# Jan 2018 (CML)

I will post seven new problems every weekend.  If you like you can try to finish the first 6 problems in 30 minutes, just like in a typical CML test.  The 7th problem is a bonus. 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 Jan 28 – Feb 3

Jan 28: A time capsule was sealed in 1940 and will be opened on the same date in 2018. How many years will the capsule remain sealed?

Jan 29: What is the absolute difference between 1/2 and 1/3 ? Express your answer as a common fraction.

Jan 30: When the integer n is squared, the result is less than 150. What is the sum of all possible values of n?

Answer: 0 (1^2, (-1)^2, 2^2, (-2)^2, 3^2, (-3)^2…  12^2, (-12)^2. So the sum is zero.

Jan 31: Minnie paid a one-time registration fee of $30 for dance lessons. Additionally, she paid$20 per lesson. If she took seven lessons, how much did she pay altogether?

Feb 1: How many pieces that are exactly 5 inches long can Sue cut from a string that is 7 feet long?

Feb 2: How many units away from the origin is the point (−3, −4)?

Feb 3: In Oregon, which has no sales tax, Gloria bought three notebooks for $1.57 each. If she paid$5.00, what is the least number of U.S. coins that she could get in change?

Answer: 5 coins.   5-1.57*3 = 0.29.   1Q4P

Week of Jan 21 – Jan 27

Jan 21: How many 4-digit palindromes contain both the digits 1 and 2?

Jan 22: What is the radius of a circle with a circumference measuring 24π cm?

Jan 23: On a number line, what number is two-thirds of the distance from one-half to 2.25? Express your answer as a common fraction.

Answer: 2.25-0.5 = 1.75 = 7/4;   2/3*7/4 = 7/6

Jan 24: The length of a rectangular sports field is three times its width. If the perimeter of the field is 880 meters, what is the area of the field?

Jan 25: Josh mowed one-third of a 2000 square foot lawn in 18 minutes. At the same rate, how many minutes would it take him to mow a 4200 square foot lawn? Express your answer to the nearest whole number.

Answer: number of square foot per minute = (2000/3 )/ 18 = 1000/27 sq-ft/min;  4200 / (1000/27) = 27*42/10 = 113.4min.   Nearest whole number is 114.

Jan 26: A bag contains 20 purple and 40 green marbles. How many purple marbles need to be added so that 5/12 of the marbles will be green?

Answer: 40/(60+x) = 5/12   x= 36 purple marbles

Jan 27: Jesse added all but one of the first ten positive integers together. The sum was a perfect square. Which one of the first ten positive integers did Jesse not include?

Answer:  6  (sum of first 10 positive integers is 55. 55-6 = 49.)

Week of Jan 14– Jan 20

Jan 14: If a # b = a + 2b, for integers a and b, what is the value of 3 # 4?

Jan 15: Kim chewed a piece of gum 42 times in one minute. If she continued to chew
at the same rate, how many times would she chew her gum in 100 seconds?

Jan 16: What is the average of the first 99 counting numbers?

Jan 17: How many fewer minutes does it take to drive 35 miles at 30 mi/h than to drive the same distance at 25 mi/h?

Answer: 60 * (35 / 25 – 35 / 30) = 14

Jan 18: After a 20% reduction, the price of a computer was $800. What was the price of the computer before the reduction? Answer: 1000 Jan 19: On a number line, what is the nearest integer to 4π? Answer: 13 Jan 20: A shirt company charges a one-time setup fee plus a certain price per shirt. An order of 10 shirts costs$84. An order of 20 shirts costs $159. How much does an order of 30 shirts cost? Answer: 10 shirts without setup fee is$159-$84 =$75.  So 30 shirts with setup fee is $159+$75 = \$ 234

Week of Jan 7– Jan 13

Jan 7: If the digits 7, 8, 2, 3 and 0 are used, each exactly once, to form a three-digit positive integer and a two-digit positive integer that differ by exactly 288, what is the sum of the three-digit integer and the two-digit integer?

Jan 8: How many diagonals are in a convex heptagon?

Jan 9: What is the first year after 2018 that is a palindrome?

Jan 10: A standard 52-card deck of playing cards includes four aces. What is the probability that two cards selected randomly, without replacement, will both be aces? Express your answer as a common fraction.

Jan 11: What is the value of $\sqrt{2 \cdot 3\cdot 4 \cdot 5 \cdot 6 \cdot 7 \cdot 10}$? Express your answer in simplest radical form.

Answer: $60 \sqrt{14}$

Jan 12:The temperature dropped from 13 °F to −5 °F. How many degrees Fahrenheit is the absolute value of the change in temperature?

Jan 13: In rectangle ABCD, point P lies on side BC and point Q lies in the interior of the rectangle so that triangle APQ is equilateral. If the measure of angle PAB is 17 degrees, what is the measure of angle QPC?

Answer:  47 deg. Angle BPA = 73 deg; angle APQ = 60 deg. So angle QPC = 180 – angle BPA – angle APQ = 47.

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

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?