/5.  The sequence starts with either a permutation of 2,3,5 or a permutation of 4,6.  So the number of desired sequences is 3!2! + 2!3!, and the prob is (3!2!+2!3!)/5!.

Jan 31: A drawer contains five socks: two green and three blue. What is the probability that two socks pulled out of the drawer at random will match? Express your answer as a common fraction.

Answer: (2 choose 2)(3 choose 2) / (5 choose 2) = 3/10 

Feb 1: A penny, a nickel and a dime are flipped. What is the probability that at least two coins land heads up and one of them is the nickel? Express your answer as a common fraction.

Answer: 3/8.  The total number of possibilities is 8; 3 heads + heads for DN + heads for QN.

Feb 2: When the circuit containing blinking lights A and B is turned on, lights A and B blink together. Then A blinks once every 5 seconds and B blinks once every 11 seconds. Lindsey looks at the two lights just in time to see A blink alone. What is the percent probability that the next light to blink will be A blinking alone?

Answer: 50%.  (I got this one wrong when I first wrote the solution.  Let me try again —  Consider a 55-second period:  0(AB), 5(A), 10(A), 11(B), 15(A), 20(A), 22(B), 25(A), 30(A), 33(B), 35(A), 40(A), 44(B), 45(A) and 50(A). When Lindsey sees A blink alone, it must be one of the 10 green times. Among them, exactly 5 will be followed by a green time. So the prob is 1/2. 

Feb 3: What is the percent probability that a randomly selected multiple of 3 less than or equal to 3000 is also a multiple of 5?

Answer: 200 /1000 = 1/5.  A thousand positive integers at most 3000 are multiples of 3, 200 are multiples of 15.  (This problem is a little ambiguous.  It doesn’t say the multiples of 3 can be 0.  I assume positive integers only. ) 


We are going to have a Travel Stretch this week.

Click GoogleForm to submit answers by Sat, Jan 27.

Week of Jan 21 – Jan 27

Jan 21: Jack and Jill travel up a hill at a speed of 2 mi/h. They travel back down the hill at a speed of 4 mi/h. What is their average speed for the entire trip? Express your answer as a mixed number.

Answer: (2/(1/2+1/4)=8/3

Jan 22: At 2:20 p.m., Jack is at the top of the hill and starts walking down at the exact same time that Jill, who is at the bottom of the hill, starts walking up. If they maintain the same uphill and downhill speeds from the previous problem, and the distance from the bottom to the top of the hill is 1.5 miles, at what time will Jack and Jill meet?

Answer: 1.5/6=0.25h = 15 min; 2:35pm

Jan 23: When Jack and Jill meet, as described in the previous problem, how many yards will they be from the bottom of the hill?

Answer: 1/2 mile = 880 yards

Jan 24: Alysha’s average speed when walking from home to the market is 5 mi/h, and it takes her 21 minutes longer than when she drives to the market. If Alysha drives to the market, along the same route, at an average speed that is eight times her average walking speed, how many minutes does it take her to drive from home to the market?

Answer: dist=2miles; 2/40 = 1/20 hour = 6min

Jan 25: Based on the previous problem, how many miles does Alysha travel to get from home to the market?

Answer: dist = 2miles

Jan 26: Jana begins jogging along a path and, 5 minutes later, Zhao begins riding his
bicycle along the same path, which has a length of 2 miles. Zhao rides his bicycle at a speed of 10 mi/h, and Jana’s jogging speed is 6 mi/h. If they both begin at one end of the path and end at the other, how many minutes after Zhao reaches the end of the path will Jana reach the end of the path?

Answer: Jana: 20 min; Zhao:12 min; 20-12-5=3

Jan 27: Based on the previous problem, how many minutes after Zhao begins riding will he catch up with Jana? Express your answer as a mixed number.

Answer:  d/10+1/2=d/6;   d=5/3;  d/10*60=10min


We are going to have a Statistics Stretch this week.

Click GoogleForm to submit answers by Sat, Jan 20. 

Week of Jan 14 – Jan 20

Jan 14: What is the range of the following scores? {43, 7, 61, 5, 19, 73, 3, 50, 72, 39}

Answer: 73-3 = 70

Jan 15: The average age of a group of students and teachers is 20. The average age of the teachers is 35, and the average age of the students is 15. What is the ratio of the number of teachers to the number of students?

Answer: 35t+15s = 20(s+t); t:s = 1:3

Jan 16: The mean of 8 numbers is 101.  If 4 numbers are each increased by 10, and the other 4 are each decreased by 20, what is the new average of the 8 numbers?

Answer: The total is reduced by 40. Average is reduced by 40/8 = 5. 

Jan 17: Seven judges give integer scores between 1 and 10 (inclusive) for Joseph’s project. His 7-score average is 8.0.  When the highest and lowest scores are removed, his 5-score average is 7.8. What is the least possible score that could be removed?

Answer: 7*8-5*7.8 = 17.  The highest score is 10, and the lowest 17-10 = 7.

Jan 18: The owners of two food carts calculate their weekly profits for three weeks. The medians and the highest weekly profit values are the same for the two carts. The mean weekly profit of Cart A is $27 more than that of Cart B. What is the absolute difference between the lowest weekly profit values of Cart A and Cart B?

Answer: 27*3=81

Jan 19: The mean of seven distinct positive integers is 20. What is the difference between the greatest and least possible medians of the seven integers?

Answer: least median: 1,2,3,4…; greatest median: 1,2,3,32,33,34,35; 32-4=28

Jan 20: The median and the mean of the five integers 10, 12, 26, x, x are equal. What is the sum of all possible values of x?

Answer: x=16 (x is median), x=41 (26 is median); x=1 (10 is median); sum = 58

 


We are going to have Patterns Stretch this week.  Here are two useful formulae:

  • Sum of arithmetic series:  average* number of terms, where average = (first + last)/2.
  • Sum of geometric series: (next – first) / (r -1), where next is the next term in the series, and r is the ratio.  
    • For Jan 10, next = 4096, first = 1, r =4.
    • For Jan 13, next = 0, first =1, r = 1/4.

Click GoogleForm to submit answers by Sat, Jan 13.

Week of Jan 7 – Jan 13

Jan 7: The first three terms of a sequence are 1, 2 and 3. Each subsequent term is the sum of the three previous terms. What is the 11th term of this sequence?

Answer: 423  (1,2,3,6,11,20,37,68,125,230,423)

Jan 8: What is the sum of the terms in the arithmetic series 2 + 5 + 8 + 11 + 14 + … + 89 + 92?

Answer: 1457    (2+92)/2 * ((92-2)/3+1) = 47*31

Jan 9: Three consecutive terms in an arithmetic sequence are x, 2x + 11 and 4x − 3. What is the constant difference between consecutive terms in this sequence?

Answer: x+11=36   (4x+22 = 5x-3; x=25)

Jan 10: What is the sum of the terms in the geometric series 1 + 4 + 16 + … + 1024?

Answer: (4096-1)/(4-1) = 1365 

Jan 11: What is the sum of the first 51 consecutive odd positive integers?

Answer: 1+3+… +99+101 = 51*51 = 2601

Jan 12: What is the sum of the terms in the infinite series 1 + 1/2+1/4+1/8+1/16+1/32...?

Answer: 2

Jan 13: What is the sum of the terms in the infinite series 1 + 1/4+1/16+1/64+1/256...?

Answer: 1 / (1-1/4) = 4/3


Happy 2018!  We are going to work on problems related to the number 2018.

Click GoogleForm to submit answers by Sat, Jan 6.

Week of Dec 31 – Jan 6

Dec 31: Jan 1, 2018 is the first day of the New Year. The 2018th day starting Jan 1, 2018 is __/__/____.

Answer: 7/11/2023 (2018 -2022 have 365*5+1 = 1826 days — 2020 is a leap year. 2018-1826 = 192 days in 2023.  Jan-June have 181 days.)

Jan 1: Jan 1, 2018, the first day of the New Year, is a Monday.  What day of the week is the 2018th day?

Answer: Tue (2017 mod 7 = 1)

Jan 2: New Year starts 00:00am on Jan 1. The 2018th minute is  __:__ (am or pm)

Answer: 9:38am (2018-60*24=578; 578/60 = 9 R 38)

Jan 3: What is the largest prime factor of 2018?

Answer: 1009 

Jan 4: What is the largest square number smaller than 2018?

Answer:  44*44=1936

Jan 5: What is the smallest power of 2 greater than 2018?  (Examples of powers of 2: 1, 2, 4, 8, 16, 32, 64 …)

Answer: 2048

Jan 6: What is the sum of multiples of 18 that are smaller than 2018?

Answer: 113904 = (18+2016)/2*(2016/18)

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